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Tesis de Posgrado

Un problema de frontera libre enUn problema de frontera libre enteoría de combustiónteoría de combustión

Fernández Bonder, Julián

2002

Tesis presentada para obtener el grado de Doctor en CienciasMatemáticas de la Universidad de Buenos Aires

Este documento forma parte de la colección de tesis doctorales y de maestría de la BibliotecaCentral Dr. Luis Federico Leloir, disponible en digital.bl.fcen.uba.ar. Su utilización debe seracompañada por la cita bibliográfica con reconocimiento de la fuente.

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UNIVERSIDAD DE BUENOS AIRES

Facultad de Ciencias Exactas y NaturalesDepartamento de Matemática

Un Problema de Frontera Libreen Teoría de Combustión

por: Julián Fernández Bonder

Directora de Tesis: Dra. Noemi Wolanski

Lugar de Trabajo: Departamento de Matemática, FCEyN, UBA

Trabajo de Tesis para optar por el título "95 4 6de Doctor en Ciencias Matemáticas F "

Mayo 2002

W

A Agustín.

Agradecimientos.

Quisiera agradecer a mucha gente que de una manera u otra contribuyó para que este trabajo haya sido posible.

En primer lugar quiero agradecer a Noemi, por haber sido unajefa de primera y haber permitido y estimulado mi desarrollo tantomatemático como personal.

También quiero agradecer a toda la gente que ha tenido el valor detrabajar conmigo: Gabito (alias “El Chanta Mayor”), Patu, JuanPa,Sandrita y, en especial, a Julito a quien debo gran parte de lo que hoysoy como matemático.

A toda la gente que siempre estuvo a mi lado en estos años, particularmente a Carlos, Silvia y Analía (compañeros de oficina).

A Lisi, Fabi, Cris y Marce por los continuos intercambios de alfajores

A No Gala (Andres, Patu —de nuevo —,Dani, Sheldy, El Gato,Tico, Nino, etc.) por las alegrías que nos da el fútbol.

A Ricardo, Claudia y Gabriela por el continuo intercambio de ideasy su permanente estímulo al trabajo. A Segovia por haber sido elresponsable de mi amor al análisis.

No puedo dejar pasar la ocación sin agradecer a Los Grandes Idolosde la Argentina, como el Enzo, el Burrito, el Conejo, la Hormiga Diaz,...(son tantos) por las infinitas alegrías que nos regalan semana a semana.

A mis viejos y a mi hermano por el incanzable apoyo y amor queme brindaron todos estos años.

Finalmente, quiero agradecer en especial a Adri. Sin ella, todo estono tendría sentido. Gracias por compartir mi vida.

Un Problema de Frontera Libreen Teoría de Combustión

Resumen

En esta Tesis consideramos el siguiente problema de perturbaciónsingular que se presenta en teoría de combustión

Alf-uf = Y‘f¿(u‘) en ‘D,AY‘—YtE = Y‘f¿(u‘) en 'D,

dondeD C RN“, f¿(s) = con f una funciónLipschitzsoportada en (-oo, 1].

En este sistema Y‘ es la fracción de masa de algún reactante, uE latemperatura rescalada de la mezcla y e es esencialmente el inverso dela energía de activación. Este modelo es derivado en el contexto de lateoría de llamas premezcladas equidifusionales para número de Lewis1.

Probamos que, bajo hipótesis adecuadas sobre las funciones uf eYE, podemos pasar al límite (E —>0) —llamado límite de alta energíade activación —y que la función límite u = lim u‘ = lim Y‘ es unasolución del siguiente problema de frontera libre

(P) Au—u¿=0 en {u>0},IVuI = 2M(:c,t) en 8{u > 0},

en un sentido puntual en los puntos regulares de la frontera libre y en el

sentido de la viscosidad. En (P), M(:I:,t) = f1w°(x_t)(s+wo(z,t))f(s)dsye_uz

y —1 < wo = lim¿_,0

Como Ye —uE es una solución de la ecuación del calor, queda completamente determinada por sus datos iniciales y de contorno. Enparticular, la condición de frontera libre depende fuertemente de lasaproximaciones de esos datos.

También probamos que, bajo condiciones más débiles sobre losdatos, la función límite u (que llamaremos solución límite) es una supersolución clásica del problema de frontera libre. Más aún, si 'Dñ 8{u >0} es una superficie Lipschitz,u resulta una soluciónclsica de

Finalmente probamos, bajo hipótesis geométricas adecuadas sobrelos datos, la unicidad de soluciónlímite para el problema

Palabras clave: Sistemas parabólicos, reacción-difusión,combustión,estimaciones uniformes, problemas de frontera libre, solución viscosa,solución límite. solución clásica.

A Free bounday Problemin Combustion Theory

Abstract

In this work we consider the following problem arising in combustiontheory

Aus —uf = Y5f€(u‘) in D,AYE —Yf = Y‘f¿(u‘) in D,

whereD C RN“, f¿(s) = with f a Lipschitzcontinuousfunction with support in (-oo, 1].

Here Ye is the mass fraction of some reactant, uE the rescaled temperature of the mixture and e is essentially the inverse of the activationenergy. This model is derived in the framework of the theory of equidiffusional premixed flames for Lewis number 1.

We prove that, under suitable assumptions on the functions ue andYe, we can pass to the limit (e —>O) —the so called high activationenergy limit —and that the limit function u = lim u": = lim Ye is asolution of the following free bounday problem

Au—u¿=0 in{u>0},|Vu| = 2M(:c, t) on Ü{u > 0},

in a pointwise sense at regular free bounday points and in a viscosity1

sense. Here M(x,t) = f_w0(ï't)(s + w0(a:,t))f(s)ds and —1 < wo =v :U .

Since Ye —uE is a solution of the heat equation it is fully determined by its initial-boundary datum. In particular, the free boundaycondition only (but strongly) depends on the approximation of theinitial-boundary datum.

(P)

lima-¡0

Also we prove that, under weaker assumptions on the data, thelimit function u (that we call limit solution) is a classical supersolutionof the free bounday problem. Moreover, if D ñ 3{u > 0} is a Lipschitzsurface, u is a classical solution to (P).

Finally we prove, under adequate geometric assumptions on thedata, the uniquenessof limit solutionsfor problem

Keywords: Parabolic systems, reaction-diffusion, combustion, uniform estimates, free bounday problems, viscosity solution, limit solution. classical solution.

Contents

Introducción1. Descripción del modelo2. Descripción del problema matemático3. Notación4. Hipótesis y estructura de la Tesis

Introduction1. Description of the model2. Description of the mathematical problem3. Notation4. Hypotheses and outline of the Thesis

Chapter 1. Uniform Estimates1. The estimates2. Passing to the limit3. A technical lemma4. Basic examples5. Behavior of limit functions near the free boundary

Chapter 2. The Free Boundary Problem1. The free boundary condition2. Viscosity solutions3. Consequences and applications

Chapter 3. Uniqueness of limit solutions1. Preliminaries2. Auxiliary results3. Approximation results4. Uniqueness of the limit solution5. Conclusions

Bibliography

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Introducción

1. Descripción del modelo

El trabajo de esta Tesis es una contribución al análisis matemáticode un modelo termo-difusivo que aparece en teoría de combustión.

Este modelo aparece en el análisis de la propagación de llamas curvas. Para una reacción elemental de orden uno, del tipo

Reactante —)Producto,

el problema general de propagación de llamas se reduce a resolver elsistema:

(0.1.1) pt —div(pv) = 0,

(0.1.2) pvt + p(v - V)v —uAv —AV(V - v) + Vp = 0,

(0.13) pT¿+ p(v -V)T - KAT = gw,(0.1.4) pyt + p(v - V)y —KlAy = —myw,

(0.1.5) p = pRT,

donde las incógnitas son la densidad p, la velocidad v, la presión p,la temperatura T y la concentración del reactante y. Las ecuaciones(0.1.1) y (0.1.2) son las ecuaciones de conservación de masa y la deNavier-Stokes; la ecuación (0.1.5) es la ecuación de estado para ungas perfecto; y las ecuaciones (0.1.3) y (0.1.4) son las ecuaciones de lacinética química para la que adoptamos la ley de Arrhenius:

_ 1 _¿(0.1.6) w —pbB(Tb)my exp ( RT) .

Suponemos que las cantidades ,u, /\, cp, my, Q, R, K y K1 son constantes positivas. Más aún, pb y Tb representan la densidad y la temperatura del gas quemado, y E es la energía de activación.

Este último parámetro juega un rol importante debido a la dependencia exponencial en el término de reacción para la temperatura w;

3

4 INTRODUCCIÓN

Esta dependencia se incrementa cuando la energía de activación se incrementa. Más aún, es la base de los métodos de análisis asintóticoscomúnmente usados por los físicos. Es también la base para la identificación de diferentes zonas caracterizadas por la importancia relativa delos términos que aparecen en las ecuaciones. Cuando E tiene a infinito,aparece un problema de frontera libre (ver [8, 37, 38]).

Con esta generalidad, el problema es demasiado complejo. El modelo termo-difusivo, consiste en una simplificación de este problema pormedio de dos suposiciones que son clásicas en teoría de combustión. Laprimera, es la suposición de número de match pequeño, es decir considerar a la propagación de la llama como un proceso isobárico. La segunda consiste en considerar la densidad de la mezcla constante. Estasdos hipótesis permiten desacoplar el sistema en el conjunto de ecuaciones que modelan el proceso hidrodinámico del gas, y las ecuacionesque contienen el proceso de combustión. Este modelo está físicamentejustificado (cf. [29]) para altas energías de activación

i >1RTb ’

bajo la hipótesis de cuasi-equidifusividad:

E Tb — Tc 1 _(0.1.7) FET (1- Le)_ 0(1),donde Le = K/ K ¡ es el número de Lewis y Tc es la temperatura delgas frío.

Este modelo se adapta bien a la descripción del fenómeno de combustión donde la dinámica del gas juega un rol secundario en comparación con los efectos difusivos y reactivos. Este es el caso, porejemplo, en el fenómeno de inestabilidad celular [29, 33, 34].

El límite E —)+00 es, por sí mismo, de poco interés dado que eltérmino de reacción w dado en (0.1.6) tiende a cero. Para preservarla reacción, es necesario que el término B(Tb) tienda a infinito; i.e.debemos considerar el límite distinguido caracterizado esencialmentepor

(0.1.8) Bm) N

Para T < T1,el término de reacción w tiende a cero exponencialmente; esto es conocido como el límite frío. Para T > Tb, (0.1.4) y(0.1.6) implican que -—al menos formalmente —y —>0 exponencialmente y de nuevo w tiende a cero exponencialmente. Luego, el primerpaso para hacer que este método funcione, consiste en asumir que la

2. EL PROBLEMA MATEMÁTICO 5

temperatura T; en el frente de la combustión verifica una estimaciónde la forma:

(0.1.9) Riga", —Tb)= 0(1).

El análisis asintótico del sistema cuando á —>+00, conduce —por lomenos formalmente - a un problema de frontera libre (ver [20, 35]).

En esta Tesis, nos enfocamos en el análisis matemático riguroso deeste modelo y, más precisamente, en el estudio del análisis asintóticopara grandes energías de activación. Consideraremos la mezcla de gasen reposo (i.e. v = 0). Luego de adimensionalizar las ecuaciones, elproblema (0.1.1)-(0.1.4) es reducido a resolver el sistema

(0.1.10) Au —ut = w(u, Y),1

Le

donde u = 7-111-7}(Tf—T) es la temperatura rescalada (o menos la temperatura) e Y es la fracción de masa rescalada del reactante. El términow(u, Y) posee propiedades precisas que describimos más adelante.

(0.1.11) AY —Y, = w(u, Y),

Para una deducciónmás detallada del modelo, referimosa

El modelo termo-difusivo descripto, ha sido estudiado por muchosautores: existencia de ondas estacionarias (por ejemplo [4, 7, 36]),soluciones de problemas elípticos (ver [3, 4, 6]), el problema parabólico([26]), estabilidad de ondas viajeras ([5, 31, 32]), etc.

El análisis asintótico para grandes energías de activación ha sidoestudiado para ondas estacionarias por [7, 18, 28] entre otros. Paraproblemas elípticos y parabólicos, ha sido estudiado en el caso Le = 1y u = Y (que es una suposición natural en el caso de ondas viajeras).Citamos los trabajos [2, 25] para ondas viajeras y la ecuación elípticay [10, 11, 13] para el problema parabólico.

También queremos hacer mención del trabajo [27] donde el sistema(0.1.10)-(0.1.11) es estudiado en el caso Le N 1 y se obtienen resultadossimilares a los de [13] en dimensiones N = 1, 2, 3.

2. Descripción del problema matemático

En esta Tesis consideramos el problema (0.1.10)-(0.1.11) en el casoequidifusional (i.e. Le = 1). Haremos las siguientes suposiciones naturales sobre el término no lineal w(u, Y): Llamamos e al inverso de

6 INTRODUCCIÓN

la energía de activación rescalada, e" = TEA?)J—Tc). Entonces, por(0.1.6), w = we viene dado por

w¿(u, Y) = Yf¿(u).

Para evitar la llamada dificultad del bordefrío, es usual en la literaturafijar f5 como cero en las zonas donde es exponencialmente pequeña,es decir, asumimos de aquí en más que f¿(s) = 0 si s 2 e. Para unadiscusión más detallada sobre la dificultad del borde frío, ver

Debido a (0.1.8), es fácil verificar que las funciones f5 verifican que

/ sf¿(s)ds—)Mo>0, 5-)0.o

Esta constante M0 juega un rol esencial en el análisis asintótico delmodelo cuando e —)0. Una forma usual --y conveniente —de simplificarel análisis, es cambiar las funciones f5 asumiendo que están dadas entérminos de una única función f en la forma

¡e<s)=€¿, (í),e

con lo cual, la integral f; sf¿(s) ds resulta independiente de 6.Estas funciones f5 todavía capturan las características esenciales

de (0.1.6). Luego, sobre f, asumimos que es una función no negativa,Lipschitz continua, que es positiva en el intervalo (-oo, 1) y cero en elcomplemento (i.e., la reacción sólo ocurre cuando T > T¡ —5(T¡ —Tc)).

A partir de ahora, haremos explícita la dependencia en e de latemperatura rescalada u y de la fracción de masa del reactante Y, conlo cual el sistema a considerar será

{ Aug —uf = Y€f¿(u‘) en 'D,(0.2.1) Aye _ y; = Y5f€(u‘) en D,

donde 'D C RN“.

El estudio del límite cuando e —)0 fue propuesto en la década del30 por Zeldovich y Frank-Kamenetski [38] y ha sido muy discutido enla literatura (le combustión.

En el caso us = Ye el término de reacción u5f¿(u‘) tiende a una

delta de Dirac, M06(u) donde Mo = folsf(s)ds. De esta manera, lazona de reacción donde uEf¿(uE)actúa se ve reducida a una superficie,el frente de la llama, y aparece el problema de frontera libre. El hechoque Mo > O asegura que un proceso de combustión no trivial tienelugar con lo cual aparece una frontera libre no vacía.


Si bien la convergencia de las formas más relevantes de propagación, i.e. las ondas viajeras, fue ya discutido por Zeldovich y FrankKamenetski, y un gran progreso se ha hecho en esa dirección, unainvestigación matemática rigurosa sobre la convergencia de solucionesgenerales se encuentra todavía en curso. Berestycki y sus colaboradoreshan estudiado rigurosamente el problema de la convergencia para ondas viajeras y, más generalmente, el caso elíptico estacionario - cf. [2]y sus referencias. Ver también [25]. El estudio del límite en el casogeneral de evolución para la ecuación del calor fue realizado en [13]para el caso de una fase (esto es, con ue 2 O)y en [10, 11] para el casode dos fases, donde no se impone ninguna restricción en el signo de uE.

En [13] los autores muestran que, bajo ciertas hipótesis sobre losdatos iniciales y sus aproximaciones, para toda suceción en —>0 existeuna subsucesión en, y una función límite u = lim uh": que resuelve elsiguiente problema de frontera libre

Au_u,=0 en'Dn{u>0},0.2.2 /( ) = 2Moen >0}!

en un sentido débil integral. Acá M0 = f0]sf (s)ds.

En [10, 11] los autores muestran que la condición de frontera librepara el caso de dos fases (asumiendo que no ocurre ninguna reacción siu‘ S 0) es

IVU+I2 — IV’ll-I2 = 2M0

y que la función límite es una solución del problema de frontera libre enun sentido puntual en los puntos regulares de la frontera libre cuando{u = 0} tiene “densidad parabólica” cero y en el sentido de la viscosidad en la ausencia de una fase nula (i.e. cuando {u = 0}° ñ 'D = (0)

Una pregunta natural es: ¿Será cierto que si uno tiene una sucesiónde soluciones uniformemente acotadas (u‘,Y‘) de (0.2.1) con (YE —ue) —) 0 cuando e —) 0 entonces ue (o una subsucesión) converge auna solución del problema de frontera libre (0.2.2)? Es decir, ¿Será ellímite asintótico para energía de activación tendiendo a infinito, en elcaso que (Y‘ —u‘) —)0 pero u‘ 9€Ye, una solución del mismo problemade frontera libre que en el caso uE = Ye?

Observemos que en el caso en consideración, cuando el número deLewis es 1, la función we = Ye —ue es una solución de la ecuación delcalor. Luego está completamente determinada por sus valores inicialesy de contorno. Más aún, el sistema (0.2.1) puede ser reescrito como

8 INTRODUCCIÓN

una única ecuación para ue,

(PE) Aus —uf = (u‘ + w‘)f¿(u‘).

En esta Tesis consideramos el caso que we/e converge a una ciertafunción wo (o sea que, en particular, YE—ue —>0). Más precisamente,asumimos que los datos iniciales YOEy ug verifican

Y05(Ï) - U8(I)—)w0(a:) uniformemente en RN,e(0.2.3)

con wo > —1. Luego, la función we(2:,t) cs la solución de la ecuacióndel calor con dato inicial Y05(:c)—u3(:r) y por (0.2.3), satisface queexiste el límite

we(x, t)

donde w0(:r,t) es la solución de la ecuación del calor con dato inicialw0(:¡:).

De esta manera, por lo menos formalmente, el término de reaccióntodavía converge a una función delta y aparece un problema de fronteralibre. Pero en este trabajo probamos que la condición de frontera libredepende fuertemente de la función límite wo, o sea que es diferente paradiferentes aproximaciones de los datos iniciales y de contorno de u.

En efecto, probamos que para cada sucesión en —>0 existe unasubsucesión enk y una función límite u = lim ue'w que es una solucióndel siguiente problema de frontera libre

P Au—u¿=0 enDñ{u>0},( ) IVu+|=\/2M(1:,t) en D08{u > 0},

donde M(:v,t) = fjwm’t) (s + wo(z,t))f(s)ds.La presencia de la función wo en el límite de integración, garantiza

la positividad de la función M(:c, t).

En conclusión, el problema de combustión es muy inestable en elsentido que el límite asintótico para energía de activación tendiendo ainfinito depende de perturbaciones de orden e de los datos iniciales yde contorno.

En esta Tesis probamos que la función límite es una solución “viscosa” de (P), con lo cual, como consecuencia de nuestros resultadosy de los resultados de regularidad para soluciones viscosas de (P) en[17], deducimos que, cuando la frontera libre de una función límite u


viene dada por 1:1= g(z’, t), a: = (1:1,1') con g Lipschitz continua, u esuna solución clásica.

Queremos remarcar que, debido a nuestra suposición Yi —u‘ —>0y dado que Ye 2 0, la función límite u debe ser no negativa, luego elhecho de que u sea una solución viscosa de (P) es novedoso, aún en elcaso uE = Y‘.

En particular, como consecuencia de nuestros resultados vemos quefunciones límite u con u‘(z,0) construidas como en [13], e Y‘(a:,0)pequeñas perturbaciones de ue(:5,0) son soluciones viscosas de (P). Enesta construcción, wo es cualquier constante tal que wo 2 —n donden > 0 es suficientemente pequeño.

Finalmente, estudiamos la unicidad del límite u = lím ue” de (P),puesto que es una pregunta natural averiguar si la única condición quedetermina la función límite u es la condición (0.2.3).

El propósito del último capítulo de esta Tesis es probar que este esel caso, por lo menos bajo ciertas hipótesis de monotonía sobre el datoinicial U0. Estas hipótesis de monotonía son similares a las utilizadaspara probar unicidad del límite en el caso u‘s= Y‘Een [30].

Nuestros resultados pueden ser resumidos en, bajo ciertas hipótesissobre el dominio y sobre el dato inicial uo, existe a lo sumo una soluciónlímite del problema de frontera libre (P) cuyo gradiente no se anulacerca de su frontera libre, siempre y cuando las aproximaciones de losdatos iniciales —que convergen uniformemente a uo con soportes queconvergen al soporte de uo —satisfagan (0.2.3).

Más aún, bajo las mismas hipótesis geométricas, si eziste una solución clásica de (P), entonces ella es el único límite de solucionesde (Pe) con datos iniciales que satisfacen las condiciones antes mencionadas. En particular, es la única solución clásica de (P

Queremos remarcar que la unicidad del límite resulta independiente de la aproximación del dato inicial uo y de la aproximación de lafunción constitutiva wo. Más precisamente, tomemos uff, üá" distintasaproximaciones del dato inicial uo y wei/Ej,1Ï)E*/€kdistintas aproximaciones de wo, sean u‘i (resp. ü‘k) la solución de (Pci) con función wei ydato inicial uff (resp. solución de (Psk) con función u?“ y dato inicial17,8").Sean u = lim uEJ'y ü = lim u‘k. Entonces, bajo las condicionesantes mencionadas, u = ü.

Como ya hemos mencionado, en el caso uE = Y‘, resultados deunicidad para soluciones límite bajo hipótesis geométricas similares a

10 INTRODUCCIÓN

las hechas en este trabajo pueden ser encontrados en [30]. Las técnicasutilizadas en este trabajo difieren de las de [30] ya que éstas últimasrequerirían en nuestro caso, hipótesis suplementarias sobre la funciónf.

En [22] los autores estudian la unicidad y coincidencia entre diferentes conceptos de soluciones del problema (P) (nuevamente en el casoue = Ye) bajo la suposición de la existencia de una solución clásicay bajo condiciones geométricas diferentes. Ver también [23] para unresultado similar en el caso de dos fases. Usamos algunas de las ideasde esos trabajos en el estudio de nuestro problema.

3. Notación

A lo largo de esta Tesis N denotara a la dimensión espacial y,además, la siguiente notación será usada:

Para cualquier xo E IR”, to E IRy 7' > 0

31(170) E {33e RN/ II * ¿Col< 7'},

BT(:vo,t0) E {(I,t) G RNH/ —mol2+ It —tol2 < 72},

Q,(a:o, to) s B,(;z:o) x (t0 —72,110+ 72),

Q;(zo, to) BT(9:0)x (¡t0- fito],

y para cualquier conjunto K C RN“

MK) s U Qr(-”L‘0=to),(xo,to)€l{

N310; U Q;(:L'0,t0).(10.10)EK

De ser necesario, notaremos a los puntos en RN por :c = (3:1,z’), con33’E RN". Además, (-, denotará el producto escalar usual en RN.Dada una función v, notaremos 11+= max(v,0), v‘ = max(—v,0).

También, los símbolos A y V notarán los correspondiente operadores en las variables espaciales; el símbolo 6,, notará el borde parabólico.

Diremos que una función v pertenece a la clase Lip¡0c(1, en undominio D C RN“, si para cada D’ CC D, existe una constanteL = L(D’) tal que

|v(:v,t) - v(y, s)| S L(II - yl + It - sIl”)

4. HIPÓTESIS 11

para todo (z, t), (y, s) e D’. Si la constante L no depende del conjuntoD’, diremos que v e Lip(1, á) en D.

Finalmente, diremos que u es supercalórica si Au —ut 5 0, y u essubcalórica si Au —ut 2 0.

4. Hipótesis y estructura de la Tesis

Para la existencia de una función límite para una subsucesión ue”sólo necesitamos la condición más débil que para cada compacto K CN;(K) c D,

(0-4-5) “Ye —uelle(N:(K)) = O(€)

Entonces, tenemos (ver [21])

- UEIICMUO=Bajo esta suposición, somos capaces de aplicar los resultados de [9]

y obtener las estimaciones Lipschitz uniformes necesarias para pasar allímite en (0.2.1). Esto está realizado en el Capítulo 1 donde tambiés seprueban algunos lemas técnicos que son usados a lo largo de la Tesis.

En el Capítulo 2 asumimos que u‘ —)0 en {u = 0} suficientementerápido. Esta es una condición esencial que ya fue considerada en [13].Esta suposición es natural en aplicaciones, significa que la temperaturade la mezcla alcanza la temperatura de la llama sólo si alguna combustión esta siendo llevada a cabo.

También asumimos que existe lim,,_,0(YE—uE)/e =: wo y, comoconsecuencia de la hipótesis u‘ —)Oen {u = 0} suficientemente rápido,mostramos que necesariamente wo > —1 en {u E 0}°. Luego, en elCapítulo 2, asumimos que para cada K C NT’(K) C 'D compacto

ye _ us(0.4.7) T —)wo uniformementeen N;(K).Entonces,

ya _ e

(0.4.8) l —" —wo —>0.5 cum)Y, para simplificar en análisis, asumimos que wo > —1en D. También,en el Capítulo 2, mostramos que la función límite u es una solucióndel problema de frontera libre (P) en un sentido puntual, y finalmenteprobamos que la función límite u es de hecho una solución viscosa delproblema de frontera libre (P) bajo una hipótesis de nodegeneración de

12 INTRODUCCIÓN

la función límite u. Además probamos algunos resultados que garantizan la nodegeneración de u.

Nuestra presentación en este capítulo es de una naturaleza local, conlo cual nuestras hipótesis están enunciadas en términos de la solución(ue, Ye). Como puede verse en el ejemplo tratado en el Corolario 2.3.8es posible deducir nuestras hipótesis sobre (u‘, Ye) a partir de condiciones sobre los datos iniciales y de contorno.

En el Capítulo 3, nos enfrentamos con el problema de unicidad parafunciones límite de (P), bajo ciertas hipótesis geométricas adicionalesque ya han sido consideradas en el caso w‘ = 0 [22, 23, 30]. Másprecisamente, asumimos que el dato inicial U0es estrellado con respectoa algún punto. Esta hipótesis de monotonía nos permite aproximaruna supersolución clásica de (P) por una familia de supersolucionesestrictas de (PE). Probamos que el límite de una sucesión de solucionesde (PE) es independiente de la sucesión siempre y cuando el límite desus datos iniciales y de wE/E sea fijo.

Introduction

1. Description of the model

The work in this Thesis is a contribution to the mathematical analysis of a thermal-diffusive model that appears in combustion theory inthe analysis of the propagation of curved flames.

For an elementary reaction of order one, of type

Reactant —>Product,

the general problem of propagation of flames is reduced to solving thesystem:

(0.1.1) p, —div(pv) = 0,

(0.1.2) pv, + p(v - V)v —pAv —AV(V -v) + Vp = 0,

(0.1.3) pTg+ p(v - V)T —KAT = gw,

(0-1-4) Pyt + P(V ' V)?! - KIAy = —myW,

(0.1.5) p = pRT,

where the unknowns are the density p, the velocity v, the pressure p,the temperature T and the concentration of the reactant y. Equations(0.1.1) and (0.1.2) are the conservation of mass and Navier-Stokes equations; equation (0.1.5) is the equation of state for a perfect gas; andequations (0.1.3) and (0.1.4) are the equations of the chemical cineticfor which we adopt the Arrhenius law:

_ L _¿(0.1.6) w —pbB(Tb)my exp ( RT) .

We make the assumption that the quantities ,u, /\, cp, my, Q, R, Kand K1 are positive constants. Moreover, ph and Tb represent the density and the temperature of the burned gas, and E is the activationenergy.

13

14 INTRODUCTION

This last parameter plays an important role because of the exponential dependence in the temperature of the reaction rate w; this dependance is increased as the activation energy is increased. Moreover,it is the basis of the asymptotic analysis commonly performed by physicists. Also it is the basis for the identification of different zones characterized by the relative importance of the terms that appear in theequations. As E tends to infinity, a free boundary problem appears (see[8, 37, 38]).

With this generality, the problem is too complex. The thermaldiffusive model, consists in a simplification of this problem by meansof two basic assumptions that are classical in combustion theory. Thefirst one is the assumption of low match number, this is to considerthe propagation of the flame as an isobaric process. The second oneconsists in considering the density of the mixture as constant. Thesetwo hypotheses allow us to (lecouple the system into the set of equationsthat model the hydrodynamic process of the gas, and the equations thatdescribe the combustion process.

This model is physically justified (cf. [29]) for large activationenergies

L >>1RTb ’

under the hypothesis of almost-equidiffusion:

_ E Tb - Tc 1 _(0.1.!) RTb Tb (1 Le) —O(1),

where Le = K/Kl is the Lewis number and Tc is the temperature ofthe cold gas.

This model adapts well to the description of the phenomenon ofcombustion when the dynamic of the gas plays a secondary role interms of the diflusive and reactive effects. This is the case, for instance,in the phenomenon of cellular instability [29, 33, 34].

The limit E —)+00 is, by itself, of little interest since the reactionterm w given in (0.1.6) vanishes. To preserve the reaction, it is necessary for the term B(T¡,) to become unboundedly large; i.e. we mustconsider a distinguished limit characterized essentially by

(0.1.8) B(T¿,)N

For T < Tb the reaction terrn w vanishes exponentially; this isknown as the frozen limit. For T > Tb, (0.1.4) and (0.1.6) implythat —at least formally —y —)0 exponentially and again w vanishes

1. THE MODEL 15

exponentially. So the first step for making this method work, consistsin assuming that the temperature T¡ on the front of combustion verifiesan estimate of the form:

(0.1.9) ¿7m —Tb)= 0(1).

The asymptotic analysis when á —>+00 of the system, leads —atleast formally —to a free boundary problem (see [20, 35]).

In this Thesis, we will focus on the rigorous mathematical analysisof this model, and more precisely, on the study of its asymptotic analysis for large activation energies. We will consider the mixture of a gasin repose (i.e. v = 0). After adimensionalization of the equations, theproblem (0.1.1)-(0.1.4) is reduce to solving the system

(0.1.10) Au —ut = w(u, Y),1

(0.1.11) Le AY —Y, = w(u, Y),

where u = fifl, —T) is the rescaled temperature (or minus thetemperature) and Y is the rescaled mass fraction of the reactant. Theterm w(u, Y) has some precise properties that will be described below.

For a more precise description of the model, we refer to

The thermal-diffusive model described above, has been studied bymany authors: existence of stationary waves (for example [4, 7, 36]),solution of elliptic problems (see [3, 4, 6]), the parabolic problem([26]), stability of traveling waves ([5, 31, 32]).

The asymptotic analysis for large activation energies has been studied for stationary waves in [7, 18, 28] among others. For elliptic andparabolic problems, it has been studied in the case Le = 1 and u = Y(which is a natural assumption in the case of traveling waves). We citethe works [2, 25] for the traveling waves and the elliptic equation and[10, 11, 13] for the parabolic problem.

We also mention the work [27] where the system (0.1.10)-(0.1.11)is studied in the case Le N 1 and results similar to those in [13] areobtained in dimensions N = 1, 2, 3.

16 INTRODUCTION

2. Description of the mathematical problem

In this Thesis we consider the problem (0.1.10)-(0.1.11) in the equidiffusional case (i.e. Le = 1). We make the following natural assumptions on the nonlinear term w(u, Y): We call e the inverse of the rescaledactivation energy, e" = fiEflTb —Tc), then, by (0.1.6), w = we is givenby b

w5(u,Y) = Yf5(u).To avoid what is called the cold boundary difficulty, it is usual in theliterature to set fe to be zero wherever is cxponentially small, that is,we will assume in what follows that f¿(s) = O if s 2 e. For a moredetailed discussionabout the cold boundary difficulty,see

By (0.1.8), it is easy to check that the functions f5 verify that

/sf¿(s)ds—)Mo>0, 5-)0.o

This constant Mo plays a crucial role in the asymptotic analysis of themodel as e —) 0. A usual —and convenient —way of simplifying theanalysis, is to change the functions f5 by assuming that they are givenin terms of a single function f in the form

1 s

Ms) —¿21‘(e),

and so the integral f; sf¿(s) ds is independent of e.These functions f5 still capture the essential features of (0.1.6).

Then, on f we assume that it is a nonnegative Lipschitz continuousfunction which is positive in the interval (—oo,1) and vanishes otherwise (i.e., reaction occurs only when T > T, —É(Tj —Tc)).

From now on, we will make explicit the dependance on e of therescaled temperature u and the mass fraction of the reactant Y, so thesystem under consideration will be

{ Aus —uf = Y‘f¿(u‘) in 'D,(0.2.1) Aye _ Y; = ste(u5) in D,where 'D C RN“.

The study of the limit as e —)0 was proposed in the 30’s by Zeldovich and Frank-Kamenetski [38] and has been much discussed in thecombustion literature.

In the case u‘ = Y5 the reaction function u‘f5(u5) tends to a Dirac

delta, Moó(u) where M0 = folsf(s)ds. In this way the reaction zonewhere u‘f¿(u‘) acts is reduced to a surface, the flame front, and a

2. THE MATHEMATICAL PROBLEM 17

free boundary problem ariscs. The fact that Mo > 0 ensures thata nontrivial combustion process takes place so that a non-empty freeboundary actually appears.

Although the convergence of the most relevant propagation modes,i.e. the traveling waves, was already discussed by Zeldovich and FrankKamenetski, and an enormous progress in this direction has been made,a rigorous mathematical investigation of the convergence of generalsolutions is still in progress. Berestycki and his collaborators haverigorously studied the convergence problem for traveling waves and,more generally in the elliptic stationary case, cf. [2] and its references.See also [25]. The study of the limit in the general evolution case forthe heat operator has been performed in [13] for the one phase case(this is, with uE 2 O) and in [9, 10, 11] for the two-phase case, whereno sign restriction on ue is made.

In [13] the authors show that, under certain assumptions on theinitial datum and its approximations, for every sequence en —>0 thereexists a subsequence 6,“, and a limit function u = lim uE'vcwhich solvesthe following free boundary problem

{Au—u¿=0 in'Dn{u>0},0.2.2

( ) |Vu+| = \/2Mo on'Dr18{u>0},

in a weak integral sense. Here M0 = folsf(s)ds.

In [10] and [11] the authors show that the free boundary conditionfor the two phase case (when it is assumed that no reaction takes placeif ue 5 O) is

IVu‘“|2 —IVu’I2 = 2Mo

and that the limit function is a solution of the free boundary problemin a pointwise sense at regular free boundary points when {u = 0} haszero “parabolic density” and in a viscosity sense in the absence of azero phase (i.e. when {u = O}°ñ 'D = (0)

So that a natural question is: Will a sequence of uniformly boundedsolutions (u‘, Ye) of (0.2.1) with (Ye —us) —)0 as e —)O be such thatuE converges to a solution of the free boundary problem (0.2.2)? Thisis, will the asymptotic limit for activation energy going to infinity, inthe case in which (YE —uE) —)0 but uE 7€Ye, be a solution of the samefree boundary problem as in the case in which ue = Ye?

Let us point out that in the case under consideration this is, whenLewis number is 1, the function wE= Y‘ —uE is a solution of the heatequation. So that it is fully determined by its initial-boundary datum.

18 INTRODUCTIOAT

Moreover, the system (0.2.1) may be rewritten as a single equation foruE, namely

(PE) Aus —uf = (uE + w‘)f5(u‘).

In this thesis we consider the case in which wÉ/e converges to afunction wo (so that in particular, l“ —u‘ —>0). More precisely, weassume that the initial data Y; and uf)verify

YoECB)- USÜI)

e

with wo > —1. Therefore, the function w‘(.1:,t) is the solution of theheat equation with initial datum Yo‘(:z:)—¡13(3) and by (0.2.3), satisfiesthat there exists the limit

(0.2.3) —)wo(a:) uniforme in RN,

(0.2.4) lim we? t) = wo(:1:,t)e—>0

and wo(:r,t) is the solution of the heat equation with initial datumwo(:c).

In this way, at least formally, the reaction term still converges toa delta function and a free boundary problem appears. But we provein this work that the free boundary condition strongly depends on thelimit function wo, so that it is difierent for difierent approximations ofthe initial-boundary datum of u.

In fact, we prove that for every sequence 5,. —>0 there exists asubsequence enk and a limit function u = lim ue”- which is a solutionof the following free boundary problem

Au—u¿=0 in'Dñ{u>0},

|Vu+| = \/2M(17,t) on 'D03{u > 0},1

where M(:c, t) = Lawn,” (s + wo(a:,t))f(s)ds.The presence of the function wo in the limit of integration gives the

necessary positive sign of the function M(:L‘,t).

(P)

In conclusion, the combustion problem is very unstable in the sensethat the asymptotic limit for activation energy going to infinity dependson order e perturbations of the initial-boundary data.

In this Thesis we prove that the limit function u is a “viscosity”solution to (P), so that, as a consequence of our results and of theregularity results for viscosity solutions to (P) of [17], we deduce that,when the free boundary of a limit function u is given by :rl = g(:v’,t),a: = (x¡,:1:’) with g Lipschitz continuous, u is a classical solution.

2. THE MATHEMATICAL PROBLEM 19

We want to stress, that because of our assumption that Y‘ —uE—>0and since Yi 2 0, the limit function u must be nonnegative, so ourresult that u is a viscosity solution to (P) is new, even in the caseu‘ = Y‘.

In particular, as a consequence of our results we see that limit functions u with ue(1:,0) constructed as in [13], and Y€(a:,0) small perturbations of u‘(a:,0) are viscositysolutions to In this construction,wo is any constant such that wo 2 —nwhere n > 0 is small enough.

Finally, we study the uniqueness of the limit functions u = lim uhk,since it is therefore natural to wonder whether the only condition thatdetermines the limit function u is condition (0.2.3).

The purpose of the last chapter of this Thesis is to prove that thisis indeed the case, at least under some monotonicity assumption on theinitial value ug. This monotonicity assumption is similar to that usedto prove uniqueness of the limit for the case ue = YE in [30].

Our result can be summarized as saying that, under suitable assumptions on the domain and on the initial datum uo, there ezists atmost one limit solution to the free boundary problem (P) with nonuanishing gradient near its free boundary, as long as the approximateinitial data —converging uniformly to uo with supports that conuerge tothe support of U0 —satisfy (0.2.3).

Moreover, under the same geometric assumptions, if there ezists aclassical solution to (P), this is the only limit of solutions to (PE) withinitial data satisfying the conditions above. In particular, it is the onlyclassicalsolutionto

We want to stress that the uniqueness of the limit turns out tobe independent of the approximation of the initial datum uo and theapproximation of the constitutive function wo. More precisely, letus take uffflf," different approximations of the initial datum U0 andwei/¿sJ-Jíz‘ik/e,cdifferent approximations of wo, let uEJ' (resp. u“) bethe solution of (Pej) with function wei and initial datum uff (resp.the solution of (Pq) with function w” and initial datum üá"). Letu = lim u‘i and ü = lim ü‘k. Then, under the same conditions statedbefore, u = ü.

As already stated, in the case uE= Y‘, uniqueness results for limitsolutions under geometric hypotheses similar to the ones made here canbe found in [30]. Nevertheless, in our work we use a different techniquesince, in our situation, the method used in [30] would require severaladditional hypotheses on f.

20 INTRODUCTION

Also in [22] the authors study the uniqueness and agreement between different concepts of solutions of problem (P) (again in the caseuE = Y‘) under the assumption of the existence of a classical solutionand under different geometric assumptions. See also [23] for a similarresult in the two-phase case. We use some of the ideas in these worksfor the study of our present situation.

3. Notation

Throughout this Thesis N will denote the spatial dimension and,in addition, the following notation will be used:

ForanyzvoelRN,toeRandT>0

31(10) E {3: e RN/ la: —¡BOI< T},

B,(:ï:o,to) E {(x,t) E RN'H/ la: —ÍL‘OI2+ It —tol2 < T2},

QT(a:o, to) E 37(10) x (to —72, to + T2),

(QT-(10,150)2 B,(zo) x (t0 —r2, to],

and for any set K C RN“

N,(K) a U QT(.1:0,to),(Io,io)€ K

N310; U Q;(Io,to).(Io,to)€K

When necessary, we will denote points in RN by a: = (31,1’), with:r’ e RN“. Also, (-, will mean the usual scalar product in lRN. Givena function v, we will denote 12+= max(v, 0), v“ = max(—v, 0).

In addition, the symbols A and V will denote the correspondingoperators in the space variables; the symbol 8,, will denote parabolicboundary.

We will say that a function v is in the class Lip¡0c(1,á) in a domainD C RN“, if for every 'D’ CC 'D, there exists a constant L = L('D')such that

IU(I, t) - v(y, S)I S L(|I - yl + It - SII/2)for every (11:,t), (y, s) e 'D’. If the constant L does not depend on theset D’, we will say that v e Lip(1, á) in D.

Finally, we will say that u is supercaloric if Au —u, 5 0, and u issubcaloric if Au —u, 2 0.

4. HYPOTHESES 21

4. Hypotheses and outline of the Thesis

For the existence of a limit function for a subsequence u‘Wcwe onlyneed the weaker condition that for every compact K C N;(K) C ‘D,

(0'4-5) “Ye - UEHLoo(N,-(K))= 005)

Then, we have (see [21])

—UEHCZ,1(K)=

Under this assumption, we are able to apply the results of [9] andget the uniform Lipschitz estimates needed to pass to the limit in(0.2.1). This is done in Chapter 1 where we also prove some technicallemmas that are used throughout the thesis.

In Chapter 2 we assume that uE -) 0 in {u = 0} fast enough.This is an essential condition that was already considered in [13]. Thisassumption is a natural one in applications, roughly Speaking it meansthat the mixture temperature reaches the flame temperature only ifsome combustion is taking place.

We also assume that there exists lim¿_,o(Y5- uE)/s =: wo and, asa consequence of the hypothesis that u‘ —>0 in {u = 0} fast enough,we show that necessarily wo > —1in {u E 0}°. So that, in Chapter 2we assume that for every K C ¡VT-(K) C 'D compact

Y‘ —u‘(0.4.7) —)wo uniformly in N;(K).

Thus,

(0.4.8) l Yi —wo —>o.5 02.1(K)

And, for the sake of simplicity, we assume that wo > —1in D. Also, inChapter 2, we show that the limit function u is a solution to the freeboundary problem (P) in a pointwise sense, and finally we prove thatthe limit function u is in fact a viscosity solution of the free boundaryproblem (P) under a nondegeneracy assumption on the limit functionu. We also prove some results that give the necessary nondegeneracyof u.

Our presentation is of a local nature, so that our hypotheses arestated in terms of the solution (ue, YE As can be seen in the exampletreated in Corollary 2.3.8 it is possible to deduce our hypotheses on(u‘, Y‘) from conditions on its initial-boundary datum.

22 INTRODUCTION

In Chapter 3, we deal with the uniqueness nature of the limit ofsolutions to (Pe), under some additional geometric assumptions thatwere already considered in the case wE= 0 [22, 23, 30]. More precisely,we assume that the initial datum uo is starshaped with respect tosome point. This monotonicity assumption allows us to approximatea classical supersolution to (P) by a family of strict supersolutions to(PE). We prove that the limit of a sequence of solutions of (Pe) isindependent of the sequence as long as the limit of their initial valuesand of wE/e is fixed.

CHAPTER 1

Uniform Estimates

In this chapter we consider a family (us, Y‘) of solutions to

Aus-115:)” u‘,(1.0.1) e te {J 1AY 41 =Yfe(u),

in a domain 'D C RN+1 which are uniformly bounded in L°° norm inD and satisfy that for every compact K C JV,-(K) C D,

(1-0-2) “Ye —uEHL°°(N,'(K)) = O(€)

Then, we have (see [21])

—UEIIC2.1(K)=In Section 1, we show that the functions ue, Ye are locally uniformly

bounded in the seminorm Lip(1, Then, in Section 2, we get furtherlocal uniform estimates and pass to the limit as e —>O. We also showthat the limit function u is a solution to the free boundary problem (P)in a very weak sense. In Section 3, we prove an approximation lemmathat will be used throughout the rest of the work.

In Sections 4 and 5, we further assume that for every K C NT'(K) CD compact there exists a function wo such that

(1.0.4) Y _ u —)wo uniformly in NT“(K).

Thus,

YE _ 5(1.0.5) u — o —>o.

C2v1(K)

We will see that is natural to impose that wo > —1in ’D. We observethat, as Ye —uE is a solution of the heat equation, condition (1.0.4) (aswell as condition (1.02)) can be deduce from initial-boundary data.

In Section 4 we prove some lemmas concerning particular limit functions in the particular case where wo is constant. These lemmas willbe useful in the next Chapters. Finally, in Section 5, we begin our

23

24 1. UNIFORM ESTIMATES

study of the limit functions and prove that every limit function u is asupersolution to the free boundary problem (P).

1. The estimates

In this Section, we show that uniforinly bounded solutions to (1.0.1)are locally uniformly bounded in Lip(1, á) norm. First, we manage toapply the results in [9] and obtain a uniform bound on the gradients of(ue, Ye) and then (as usual in parabolic rcgularity theory) we get theHólder 1/2 bound on t.

For convenience, let us define the following function

(1.1.1) w‘(:r,t) = Y5(.r,t) —u€(.1:,t),

then, w‘ is a caloric function and, by (1.0.3), ||w‘||02.1(K) = O(e) forevery compact set K C D.

For further references, let us now state the followingTheorem provedin [9]

THEOREM 1.1.2 ([9], Corollary 2). Let u be a bounded solution inQ1 Of

C0 S AU —Ut S :X{0<u<e}

Then u is Lipschitz (in space) in Q1¡2 with bounds independent of e.

We begin with a proposition (which is a consequence of Theorem1.1.2) that gives us the uniform control on the gradients of solutions of(1.0.1).

PROPOSITION 1.1.3. Let (uE,Y‘) be solutions of (1.0.1) such that“uEllooS A, Y‘ 2 0 and verify (1.0.2). Let K C D compact and 7' > 0such that NT’(K) C D. Then, there ezists L = L(T,A) such that

IVu5(a:,t)| S L, IVYE(.7;,t)| 5 L.

PROOF. Let us start by making the following observation

u‘ = Ye —we 2 —w€ 2 —Ce.

Then, let z":= ¿(us + Ce) and define, for (:ro,to) e K

1

25(1',t) = ;z‘(zo + Trr,to + T2t).

1. THE ESTIMATES 25

In B1(0)x {-1,0},z; verifies(withB 2325 T 1 u‘< E_ T E _ _o_AzT a, _ c+1(Ce+Iu I)€2f(¿)

1 E B E

S BTEX[—CE.E](U) = —Ehaving/11(4)

On the other hand

|u‘(:1:,t)|+C <1.Á+Cr(1+C) —r 1+C'

Therefore, by Theorem 1.1.2, it follows that

|Vz:(z,t)| 5 L = L(r, A) in 31,2(0) x (-1/2,o].

In particular,

IVu‘(:ro,to)| = (C +1)IVZE(Io,to)I = (C +1)|VZÍ(0,0) S (C +1)IÉ,

|VY‘(:vo,to)| 5 |Vu""(a:o,t0)|+ IVwE(:z:o,to)|5 (C +1)L + C.

The proof is finished III

Izï(1=,t)| S

As is usual in parabolic regularity theory, Lipschitz regularity inspace, gives Holder 1/2 regularity in time. For the proof we need thefollowing result

PROPOSITION 1.1.4 ([10], Proposition 2.2). Let u e C(Él(0) x[0,1/(4N + be such that IAu —utl S A in {u < 0} U {u > 1}, forsome A > 0. Let us assume that |Vu| 5 L, for some L > 0. Thenthere ezz’stsa constant C = C(L) such that

-— < ' < < .|u(0,T) u(0,0)|_C tf 0_T_4N+A

PROPOSITION 1.1.5. Let (uE,Ye) be solutions of (1.0.1) such thatHuellooS A, Y5 2 0, and verify (1.0.2). Let K C D compact andT > 0such that N,(K) C 'D. Then there czists C = C(T,A) such that

|u€(z,t+At)—u‘(z,t)| g CIAtll/2, ¡W(z,t+At)-Ye(x,t)l S C|At|1/2,

for every (1:,t), (:r, t + At) e K.

PROOF. As in Proposition 1.1.3 we define z‘E= C+H(ue+ Ce) and

1

23(1, t) = Xzeuo + Ax,to + Azt),

for0</\<Tand (1:0,t0)GK.

By a simple computation we get, as in Proposition 1.1.3

32‘ B0 < A e ——* < —X E .

- 2* at - s/A [°"/*](z*)

Now, zi 2 O, and in {zi > 1} we have

Üzil{ 5 B ifs/AZI¡AZA‘ at =o ife/A<1.

Moreover, we have that

1 _

Isz\(:c,t)| = C—+1|Vu‘(:co+ AI, to + Azt)l s L

in BT/A(0) x [0,72/A2]. Then, by Proposition 1.1.4, we have

Izi(0,t) - zí(0,0)l s C(L) v 0 51:5 4N + B

which, in terms of ue, is

|u‘(1:o, to + Azt) —u€(1:o,t0)| 5 C(L)¡\.

In particular

A2mIu‘(a:o, to + )— u5(1:o,t0)| S C(Í,)/\.

Let (zo,to + At) e K. If 0 < At < 72/(4N + B), we take /\ =Atl/zx/4N + B < 7' to get

|UE(I0,to+ At) —u€(110,t0)|S + BAtl/2.

If At 2 4N113, we have

2

IUE(IL‘o,t0+ At) - U€(120,t0)lS S TAV4N + BAtl/2.

The analogous inequality for Y‘ is an immediate consequence of(1.0.3).

REMARK 1.1.6. Under the hypothesis of the previous propositions,we have that

uE E Lip¡oc(1, 1/2).

2. PASSING TO THE LIMIT 27

2. Passing to the limit

In this Section, we prove further uniform estimates on the solutionsof (1.0.1) and pass to the limit. Then we show that the limit functionu is a solution to (P) in a very weak form.

PROPOSITION 1.2.1. Let (uE,Ye) be solutions of (1.0.1) such that“uElIoo S A, YE 2 O and verify (1.0.2). Then, for every sequenceEn —-)0, there exists en: —-)0 a subsequence and u E Liploc(1, 1/2) suchthat

(1) uEn’ —-)u uniformlzy on compacts subsets of D.2 Vuen' —>Vu in L .

E3; ¿usd —) áu weahïy in LÏOC.(4)Au-%=Oin {u>0}(5) For every compact K C D, exists CK > O such that

ausÉL2(K) _

for every e > 0.

PROOF. Let K C D be a compact set, and r > 0 such thatN31(K) C D. Let L = L(K) such that

Iu‘(1ï,t)- UEG/as)| S L (II - yl + It - sIl”),where (1:,t), (y,s) e NT(K).

Then, by Arzela-Ascoli’s theorem, there exists en, —) O and u eLip(1,1/2) in NT(K) such that uen’ —>u uniformly in NT(K). By astandard diagonal argument, (1) follows.

Let us now find uniform bounds for 6;; in LÏOC(D). In fact, u‘verifies

at

Now, let (1:0,to) E K and let us multiply the equation by uïifiwhere 11)2 0, ¡b = ¡“1) E CÏ°(BT(:co)), 1p E 1 in BT/2(a:o). Then,integrating by parts, we get

f/ (uï)21,b2dzdt+ á / / (¡Wim/FdwQr(1=o.to) r(Io.to)

+ 2 Vu‘ufrszzJ)dxdt = — Y‘f,5(u€)u'f1/12dzdt.r(Io.to) r(1=o.to)


Now we use Young’s inequality to obtain

1 1

5 (uï)21,/22dzdt + |Vu‘(:co,to+ 7'2)|21/)2dz 5Qf(a:o.to) 31(10)

l / IVu5(xo,to—72)|21/)2dx—// Y‘¿(um/,31}?dasdt2 B,(zo) r(1—'o.t0)

+ C// |th|2|V1p|2dzdt.1(301t0)

Then, by Proposition 1.1.3

to+1‘2

f dzdt 5 f |Vu5(:co,to- 72)|21,/)2da:Br/2(Io) to-‘r2 37(10)

// Y‘fs(u‘)uïw2dzdt + C f/ IVu‘IZIquIZdzdtr(Io.lo) 14130.10)

5 C(T)+ 2 Y‘f.5(tf)uïi,/J2dmdtr(30ut0)

+2

Hence, it only remains to get bounds for

// Ú2UÏY€f¿(u‘)da:dt= I.

Let

g¿(u,a:,t) = /u(w5(x,t) + s)f¿(s)ds,o

then

a e _ aus /e e agí e,xat))_ 1 fE(u (uaz)t)’so that we get

I=/ w2ï(ge(uax,t))dzdt—// w?Qr at . Q7

39's E _ _at (u ,x,t)da:dt —A B.


Let us first get bounds on A:

to+'r2 a

A = / f 1%- (95(u‘,:c,t))dxdtt 'r(:0)0-12

to+'r2 a

=/ 1p? f —(Q¿(u€,:v,t))dt da:BT(Ï0) to-T2

=/ 1,122[95(u‘(:v, to + 72),:5, to + 7-2)81(30)

Q¿(u‘(z, to —T2), 1:, to —72)] dx.

Since u‘ 2 —C€, f¿(s) = O if s 2 e and Iw‘l = O(€), we have

Kuwamg&/'twm+/.mmasa—Ce -Ce

so thatIAI S C(T)

It only remains to get bounds on B. For that purpose, let us firstmake the following observation:

895 auf "et d

By (1.0.3),

CE

awe

ECD, t)‘ .(u‘, I, t) S

la“ 505 for(z,t)eN,(K).atTherefore, using the fact that 0 S 1,!)5 1, we get

BSg// aw‘ Üw‘5 QTat at

Thus,

uasádl sama.(I, t)

io+T2

/ (uf)2dxdt 5 C,Br/2(IO) lo-1'2

with C independent of e and (xmto) e K. Now, as K is compact,

// (uï)2da:dt5 C,K

so that, for a subsequence, áuen' —)¿u weakly in L2(K) and by astandard diagonal argument, (3) follows.

Let us see that u is a solution of the heat equation in {u > 0} .In fact, from the fact that u‘ ——)u uniformly on compact subsets ofD, we deduce that every point (2:0,to) e {u > O} has a neighborhood

V such that u5(9:,t) 2 A > 0 for some /\ > 0. Therefore, for e < A,f¿(u€(:c, t)) = 0 in V. Thus u‘ is caloric in V for every e < /\, and then,the same fact holds for u.

Let us finally analyze the convergencc of the gradients. We alreadyknow that “VUÉIILGO(NT(K))S L. So we can assume that VuE —>Vuweakly in L2(NT(K)). In particular

f/ d>|Vu|25 liminf qSIVuEI2,,(K) HO . .N,(K)

for every nonnegative ó e L°°('D).

We follow here ideas from [2] and [13] in order to prove that wehave strong convergence.

Since Au —ut = 0 in {u > 0}, if wc take ó > 0 and multiply thisequation by (u —ó)+1[¡(1:)with 1,!)e L°°(D) and nonnegative, we getafter integration by parts in Q,(a:0, to),

// IVu|2dJ= - uVqu/2+6 Vqu){u>6} {u>ó} - {u>6}

1 1_ _ _ 2 7-2 _ u_ 2 a: _7_2 ) .

2 AMM <5)(x,to+ )w(z)+2/{u>ó}( ¿)( ,to wm

Now, letting 6 —>0, we get

l

// |Vu|21/)= — uVuV'L/J——/ u2(:v,to+ T2)’l/)(I){u>0} {u>0} 2 {u>0}l

+-/ u2(:c, to —T2)1,/)(1:).2 {u>0}

On the other hand, since 1!)2 0, f5 > 0 and ue 2 —C€, multiplying(1.0.1) by (ue + Cs)1j) and integrating by parts we get

f/ ¡Wa/,5-// nevuwcg/f WW1410,10) 4170,10) r(Io.to)

_ (uE+ CE)2(:L‘,to + T2)w(z)2 34:0)

+ (ui + C5)2(:I:,to- 72W“)2 Br(10)

limsup IVuEI2't)S IVu|21j2,540 r(-‘Eo.to) - Qr(IoJo)

Thus,


so that

llwl/zquIIL2(Q1-(Io,to)) —*llïpl/ZVuHL2(Qr(zo,zo))

Since, in addition,

wl/zVu‘ —>wl/QVu weakly in L2(Qr(xo,to)),

it follows that

wl/zVuE —)wl/ZVu in L2(Q.,(a:o,to)).

Therefore, as 1pE 1 in 31/2050),

VuE —)Vu in L2(QT/2(zo,to))

and since K is compact, this implies that

Vue —)Vu in L2(K).

By the same standard diagonal argument used before, the assertion ofthe Theorem follows. El

Next we show that the limit function u is a solution of the freeboundary problem in a very weak sense.

PROPOSITION1.2.2. Let (uEi,YEJ')be afamin ofsolutions of (1.0.1)z'na domain D g RN+1 such that u‘í —)u uniformly on compact subsetsof D, YEJ'2 0 and verify (1.0.2). Then, there ezists a locally finitemeasure u supported on the free boundary D n 8{u > 0} such thatY‘J'f¿j(u‘¡) —),u weaka in D and therefore

au _

Au —E- —u m D.

That is V d)G Cg°(D)

(1.2.3) f/D(ud>¿ —VuVó) dzdt = //Dd>d,u.

PROOF. Let us multiply (1.0.1) by 45e Cg°(D) and integrate byparts. We obtain

(1.2.4) //1;(u‘d)¿—VuEqu)dxdt: //1)Y‘f¿(u‘)45dzdt.

We want to pass to the limit in (1.2.4). We now that u‘J' -> u uniformly on compact sets of D and thus from Proposition 1.2.1, VuEJ'—)Vu in L2 (D), so the convergence of the left hand side follows. Nowloc

let K C D be a compact set and choose o/ = óK E C8°(D) such thatdm = 1 in K. Then (1.2.4) yields

//K Ysjfej(u51) (lardt S C(<Í>K)'

This Lllocbound implies that there exists a locally finite measure u inD, such that (for a subsequence) YElf¿j(u5i) —)u as measures in D.Now, passing to the limit in (1.2.4) we get (1.2.3). In addition, we seethat (1.2.3) implies that the whole sequence Y51'f¿j(u51') converge to uand that

Au—u¿=,u in D.

Finally, since we know that Au —u, = 0 in {u > 0}, we eonclude that

support u C D ñ 8{u > 0},

and the proof is complete. El

3. A technical lemma

In this section we state an approximation lemma that will be usedthroughout the rest of the Thesis.

LEMMA 1.3.1. Let (u‘i,Y5i) be a family of solutions of (1.0.1) ina domain D g RN+1 such that ue! —>u uniformly on compact subsetsof D, Y“ 2 0 and verify (1.0.2). Let (:r0,t0) e DO 6{u > 0} and let(zmtn) e D ñ Ü{u > 0} be such that (Imtn) —) (zo,to) as n —)oo.Let A" —)0, u,\n(:c,t) = ¿Maru + /\,,a;,t,, + Afit) and (u‘i),\n(:c,t) =

¡lïuíi (.73,l+ Anar,tn + Aít). Assume that tu" —>U as n —)oo uniformlyon compact sets of RN+1. Then, there artists j —)oo such that foreveryjn Z there holds that ——)0 and

(1) (u‘in),\n —)U uniformly on compact sets of RN“,(2 V(u‘in),\n —)VU in LÏOC(R""+'),

(3 %(uein),\n —)áU weaka in. L2(IRN“).Also, we deduce that

4) Vu,\n —)VU in L2(]RN+‘),

¿uh —>áU weaka in L2(RN+').

VV

(

(

PROOF. The proof is a rather straiglitforward adaptation of Lemma3.2 of [10] but we include here the proof in order to make the thesisself contained.

3. A TECHNICAL LEMMA 33

Let us find the sequence j (n). In order to verify (1),

ueï A x,t —U(:r,t = uii zn+/\na:,tn+/\2t( )n( ) n1

A:

—u(:1:n+ Aux, tn + A310]

+ (u¿n(a:,t)- U(a:,t)) = I + II.

Let us fix r > 0 such that Q3,(Io, to) CC D, so for n large

Qr(170,to) C Q2,(:cn,tn) CC 'D.

Let k > O be fixed and ó > 0 be arbitrary. We know by hypothesesthat IIII < 6 in Qk(0,0) ifn 2 n(k,6). Let us bound |I|.

For each n there exists j(n) such that, if j > j(n),

¡mi(21:,t) —u(I,t)| 5 for (z, t) e Q,(z,.,t,.).

Therefore,if j > j with n large so that An< 'r/k then,

¡1| 5% for (z, t) e Qk(0,0).

So that if j > j (n) and n large,

|(u‘1'),\n(a:,t) —U(a:, t)| < ó +á for (3:,t) e Qk(0,0).

Therefore, if jn _>_j(n), then (u‘ín),\n —)U as n —>+oo uniformly inQk(0,0). In particular (115),“.are bounded uniformly in n and j inQk(0,0)forj 2 and n largeenough.

It is easy to see that (115),“ are solutions to

Ü u‘i . . .

n — ( at)“ = «um + (w51)in)¡e,./in(<ue)in),

where (wei),\n = iw‘üxn + Anz,t,. + Afit), in Qk(0,0) for n large,and we may assume without loss of generality that sj/An < 1/n forj > j(n)

By Proposition 1.2.1, for every choice of a sequence (ju) with jn 2j there exists a subsequencejn: such that for the corresponding Ani,

V(u€j,.l)l\nl _) VU in leoc(Qk(020))

A(’U.Ej ),\

and a aE(u"'m),\n, —>EU weakly in LÏOC(Qk(O,

By the uniqueness of the limit we see that the whole sequence(ifJ'),\n converges. At this point we want to remark that the sequence

is independent of k. Therefore (1), (2), and (3) are proved.

Let us see that we also have (4). In fact,

lquAn - VUllL2(Qk(0.0))SlquAn - V(“EJ)z\nllL2(Qk(0.O))

+ llVÜte’lxn —VUllL2(Qk(0,O))= I + H

We know that IIII < 6 ifj 2 j(n) for n large enough. Let usestimate I.

“VU/M. — V(usj)z\nllï2(ok(o,0)) =

// IVu —Vu” |2(.1:,¡+ /\".1:,tn+ Azt)dzdt =Qk(0a0)

1 f/ ' 2— [Vu —Vuell (rc,t) dzdt.Mi”? amour")

By Proposition 1.2.1 and the fact that the whole sequence u‘i converges to u, Vueí —)Vu in L2(Q,(0, 0)), where Q,(a:0, to) C Q2T(a:n,tn)CC D. Therefore if j is sufficiently large and n is large enough so thatAnk S 1‘,

// IVu —Vu51|2(:c,t) dxdt < ¿7‘96?ank(-Ï"Itn)

Therefore,

IIVuxn - VU “muak-(0,0))í 25

if n is large and thus (4) follows.

Finally, let us show that (5) holds. Givcn k > 0, we want to boundlláuxnllqumpn

We first see that the uniform bound for (u‘i),\n shown above, together with Proposition 1.2.1, implics that there exists C > 0 such thatfor j 2 j (n) and n large

SC.8 .

t L2<Qk(o,o))

Next, it is easy to see that for every function v such that vt EL2(Qk(aro, t0)) and for every /\ > 0 such that /\k 5 7',

1

S Wllvtlll«2(ka(Io,lo))ll a v_ /\at L2(Qk(0:0))

4. BASIC EXAMPLES 35

(where v,\(:c,t) = iv(xo + Ax,to + A20) and therefore, for n large

H2m =|luz|lmolnk<xmtnn- IIUÏÍ“women,anat n Mmmm) A17+23 .

+ lla-(um = I + II.t Luouom)

We already know that for j 2 j (n), III | S C. On the other handsince uïj —)ut weakly in weakly in L2(Q3,(xo, t0)),

. . E‘

llutllümlmzmzn» S 11111915,?llut’ ||L2(Ql..k(zmtn))

Thus for 6 > 0 and n large,

llutllümlnuzmenn - IIUÏ’IIL2(ol,k(=n.tn))S AÏHÓ

if j is large enough. So that

lláuÁnllL2(Qk(0,0)) S C

Therefore, for a subsequence An: —>0,

Qui“, —)Ut weaklyin L2(Qk(0,at

By the uniqueness of the limit, the whole sequence (áuxn) convergesto Ut weakly in L2(Qk(0, 0)), and therefore in LÏOC(RN“). E]

4. Basic examples

In this section, continuing with the local study of the problem, westudy the special cases in which the limit function is the difference oftwo hyperplanes and the limit function wo = lim ¿(Ye —uE) in (1.0.4)is constant. First, we show that if u = azÏ, there holds that 0 5 a S‘/2Mwo where

Mu,o= /l (s +wo)f(s) ds.wo

Next we prove that if u = aan}L+ ázl‘ with a,á > 0 then a = á 5‘/2Mwo.

These lemmas, will be helpful in the remaining of the thesis wherethe situations covered by these lemmas appear as a blow-up limit of(1.0.1) (see, for example, Proposition 1.5.1).

LEMMA 1.4.1. Let (uei,Y‘i) be a solution to (1.0.1) in a domainD C RN“ such that YEJ' 2 0, and verify (1.0.4) in 'D with wo =constant. Let (zo, to) G D and assume that u‘i converges to u = a(a: —11:0);Luniformly on compact subsets of D, with a e IR and ¿J- —>0.Then,

< V 21Vw0.(1.4.2) 0 5 a _

where Mu,o= flwo(s + wo)f(s) ds.

PROOF. The proof is an adaptation of Proposition 5.2 of [10].

Without loss of generality we may assume that (9:0,to) = (0,0).

First we see that necessarily a 2 Osince u is subcaloric in D andu(0, 0) = O. If a = 0 there is nothing to prove. So let us assume thata > O.

Let w G Cf°('D). Multiplying (PE)by nílw and integrating by partswe get(1.4.3)

f/ “ti-jua? IVqulzwx.- 1€qu”de‘D 1) 'D V

+//[351.(u81,:r:,t)1,/;Il+//w:11(/u ij(s)ds)w,'D D -wo

where B¿(u,:v,t) = fÏon(s + w5)f¿(s)ds.In order to pass to the limit in (1.4.3) we observe that, by Propo

sition 1.2.1

(u5¡)¿ —>0 weakly in LÏOCUD),

Vu” —)aX{¡l>o}e1 ¡n LÏOC(D).

On the other hand,

VwEJ'

Ej-—>0 uniformly on compact subsets of D.

Therefore, in order to pass to the limit in (1.4.3) we only need toanalyze the limit of B¿j(uei,z, t). On one hand, it is easy to see that

(1.4.4) BE].(u51(1:, t),x, t) —) Mwo

for every (:r, t) such that 2:1> 0. In fact,

wei

B¿j(uei,x,t) = (s + —)f(s)(ls =/_ (s + w—5j)f(s)ds- wo 61' wo

ifj is large enough. Since IB¿j(uEí,1:,t)| 5 C there holds that (1.4.4)holds in Llloc({:r¡ 2 0}.

On the other hand, there exists M (1:,t) e L°°(‘D)such that

Bej(u‘1',:r, t) —)M(:r,t) weakly in LÏOCCD).

Clearly, M(:r, t) = M1,,0in {1:1> 0}. Let us see that M(z,t) = M(t)in {:cl < 0}. In fact,

63€.V(B¿j(u‘ï(z,t),a:,t)) = a—u’(u‘i,:r,t)Vu‘¡ + VB¿j(u‘J',a:,t)

Cu j

= (ueí+ w”)fe,-(u‘j)Vu‘í+ Vw‘Í/ f¿¡(s)ds—woe¡

u‘j6' T= Y‘Íf5j(uEÍ)Vu€¡+ J f(s)ds.

.7 —w0

Since Y‘J'fejüfi) —>0 in Llloc({:cl< 0}), Vu‘i is uniformly bounded in

L°°(D') if D’ CC D and V2”: —>0 uniformly on compact subsets of 'D,there holds that

V(B¿j(uej(x,t),1:,t)) —)O in Llloc({1:1<

So that, passing to the limit in (1.4.3) we get

a2 _

'2- / Ús,=Mwo ¡bn+/ '{1:¡>0} {Ii>0} {1:1<0}

Thus, integrating in the variable :cl we get

a2 —

/ (— —Mwo+ M(t))1,b = o.{21:0} 2

Since 1/)is arbitrary, we conclude that2

0lï—MwC,+1\7I(t)=O.

Finally, we notice that A71(t) 2 0. In fact,

e. w‘i w‘íB¿j(u1,1:,t)= e, (s+—)f(s)ds+ (s+ )f(s)ds

_% 61' _wo Ej_w_‘j

2 [me]. (s+%)f(s)ds+0Jn a

smce “2’ —->wo uniformly on compact subsets of D.J

a = «201/111,o—M(t)) s 21mo

and the proof is complete. Ü

Thus,

LEMMA 1.4.5. Let (u‘i,Y‘J') be a solution to (1.0.1) in a domainD C RN“ such that Y‘J' 2 0 and verify (1.0.4) with wo = constantin D. Let (zo,to) E D and assume that usi converges to u = a(:c —3:0)? + á(:v —1to); uniformly on compact subsets of D, with a,á > 0and ej —>0. Then,

(1.4.6) á = a 5 M2111“,

where Mu,o= flwo(s + w0)f(s) ds.

PROOF. We argue in a similar way as in Proposition 5.3 of [10].

We will denote Q, = Q,(0,0). Without loss of generality we willassume that (zo, to) = (0,0) and that Q2 CC D.

As before, uE satisfies

//D uïuïn‘P=%//19|Vu‘l2ibzlugvuivtp

+ //D BE(UE,I,t)Ï//‘r¡+ //D wi] (font f5(3)d3) ¡IL

We want to pass to the limit. By Proposition 1.2.1 and the factthat uEi converge to ax]L+ ázf we have that

uïj —>0 weakly in Lïocw),

Vu“ —>01X{I¡>o}€1 - ÓX{11<0}€1 in Ll20c(D)'

Clearly, as a, á > 0, B(uej,13,t) —*M ¡n Llloc(D)'

So, passing to the limit in the latter equation for the subsequenceej, we get

a2 62 ,

—ï / wn—ï/f un + Mwo/ wn= 0.{Il>0} {21(0}

Integrating in the 3:1variable, we conclude that

a = á.

Next, we will assume that a > «231.00 and arrive at a contradiction.

First, let us consider z‘i, defined in Q2, the solution to

(1-4'7) AzEj _ zïj = (fis,-(Zsj) + WÉjfÉj(261.» pEj(zEj/Ej) in Q2

with boundary conditions

z‘i = u —b‘i on 6pQ2

where fi¿(s) = sf¿(s), WE= supQ2 w‘, b‘i = supQ2 Iu‘i —ul and pe]. is asmooth cutofl' function with support in [-(wo + 2Cej), 3] and psi E 1in [-(wo + Cej), 2] (Here CE}.—)0+ is such that Iw‘i/ej —wol 5 CE].inQ2 so that u‘i/ej 2 —(wo+ Cel.) in Q2).

Observe that 251(x1,x', t) = in(-:v¡,:v’,t) in Q2.

It is easy to see that the proofs of Propositions 1.1.3 and 1.1.5 canbe adapted to z‘i so that, for a subsequence, that we still call Ej, thereholds that z‘i —>z uniformly on compact sets of Q2. We will show thatz = u.

First,

Au‘ï —uïj = (uEÍ+ wE¡)f¿j(uEÍ) 5 fiejüfï) + Wejf¿¡(u‘¡)

= (fiejüfj) + Wejfsj (11"))pej (u‘j /5j) in Q2

From the fact that zii 5 u” on ópQg, we deduce that z‘ï 5 u‘í inQ2 and therefore z 5 u.

In order to see that u 5 z, we consider aii e C2(R) such that

ai; = fi(a‘¡)+ a‘j))p¿j(a51'),s e IRa‘i(0) = 1, ¿"(0) = a

Integrating the equation we get, for every s e R, that

0<7_Kej 50:1.(5)Sa

¿a? — M1,,o > 0 and KE].—>0 when j —) oo.

It follows that there exists se}.< 0 such that

1 2 _where ¡7 =

aj(s):{1+as _ 329(7 — n51.)(s — 35,.) s 5 351.

and it is easy to see that se]. are uniformly bounded by below andmoreover, there exists 3 < 0 such that 351.—>3.

Now let

. . bei¿51(3) = eja51(fi —

—+s. .61‘ ('Y-Kej)€j 5’)

40 l. UNIFORM ESTIMATES

Using that áeí(0,1:',t) = —b5iand the bounds on aii, we deducethat

á‘j 5 u —bei in Q2.

Now, since áEi 5 ze! on 3,,Q2, and (¿51'is a one dimensional stationary solution to (1.4.7), we have that ás! 5 zii in Q2. Since áei —)uuniformly on compact subsets of {x1 > 0}, we deduce that u 5 z inQ2ñ {Il >

Finally, we notice that z‘i (2:1,x’, t) = 251(—:¡:1,a:’,t), so we concludethat u 5 z in Q2.

Now, let

R = {(zr,t)|0 < 2:1< 1, |:¡;’|< 1, |t| < 1}.

Let us multiply (1.4.7) by and integrate in 'R. Then, we have

= Mii/2:1; -//RAzlzez:¿ = F3—GJ-,

where f¿j(z) = ffwcj (s + We,)fe,-(5)Pe¡(3/5j)dsSince every z‘i is symmetric in the :rl variable, we deduce that

zí’,(O,z’, t) = 0 and therefore,

Ej2 / )2_ fíj (250)dmldt8R0{z¡=l} 2

Since 251—)u = ax? + azl’ uniformly on compact subsets of Q2,we deduce that, in ¿23/2ñ {xl > á}, 2512 s]- isj is large and

—)a uniformly,_ 1 WE.

IFEJ.(zEJ)=/ s + —’ f(s)p¿j(s) ds —>Mwo.-w,,./e,- Ej

Then we get

liminfEj 2 f (la? —MW) dx'dt.J—’+°° ann{:¡=1} 2

On the other hand, we know from Lemma 1.2.1 that

z? —)u, = 0 weakly in L2(Q3/2),

—*un in L2(Q3/2)«,

5. BEHAVIOR OF LIMIT FUNCTIONS 41

which implies that —>0.

Finally, integrating by parts, we get

1

—GJ-5/ Iz;{||VI:z°¡Idet+/ -IVacrzeJ'|2dx'dt.ann{|:l¡=1} ann{:¡=1} 2

From the convergence of in -—)u it follows that

IVIIZE-¡I—>0 pointwise in Q2 ñ {2:1 > 0}.

If we now use that 2‘1' are locally uniformly Lipschitz in space, wededuce that

lim sup(—GJ-) 5 0,j->+oo

which gives that ¿az —Mu,o 5 0, a contradiction. This finishes theproof. El

5. Behavior of limit functions near the free boundary

In this section we analyze the behavior of a limit function u =lim u‘í near an arbitrary free boundary point.

First we show that every limit function u is a supersolution ofproblem (P) under the assumption (1.0.4), and then we show thatif U is a global limit and wo in (1.0.4) is constant, there holds that

1

|VU+| S ‘/2Mwo, where Mu,0= f_wo(s + wo)f(s) ds.

PROPOSITION 1.5.1. Let (u‘i, Yei) be a solution to (1.0.1) in a domain D C RN“ such that Y“ 2 0 and verify (1.0.4) with wo > —1.Assume that u‘i —)u uniformly on compact subsets of D. Then u is asupersolution of (P) in the sense that

i. Au—u¿=0 in {u>0}ñD

ii. limsup |Vu(a:,t)| 5 x/2M(1:o,to) for (3:0,to) e 6{u > 0} ñ(I,t)-)(zo,to)

D, u(:c, t) > 0.

PROOF. We only have to show ii. The proof is a rather simplemodification of Theorem 6.1 of [10].

|Vu(:r,t)| with u(:c,t) > 0.

Since u e Liploc(1,á) in ’D,we know that a < +oo. Ifa = 0 there isnothing to prove. So let us assume that a > 0 and let (zm tn) —)(3:0,to)

Let a = lim Sup(1,t)->(zo,to)

be such that u(zn,tn) > 0 and |Vu(.1:,,,t,,)| —)a. Let (zmsn) e ’Dñ8{u > 0} be such that

d" E max{|:cn —znl, |tn —3,,Il/2}

= ' f . — . t — 1/2 .(z,s)ég{u>0}{maxfitn 2|, I n SI

Let us consider the sequence1

udn(:r, t) = d—u(zn+ dnm,su + dit).n

Since u e Liploc(1,%) in ‘Dand dn —>0, given a compact set K C RN“the functions ud" are uniformly bounded in Lip(1, á) seminorm in K, ifn is large enough. On the other hand, ud"(0, 0) = 0 for every n. So thatud" are uniformly bounded on compact sets of RN“. Therefore, for asubsequence (that we still call ud"), ud" —)U0 uniformly on compactsets of RN“, where U0e Lip(1, %)in RN“.

Let Ín = (mn-zn)/dn, ín = (tn —s,,)/d,2¡.Then (cimt-n)e ÜQ1(0,0)so that (for a subsequence) (in, ín) —>(:í',í) e 8Q1(0,0).

On the other hand, since ud" > 0 in Q1(:ín,ín) we deduce that inQ1(ï:, í), uo(a:, t) Z 0 and U0is a solution of the heat equation.

Let us consider the sequence

V __ Vu¿n(:ï:n,ín) _ Vu(1:n,tn)n lvudn(Ïmt-n)l lvu(-Ïnatn)l.

We may assume that un -—>1/. Let us see that

8' .

¡vuemmle #(M).u

To this end, we will show that Vud" —>Vuo uniformly on compactsubsets of Q1(í,í). But this is a consequence of the fact that anysuch compact set is at a fixed positive distance from the boundary ofQ1(ín, t-n),in n is large enough. In fact, let K be any such compact setand let T > 0 be such that N2,(K) C Q¡(:ï:,,,í") for n large. We have,for n, m large,

lvudn(x, _ Vudm(I, = IV(udn_ udm)(:v?< C a ', — fo a ,t K,_ N20?) lu 1.. udml r ny (a: )€

since every ud" is a solution of the heat equation in Q1(Ïn, ín). Therefore we have the uniform convergence of the gradients, so that

auo __ _ lECMD —(.1.

On the other hand, it is easy to see that IVuJI 5 a in RN“. Infact, let R > 0 and ó > 0 be fixed. There exists Aosuch that

|Vu+(3:,t)| 5 a + ó for (2:,t) E Q,\R(zo,to)

A< /\o.

Since anR(zm3n) C annRÜJoJo) if /\n = maXÜ-Tn- 330th toll/2}(Z dn) and R > 1 and since An —>0 as n —)+oo, we deduce that

IVujn(3:,t)| 5 a + ó for (3:,t) e Q,\3(O,0)

if n is large enough. Thus Vu}: —)Vuó" *-weakly in L°°(QR(0, 0)) andtherefore IVUEÏI S a+ó in QR(0, 0). Since ó and R where arbitrary wededuce that

IVuÉI S a in RN“.

Let V = auo/ 61/. We know that V is a solution of the heat equationin {ug > 0} since U0is a solution of the heat equation in this set. Onthe other hand, we know that V 5 a in {uo > O} and V(:ïc,í) = a.Since a > 0 we must have uo(:ï, í) > 0 (otherwise uo E 0 in Q1-(¡5,and thus uo > 0 in Qp(í:, í), for some p > 0. It follows that V E a inQ;(:ï:, Moreover, if we call R the set of points in {uo > 0} ñ {t < flwhich can be connected to (:ï:,í) by a continuous curve in {uo > 0}along which the t-coordinate is nondecreasing, we see that V E a inR.

Since IVuol 5 a in {U0 > 0} we deduce that Vuo = Vu in R.Let us assume, for the sake of simplicity, that 1/ = el. Then by theconsiderations above

uo(:c,t) = azl + b(t) in R.

Since U0 is caloric in R, b(t) must be constant. Thus there existsí e RN such that

uo(:c,t) = a(z —í)1 in R.

It is not hard to see that R = {(a:—:E)1> 0, t < Hence,

uo(m, t) = a(z —5:)1 in {(115—í‘)¡ > 0, t <

By Corollary A.1 of [10], we get for some á 2 O

u0(a:, t) = á(:t: - Si); + o(|:l: —il + It —ill/2) in {(12—:ï:)1< 0, t <

Let us consider for A > 0 the function (U0),\(IE,t) = (1//\)uo()\:l: +Í,/\2t + Now, one can check that (ug),\ converges uniformly oncompact sets of RN+1 to uoo where

uoo(:v,t) = azïF + áxf in {t 5 0}.

44 l. UNIFORM ESTIMATES

Let 1

(Uej)dn(1:, = d_uej(Zn+ (lux,Sn+n

By Lemma 1.3.1, there exists a sequence ju —>+oo such that (uEJ'n)d,, —>U0 uniformly on compact sets of RN“ and ein/du —)0. It is easy tosee that (u‘in)d,_is a solution to

_ (9 _ _ . .

A(IIE’")d..- ¿("mln = ((ue’")d,. + (1115“)d..)fe¡,./d,.((ue’")d..)

in Q¡(a':,t) for n large, where (wein)d" = (1/d,.)w"'i"(zn +dnrc, sn +dfit).

By calling 52 = ej" /dn, usa = (uein)d" and wea = (wíin)¿n, then u‘gare solutions to (P52) with

w591 .

É —)w0(:co,to) uniformly on compact sets of Q¡(:c, í),

usa —)ug uniformly on compact sets of Q1(a":,t),

(ug),\k —>uoo uniformly on compact sets of RN“,

eg —)0 and /\k —)0. Therefore we can apply Lemma 1.3.1 again and

find a sequence 62° —) 0 and solutions 11,5910to (Pego) in Q¡(0,0) suchthat

w00sn

52°

—>wo(1:0,to) uniformly on compact sets of Q1(0, 0),

ue")lo—)uoo = our]L+ das]- uniformly ou compact sets of Q1(O, O).

Finally, if á = 0 we apply Lemma 1.4.1 and if á > 0 we apply Lemma1.4.5 to deduce that

a S 2A/I(ÏL‘0,to).

So the Proposition is proved. El

LEMMA 1.5.2. Let (u‘i,Y‘i) be a solution to (1.0.1) in a domain'Dj such that YEJ'Z 0 and satisfies (1.0.4) in 'DJ-with wo = constant.Here ‘Dj is such that 'Dj C 'Dj.“ and UJ-Dj= RN“. Let us assume thatuei —)U uniformly on compact subsets of RN“ asj —)oo and ¿J-—)0,with U 2 0, U E Lip(1, 1/2) and 8{U > O} 7€(0. Then,

(1.5.3) IVUI 5 2M",0 m RN“

with Mwo= fjwo(s + w0)f(s)ds.

PROOF‘.The proof is similar to that of Theorem 6.2 in [10]. Herewe use Lemmas 1.4.1 and 1.4.5 instead of Propositions 5.2 and 5.3 in[10].

Let a = supIVU+|. By assumption, a < +oo. If a = 0 thereis nothing to prove. Let us assume that a > O and let (zmtn) besuch that U(:vn,tn) > 0 and |VU(:vn,t,,)| —) a as n —) +oo. Let(zu, sn) e 3{U > 0} be such that

dn := max{|:1:n —znl, Itn —snll/z}

= . f n _ a tn _ 1/2 '(z.s)eual{u>0} zl I SILet

Ldu

Then the family Ud" is uniformly bounded in Lip(1, seminorm inRN+1 and since Ud"(0,0) = 0, the family is uniformly bounded oncompact sets of RN+1. So that we may assume (by taking a subsequencethat we still call Ud") that Ud" —) U0 uniformly on compact sets ofRN“, whereU0e Lip(1, in RN“.

Let a?” = (33,.—zn)/dn, ín = (tn - sn)/dfi. It is easy to see (bytaking a subsequence) that (in, ín) —-)(:ï:,ï) G 3Q1(O,0). Also,

VUd,l (Ïm {11)

IVUdn(Ïnaín)l

We will assume without loss of generality that y = el.

U¿n(:c,t) = (zn + dnz, sn + dit).

ll.

Proceeding in a similar way as in the proof of Proposition 1.5.1 wesee that necessarily

U(It)_ (ICE-Í):- in(a:—í')¡>0,t<t_° ’ “ aca-5);+o(|x—;í|+|t—í|1/2) in (zu-501<0, t<t'for some point fi e RN and some á 2 0.

Let Uoo= lim,\_,0(U0),\where (U0),\(I,t) = (1//\)U0(Í+/\Z,í+ Azt).Then, Uoo(a:,t) = 031++ áml‘ and the proof follows as in Proposition1.5.1

CHAPTER 2

The Free Boundary Problem

In this chapter, we find the free boundary condition for the limitproblem and we show that the limit function u is a solution to thefree boundary problem (P) in a pointwise sense, under the assumptionthat the free boundary admits an inward spatial normal in a parabolicmeasure theoretic sense (Definition 2.1.3). Then we show that, undersuitable assumptions, the limit function u is a viscosity solution of thefree boundary problem (P).

Finally, we end this Chapter with some applications of the resultsand construct a family (usd/5) of solutions to (1.0.1) with wE/E —>wo 96Osuch that u = lim uE is a viscosity solution to (P), by showingthat the local assumptions in Theorem 2.2.9 can be fulfilled by imposingconditions on the initial data (ug, YO‘”).

1. The free boundary condition

Throughout this section we will assume that (1.0.4) holds and thatfor every K C {u E 0}° compact there exists 0 < n < 1 and ¿o > 0such that, for 5 < so

(2.1.1) 5 7] in K.mIïm

This assumption is a natural one in applications, roughly Speakingit means that the mixture temperature reaches the flame temperatureonly if some combustion is taking place.

As a consequence there holds that

v‘ — uE _ ue ,2 —limsup— 2 —n> —1 m K.

5 6-?0 5= l'm

wo El->0

So that, for the sake of simplicity we will assume from now on thatwo > —1 in D.

We start this section with a lemma that is the essential ingredientin the subsequent proofs.

47

48 2. THE FREE BOUNDARY PROBLEM

LEMMA 2.1.2. Let (u‘*,Y‘k) be a solution to (1.0.1) in a domainD C RN“ such that Y“ 2 0 and (1.0.4) and (2.1.1) are satisfiedwith wo > —1. Let u = limu“, with ek —) 0, and B¿k(u,:1:,t) =

ffwoek(s + w‘*)f¿k(s)ds. Then,

35k(UH, 37, _) MCE)t)X{u>0}7 L;oc(D)'

where M(a:,t) = fl (s + wo(x,t))f(s)ds.—wo(::,t)

PROOF. First, let us observe that

/E (w‘+s)f¿(s)ds= /E (we+ ds—woe —wos

=/_l (w—€+s)f(s)ds.wo E

Therefore,

lim /Ek (w‘k+ s)f5k(s)ds = M(x, t).Ek->0 _w05k

uniformly on compact sets of D.

Let us now see that Bum”, z, t) —)M(:c, t) uniformly on compactsubsets of {u > 0}.

Let K CC {u > O}, then there exists /\ > 0 and eo such thatu‘*(x,t) > /\ Vek < eo, (9:,t) e K. Thus, we have

u‘k (1:,t)

lim / (wE+ s)f¿k(s)ds. t:¿MW (I’M) M

—onk5k

= klim/ (u:5+s)f¿k(s)ds=M(:c,t).—)oo _w05k

Since IB“,(usk, 1:,t)| 5 C on every compact subset of 'D, there holds,for a subsequence that we still call ek that

B¿k(u€*,a:, t) —>M(a:, t) weakly in LÏOC(D).

Clearly, M(x, t) = M(1:, t) in {u > 0}. Let us see that M(m, t) = 0 in{u E 0}°. In fact, let K be a compact subset of {u E 0}°. For every

1. THE FREE BOUNDARY CONDITION 49

61,62 > Othere holds that, for k large enough

|{(:z:,t) e K/61< B¿k(u5*,:c,t)< M(a:,t) -62}I S

uaneK/gfina>—muixut_k

í</“@+Wymu<M—3.2 _wo 2

In fact, let 6 > 0 be such that

—wo-6 61

/ (s + wo)f(s)ds < —._wo 2

Since uek/e;c is bounded in K ,

u‘k/Ek

Bu(u‘*,z,t) - j (8+wo)f(8)ds < mimi-2%)_wo

if k 2 ko. On the other handE E

Ï¿2—ïï2flm—ó6k 6k

if k 2 k1. Thus, if k 2 max(k0, k1) and u"°/e,c 5 —wothere holds that

B¿k(u"‘,1:,t) < 61.

Therefore,u‘k

{(2, t) e K/ E—(:r,t) > —wo(:r,t),k fií</“@+%Vwm<M—3}2 _wo 2

Ek

smaüeK/—wuw+ns%—51—wlk

Y

esI{(z,t)eK/ 2C

S I{(I,t) GK/Y‘*fek(u"‘) Z ÉH

Since Y‘kfeküfk) —)0 as measures in K and Y“ 2 0, fek 2 0 thereholds that

Yskfeküf") —>0 in L1(K).

Therefore,

|{(:v, t) G K/El < B¿k(u5",:c, t) < M(a:, t) —62}| —)0.

On the other hand, let 1 > n > supK(—wo) bc the constant in (2.1.1)in K, there holds that

Ck 8k“_ 5k _w_

tk (3+ 6k)f(s)ds+/ tk (3+_wo 6k35km“,sv,t) =/ _fiL _fl

s f (s+‘:—:)¡(s)ds+l/_ (8+wsk)f(s)ds_fi‘k

=/17 (s + 1:5k)f(s)ds -—) n (s + wo)f(s)ds < M(z, t),_w0 k —wo

_ t . e . .

smce 4:: 5 fi 5 77m K. Therefore,n

limsup B¿k(u€"',x,t)5 / (s + 'u;0)f(s)ds < M(z,t)._w0

So that, for 62 > 0 small we get

|{(:c,t) e K/El < Bek(u9k,.r,t)}|=

E ¡{/51 < B¿k(u€k,.’ll,t)< Al —E2}I—)

Let us now see that M(:v,t) = 0 in K. As in Lemma 1.4.1 wesee that M(1:,t) 2 0. Now assume that for some el > 0 we have|{117I(:c,t)> 61H > 0. Then, there cxists m such that |{M(a:,t) >61+ S= >

Now,

/ B¿k(u€",:1:,t)—) NICK,t) > (61+ l)IAmIm Am m

but,

/ B¿k(u“‘,z,t)=/ B¿k(u”',a:,t)A". Amn{ng(1L‘k,I,i)>E|}

+/ B¿k(u‘k,a:,t).Amñ{B¿k(It’k ,I,t)SE¡}

Since the first term in the right hand side goes to zero and the secondis bounded by EllAmI, we get a contradiction.

The proof is finished. El

Let us give the definition of a regular point.

1. THE FREE BOUNDARY CONDITION 51

DEFINITION2.1.3. We say that 1/is the interior unit spatial normalto the free boundary 3{u > O} at a point (zo,to) E 8{u > 0} in theparabolic measure theoretic sense, if V e RN, |11|= 1 and

. 1

“13%71m f/Qrwo) |X{u>o) - Xuzn/ <:—:o.u>>o}ld9=dt— 0—r

DEFINITION2.1.4. We say that (zo, to) is a regular point of 3{u >0} if there ezists an interior unit spatial normal to 8{u > 0} at (zo, to)in the parabolic measure theoretic sense.

We can now prove the main result of this section.

THEOREM 2.1.5. Let (u5¡,YEJ') be a family of uniformly boundedsolutions of (1.0.1) in a domain 'D C RN“ such that u‘i —)u uniformlyon compact subsets of 'D, Y” 2 0 and verify (1.0.4) and (2.1.1), withwo > —1. If (zo,to) is a regular point of'D ñ Ü{u > 0}, then u has theasymptotic development

u(a:, t) = a(:r —1:0, 1/)+ + o(|:1: —zo] + It —¡fall/2),

with a = 2M(xo,to), where M(a:,t) = fjwo(“)(s + wo(x,t))f(s) ds.Here 1/is the interior unit spatial normal to the free boundary at (3:0,to)in the parabolic measure theoretic sense.

PROOF. We assume, without loss of generality, that (zo, to) = (0,0)and 1/= el = (1,0, ...,O).

Let 1,11E CS°(D). We proceed as in Lemma 1.4.1. Let us multiplythe equation for ue by uíld) and integrate by parts. We have=á/t —//,, +¡Á

+ _ wo)f(_w0)(wo)zn/J+ //D 10?; f(s)ds)1,0.Since

BEJ'(¡firma = +w0)f(s)ds+ _w0)f(s)dsa—wo -w0 .7

and B¿j(u5i,:1:,t) —)0 weakly in L1(K) for every K C {u E 0}° compact, there holds thatg

FE}.(1:,t) := / e] (s + wo)f(s)ds —)0 weakly in L1(K)._wo

Since F51.is nonnegative there holds that. 1 ,

FE}. —) 0 m L (lx).

So that, for a subsequence that we still call s]- there holds that

FE]. —) 0 (L.(’.. K.

Thus,

É —) —wo (1,.(3.K.Ej

Therefore,

Lei 1

(2.1.6) ej f(s)ds —>( f(s)ds> X{u>o} a.e. D.-w0 ‘w0

By using Proposition 1.2.1, Lcmma 2.1.2 and (2.1.6) we can passto the limit (for the sequence Ej —>0) in the lattcr equation and get( - . )

ÍZJZWI‘ÍP=%/Álvul2wrx ’ //DUI¡Vqu

+ //{u>01 Mu’twï‘ + j/{wo} (wo)Il ( _:Uof(8)ds) 1p

for every 1/)G Cf°(D).

Now, let w’\(z,t) = MJJ(I—Ï-°, Replacing 1,0by 1,0"in (2.1.7)and changing variables, we get for u,\(a:, t) = intro + Ax,to + A2t),(2.1.8 )

//(uA)t(uA)x¡1/1= |Vu,\|21‘/;,Cl_ .//(uA)IquAVdJ

+ / /{wo} Mm, A21911211+ f /{M} (100)“(¿ya NSW) wi

Let r > 0 be such that Q,(1:o,t0) CC D. We have that u,\ GLip(1, 1/2) in (QT/¿(0,0) uniformly in A, and u,\(0,0) = 0. Therefore,for every An —)0, there exists a subsequence An: —)0 and a functionU e Lip(1,1/2) in RN“ such that u,\", —>U uniformly on compactsets of RN“.

By our assumption on (2:0,to), we can easily see that for every k > 0

(2.1.9) |{uA>0}f'l{:l:1<0}r‘le(0,0)I—->0aS/\—)O,and

(2.1.10) |{u,\=0}n{a:1 >0}0Qk(0,0)|—>0as/\—)0.

l. THE FREE BOUNDARY CONDITION 53

Now, using lemma 1.3.1 and the fact that 10"—>0 uniformly in Dand supp zp"C supp w, we can pass to the limit in (2.1.8) and get(2.1.11)

// UgUz,1p=lf/ NUM! —f/ U1,VUV1/1{mo} 2 {mo} {me}

+M(o,0)// 11,.{11>0}

Our aim is to prove that U = azÏ. First, by (2.1.9) and (2.1.10),we deduce that U = 0 in {1:1< O}. On the other hand, U is a solutionto the heat equation in {U > 0} C {3:1> 0}. By Corollary A.1 in [10],for every :ï:’e RN", í e R there exists a 2 Osuch that

U(:r,t) = azrf' +0(|(z¡,a:') —(0,:Ï’)I + It —t-Il/z) in {srl > 0} ñ {t <

Replacing the test function tb by "(:r,t) = MDF},#, with eCg°({t < 0}) and proceeding as above we get

2

(2.1.12) ¿2- / cpm+M(0,0) <1>n=o.{-Tl>0} {21>0}

In order to pass to the limit for a sequence An —>0 we have used Lemma1.3.1.

Thus, a = 2M(0,0).

In order to see that U = ami" we use Lemma 1.5.2. In fact, byLemma 1.3.1 there exists a sequence jn —)oo such that

1

u‘s" := A-u‘í"()1na:, Ait) —>U(:1:,t)

uniformly on compact subsets of RN+1. We recall that (u'5",Yó") is asolution to 1.0.1 with e replaced by ón. Moreover,

w‘s" w‘in(/\nz,)1,2,t)=— 00uniformly on compact sets of RN+1.

On the other hand, U 2 0 and 3{U > 0} 96(Z).By Lemma 1.5.2 wehave that IVUI 5 a = \/2M(0,0). Since U E 0 in {:rl = 0} we deducethat

U 5 axl in {1:1> 0}.

By Hopf’s Principle, we deduce that

U = azl in {3:1> 0}.


Since the limit of u,\n, is 01:12;lwith a independent of the sequenceAni, we deduce that u,\ -) am;Lso that

u(a:, t) = cual++ o(|:I:| + Itlm).

The Theorem is proved. El

REMARK2.1.13. It is clear from the proof that the result isstilltrue if we replace condition (2.1.1) by the following property: ——)—wo a.e. {u E 0}°.

2. Viscosity solutions

In this section we prove that, under suitable assumptions, the limitfunction u is a viscositysolution of the free boundary problem

For the sake of completeness, westate here the definition of viscositysolution that was introduced in [11] for the two phase case of thisproblem when wo = 0.

DEFINITION 2.2.1. Let Q be a cylinder in IR” x (0,T) and let v eC(Q). Then u is called a classical subsolution (supersolution) of (P)in Q ifu 2 O and

(1)AU—U¿ZO(SO) inQ+EQñ{u>0}.(2) v e CRF).(3) For any (1:,t) e 89+ ñ Q, Vu(.r, t) 760, and

|Vv(:c,t)IZ 2M(:c,t) (5\/2M(a:,t)).

We say that 'u is a classical solution in Q if it is both a classicalsubsolution and a classical supersolution.

DEFINITION 2.2.2. Let u be a continuous nonnegative function inQ; u is called a viscosity subsolution (supersolution) of (P) in Q if,for every subcylinder Q’ CC Q and for every classical supersolution(subsolution) v in Q’,

u 5 o on ÜPQ’ (u 2 v on ÜPQ’) andu > 0 on {u> 0}03,,Q' (u> 0 on {v > 0}03pQ')

implies that u 5 o (u 2 v) in Q’.

The function u is called a viscosity solution if it is both a viscositysubsolution and a viscosity supersolution.

2. VISCOSITY SOLUTIONS 55

DEFINITION2.2.3. Let u be a continuous nonnegatiue function in Dand let (1:0,to) e 6{u > 0} ñ D. We say that (9:0,to) is a regular pointfrom the nonpositive side, if there ezists a regular nonnegati'ue functionv in D such that v > u in {u > 0} fort < to and u(a:o,t0) = u(zo,to).

Finally we need the following definition of nondegeneracy.

DEFINITION2.2.4. Let u be a continuous nonnegatiue function inD. Let (1:0,to) e D be such that u(a:o,to) = 0. We say that u does notdegenerate at (1:0,to) if there exist r0 > 0 and C > 0 such that

sup uZCr for0<r5ro.ÜPQHïoJo)

We now prove that, under suitable assumptions on the limit function u, there holds that u is a viscosity solution to the free boundaryproblem.

THEOREM 2.2.5. Let u = limuE", where (u5*,YE'=) are uniformlybounded solutions to (1.0.1) with Y“ 2 0, satisfying (1.0.4) in D, withwo > —1, and such that u‘k either satisfies (2.1.1) or uï" 5 0 in D.

If u+ does not degenerate at every point of the free boundary whichis regular from the nonpositive side, then u is a viscosity solution of(P)

In order to prove Theorem 2.2.5, we need to show that u is both aviscosity super- and subsolution. We perform this in Theorems 2.2.6and 2.2.9 respectively.

We want to remark that every limit function u is a viscosity supersolution to problem (P) (i.e. we do not need the nondegeneracy,monotonicity nor condition (2.1.1)).

Let us first show that every limit function u is a viscosity supersolution.

THEOREM 2.2.6. Let u = lim u‘k, where (u‘*,Y‘*) are uniformebounded solutions to (1.0.1) with Y“c 2 0, satisfying (1.0.4) in D, withwo > —1.

Then u is a viscositysupersolution of (P

PROOF. The proof of this Theorem is analogous to that of Theorem4.1 in [11]. We include the details here for the sake of completeness.

Let Q CC D be a cylinder which will be assumed to be 31(0) x (0, T)and let v be a classical subsolution in Q satisfying

quonÜPQ

and

u>00n {v>0}nópQ if {'v>0}03pQ;É(0.We will show that u 2 v in Q.

If {v > 0} ñ ÜPQ= (D,then v 5 0 in 3,,Q and therefore v 5 0 in Q.As u 2 Oeverywhere, we see that u 2 v in Q.

If {v > O}OÜPQ 7€(0,it follows from the continuity of u and v thatu > 0 in {v>0}ñQ forO S t < 7', for somo small T > 0. It is nothardtoseethatifu>0in{v>0}ñQforOSt<s,thenquinQñ{0 5 t S s}. We set

t0=sup{0<s<T: u>0in {v>0}ñQñ{OSt<s}},and we will arrive at a contradiction assuming that to < T.

We have that to > 0 and u 2 v in Q ñ S t S to}. In addition,there exists a sequence (In, tn) —)(xo, to) e Q such that u(a:,,, tn) = 0,(Imtn) e {v > 0} 0Q. Then, u(1:o,to) = v(:¡:o,to) = 0 and (zo,to) E3{v > 0} OQ. Since v is a classical subsolution, there exists a sequencey" —>2:0such that 0 < v(yn, to) 5 u(y,,, to), so we have proved that

uzvinQñ{05tSto},(1:0,to) G 3{u > 0} ñ 8{v > 0} ñ Q.

Now consider for /\ > 0

u,\(:1:, t) = ¿Mazo + /\.1;,to + A2t),

’U,\(ilï,t) = ¿00130+ to + A2t).

Since u,v e Lip(1, in Q, and u,\(0,0) = v,\(0,0) = 0, there exists asequence A" —>0 and uo,vo e Lip(1, in RN“ such that UA"—)v0and uAn—)uo uniformly on compact sets of RN“. Since v is a classicalsubsolution, if we assume that V'U+(.”L'0,t0)/IVU+(:CO,t0)I= el and weset á = |Vv+(:vo,to)|, we see that (as v 2 0)

v0(a:,t) = áIT, á 2 V2M(1:o,t0).

Moreover, uo 2 vo when t < 0, so that no is caloric in {xl > 0, t < 0}.In addition u0(0, 0) = o.

There are two possibilities depending whether the following assertion holds or not:(2.2.7)

There exists 6 < 0 such that U0—v0 > 0 when 1:1> 0, 6 < t < 0.

CASE I. Suppose that (2.2.7) does not hold. Then there exists asequence (zmtn) in {1:1> 0, t < 0} such that tn —>0 and uo(:cn,tn) —vo(zn, tn) = 0. From the strong maximum principle, it follows that

uoEvo=ázrf in {11:120, tSO},

implying that

(2.2.8) ¿(u —v)(:co + Anel, to) —>0 as n —)+oo.

We denote (z, t) = (31,33’,t) and for small p, 7‘> O we define

E = {gb/,15) < 221< 9(I',t) +p, II' - 176|< T, It - tol < 72},

where g is a Cl function in a neighborhood of (9:3,to) such that for asmall ro > 0

Bro(ivo,to) ñ 3{v > = Bro(.'lio,to) n {(13,t) I 121= g(.’17’,t)}

and

Bro(Io,to) n {'U> = Br0(l‘0,t0) ñ {(2,15) I ¿El> g(.’17',t)}.

Ifr and p are small enough, then E C {v > 0} and therefore, u-v ispositive and supercaloric in E ñ {t < to} and in addition, u-v 2 ,u > 0in É, for some small ,u and some ball É with center in ÜPEñ {t < to}and not touching 8{v > 0}.

Let wl be a caloric function in E with smooth boundary data satisfying

w1=00n6pE\É, 0<w15uon8pEñÉ,and let wz be the caloric function in E such that wz = v on ópE.

We have

u-vaIZng inEñB,2(a:0,to)ñ{t5to}for some constants C > 0 and 7'2> 0 small, the last inequality followingfrom Theorem 3 in [14]. Hence,

u-vZCv inEñB,,(zo,to)ñ{tSto}and therefore,

. . 1 _

largánf —v)(a:o+ Achto) 2 Ca > 0,

which contradicts (2.2.8).

CASEII. The argument above implies that necessarin (2.2.7) holds.Then, from Lemma A.1 in [10], it follows that

(uo - vo)(1‘,t) = 01‘? + 0(II| + ¡ill/2)

when 11:1> 0, t < 0, for some a > 0. That is,

uo(a:,t) = art;L + o(|:z:|+ |tll/2) in {:cl > 0, t < 0} with a > á.

Now consider for A > 0,

1 1

(uo),\(a:,t) = Xuo(/\a:, A215), (v0),\(:c,t) = Xv0(/\:v,)\2t).

As before, there exists a sequence A" —>0 and uoo,voo e Lip(1, inRN“ such that

(U0)An _’ “00a (U0),\,. —) Uoo

uniformly on compact sets of RN“. Clearly uoo2 voowhen t < 0 andmoreover,

voo(z,t) = áxf and

uoo(a:,t) = ax;L in {.731> 0, t < 0}.

Since uoo is caloric in {uoo > 0}, we can apply Corollary A.1 in [10]to uoo in {Il 5 0, t < O} and hence,

+ ._ azl ln {131> 0, t < 0},

Uoo(33at)_ { 73:1’+o(|:1:|+lt|1/2) in {1:1< 0, t < 0},

for some 'y 2 O. We consider

1 1

(u00)¡\(13, t) = XU00(/\.’IJ,/\2t), (U00),\(I, t) = XU00()\.’13,/\2t).

There is a sequence An —>0 and uooo,voooe Lip(1, á) in RN“ suchthat

(1100)». —) Uooo, (‘Uoo),\,. —> Uooo

uniformly on compact sets of RN+1 and moreover,

vooo(.'L‘,t) = ¿II-1*-and

uooo(:z:,t) = axÏ + 711

for t S 0.

Applying Lemma 1.3.1 three times, wc find a sequence ¿“J-000—>0and solutions ueiooo,Yeioo'Jof (1.0.1) in Q¡(0,0) such that u‘iooo —>uooo

and w51000/5j000—>w(3:o,to) uniforme on compact subsets of Q1(0, 0).

We recall that

uooo(z, t) = am? + 7.7:,- with a > 0,7 Z 0

for t 5 O.

If 7 = O, we apply Lemma 1.4.1 to uoooin a neighborhood of somepoint (0, í) with í < 0 and deduce that

a S 2M(:co,to).

If 7 > 0, we apply Lemma 1.4.5 and conclude that 7 = a and

01S V2M(zo,to)

In any case, as a > á 2 2M(a:o,t0) we get a contradiction andthis finishes the proof. El

Finally, we end this section (and the proof of Theorem 2.2.5) byshowing that, under the nondegeneracy assumption, a limit function uis a viscosity subsolution.

THEOREM 2.2.9. Let u = limu‘k, where (u‘k,Y"‘) are uniformlybounded solutions to (1.0.1) with Y“ 2 0, satisfying (1.0.4) in 'D, withwo > —1, and such that u‘k either satisfies (2.1.1) or uï" 5 O in D.

If u+ does not degenerate at every point of the free boundary whichis regular from the nonpositive side, then u is a viscosity subsolution of(P

PROOF. In order to see that u is a viscosity subsolution, let v be aclassical supersolution such that

uSvinÜPQ andv>0in {u>0}08pQwe want to see that u 5 v in Q.

If not, we define

to=sup{0<s<T: v>0in{u>0}ñQn{05t<s}}.From the definition of to, it follows that to > 0 and, from our

hypotheses wededuce that v 2 u in Qn{0 5 t 5 to}. In addition, thereexists a sequence (a:(s),t(s)) —>(zo,to) e Ó such that v(z(s),t(s)) =0, (a:(s),t(s)) e {u > 0} ñ Q. Clearly, u(a:o,to) = v(a:o,to) = 0 and(:co,to) e 3{u > 0}0Q. If (27:0,t0)E {v = 0}° then, for 7' small we haveu 5 v = O in B,(a:o,to) ñ {t < to} and therefore, u E 0 there, whichcontradicts our hypothesis. Thus

vzuinQÑWStSto},

60 2. THE FREE BOUXDARY PROBLEM

(Imto)G > ñ8{v> ñWe may assume, without loss of generality, that (:co,t0) = (0,0)

and Q1(0,0) = Q1 C Q (consider instead of u the function 7101420+A015,to + Agt) for certain A0 > 0 small, and analogously with v). Let us

take 1 1

v,\(:v,t) = Xv(/\:r,/\2t), u,\(.1‘,t) = Xu(/\:z:,/\2t).

It is easy to see that there exists a sequence /\,, —)Oand functionsU0, vo such that v,\ —+v0, u,\ —>U0.

Since v is regular, we have that z;0(.r,t) = BIT with O 5 fi 52M(0, 0) (for some system of coordinates).

Let us see that also u0(1:,t) = aztf' for some a 2 0,

We may think that in Q1, 3{v > 0} is the graph of some functionw(a:’,t) = :rl, :c = (11,15’)with 11;e Lip(1, 1/2), where w(0,0) = 0 and{v > O} = {1:1 > w(a:’,t)}.

Hence, we have that

|w(9:’,t)l S C (I-T'I+ hill/2)

Let 'R = {(1,06 Q1: 3:1< —C(|:1:’|+ [tv/2)}. Then ’R,n {U >0} = (Dand let w be the caloric function in (9 = Q1-\'R, with w = O inGPRand w = L 2 in the rest of 8,,(9.

Since u is globally subcaloric and u 5 w on 8,,(9, then u < w in O.

Now, w —u is supercaloric in (9, w —u > 0 in the interior andw —u = 0 at (0,0), then, by lemma A.1 of [10], we have that w —u =6:12}L+o(|:r| + Itll/z) and, since by the same lemma, w has an asymptoticdevelopment at (0, 0),

u(:c,t) = (1:5;L+ o(|:t| + hill/2), with a 2 0.

Since by hypothesis u+ does not dcgcncrate, there follows that a >0.

On the other hand, since v is regular, v admits an asymptotic development at the origin in the form v(:c, t) = BIB-1*-+o(|2:| + |t|1/2). Clearly,B 2 a

Now, let h be the caloric function in Ó := Qf n {_v> O}f"!{-p <t < 0} for some small p > 0, with h = U- u on 8,,(9. And, let g bethe caloric function in (9 with g = v on 8,,(9. Then, h = g = 0 inana{v>0}n{—p<t<0} andh>0,g>0in(’).

Therefore, by the results in [1], there exists a > 0 such that h 2 agian/20{v>0}ñ{—% <t<0}.

Since u is subcaloric in Ql’ and u 5 v in Q1-we deduce that v —u 2

au in QI?2 ñ {v > 0} ñ {-15 < t < O}. In particular B —a 2 aa > 0.

The theorem will be finished if we show that a = w/2M(0, 0).

Case 1: u‘k verifies (2.1.1).

As in Theorem 2.1.5, we obtain

f/D utuzl'fib= IVulzï/Izl- f/D unVqu/i

+ [jor-1990}Mhfiwx‘ + //Dn{u>o}(w0)a:¡( ¿”o f(8)ds) tb

for every test function 1,11.Then, taking 1p"(11:,t) = MPG, and changing variables, we get

//D(u/\)t(ux):flfl= IVuiI2tbn- f/D(u,\),,quAV¡/¡

+ [Ánhpm M(/\ï,)\2t)1/1:¡+ //Dn{u>0}(wo)z¡( ¿”a f(s)ds) w

By Lemma 1.3.1, we get (for some sequence /\k —>0)1

0= --a2 «[11,+ lim M(/\kz,)\it)1pn.2 Dn{z¡>0} k-’°° ’Dn{u,\k>0}

We want to check that ka >o}—)tho} a.e. or, equivalently,

(1) {xl > O} C U,°1°=lr1an {uik > 0} = liminf{u,\,c > O} a.e.(2) 0:11 Uk?" {un > 0} = lim sup{u,\k > O} C {xl > 0} a.e.

Let us see (1). If 2:1 > 0, we get that 01:51> 0 and since u,“ (1:, t) —)azl it follows that u“ (1:,t) > 0 Vk 2 k0.

Let us see (2). If exists k]- —) oo with un]. (11:,t) > 0 then it must

be z] 2 O, because if 21:1< 0, we have that Unitat) = 0 forj 2 jo(because as v is regular, {ka > 0} —>{:vl > Since uAkj 5 UM].weget a contradiction.

Therefore,1

Dn{z¡>0} Dn{z¡>0}


So that,

0 =/ (la? —M(o,0)) wdx’dt.'Dn{a:¡=0} 2

Since 11)is arbitrary, ¿a2 = M(0,0), so that,

a = V2M(0, O)

and the proof is finished

Case 2: uf“ 5 0

We already know that, if we consider u,\(z, t) = %U(AI,Azt), thenit follows that

u,\(a:, t) —)uo(:z:,t) E axf,

uniformly on compact subsets of RN“.As before

// IVuEklzwz.-// ItífiVuíka19 D D

+ Bsk(u'-’k,:1:,¡th/¡Il+ f¿k(s)ds>11)D D -w06k

+ /(wo)1.(“’:" —won-wow6

Now, as in the previous case, ifwc consider first 1/2"(:1:,t) = MPG, f5)and change variables, we obtain(2.2.10)

1

f/(uik)t(u:k)IIL/)= 279/)1.‘—El;‘k

+/:/B;\k/,\(uï\kazat)wzl +// (we-k)“ f(3)d8 ’dJA‘D 5k _wo

we"

+//1)(wo)n(í —wo)f(—wo)1/¡'\

where B:(u,:z:,t) = [I'ma/“¿2043 + w5(.1:,t))f¿(s)ds. We want to pass

to the limit as both ek and A go to zero.

Using Lemma 1.3.1, we see that for every sequence An —>0 thereexists a sequence kn —) oo such that ón := ¿kn/An —) 0 and uó" :=(u‘kn),\n —>uo uniformly on compact sets of RN“. By Proposition

1.2.1 we see that we can pass to the limit in the first three terms of(2.2.10) (with e = ek" and /\ = An).

Let us study the limit of B¿:(u‘5"(a:,t),:z:, t).

It is easy to see that in {1:1 > 0}, B¿:(u5"(z, t),z, t) —) M(0,0)uniformly on compact sets. Now, let K C {9:1< 0} be compact. Wewill show that

V(B"5\:(uó"(:c,t),z,t)) —>0 in L1(K)

In fact,

(uan(I)t), I) = Yónfón(U6")VU5"ón

+ Áano(/\na:, mx"; (z, t) _ w0(¡\n;1:,A7210)¡(Wounz’ ¿im6,.

6.. 25n

Vw s)ds¿n -wo(An:,A3,t)

Since Y's"f5" (u'5") —)0 as measures in K and is nonnegative, we deducethat the convergence takes place in L1(K). On the other hand, Vuó"is uniformly bounded. Therefore, the first term goes to zero in L1(K).

In order to see that the second and third terms go to zero uniformlyin K we only need to observe that

u‘s" u‘knt = n a 2

¿n (ar, ) 5k" (A a: Ant)

and a similar formula holds for 3:" . So that¿nw .

l (a3,t) —wo(/\na:, Afit)| —)0 umformly on compact sets of RN“,n

6 6U" 'LU"> _ > _5,, - ón ‘ C’

a" 6k"

IVÉU ICC,t) = Anlvgw_l(,\nz, Ait) —>0 uniformly on compactn k"

sets of RN+1.

On the other hand, |B¿:(u""(a:, t),:1:,t)| 5 CK, so that we have,

Bá:(u“"(z, t),:c,t) —>Mu) weakly in L2(K).

Let us now see that, actually, the convergence takes place in L‘(K).

There holds that

8 6a ([35: (11,5"(21:,t),:c, t)) = Yó"f5" (uó")(u‘s")¿ + EBA: (u'5",1:, t)

< É[3’611'(u‘s",1:,t) S CK in K._ at

On the other hand, for every (zo, to) e K, and QT(xo, to) C {xl < 0}

3— BA" ua" :c,t,:c,t =

//r(3o.to) at ( ón( ( )

/ Ba:(u‘"(:v, to + T2)a13,(t0+ T2))dr87(1'0)

_/ (u6n(z’to _ T2)1Ï7(tO_ 72))da:2 _C1'37(10)

since |B¿:(u6"(a:, t),a:,t)| 5 CK for every compact set K.Therefore there exists CK > 0 such that

“32: (“6"(ï, t),1:at)“W‘-'(K) S CK

Hence the convergence takes place in L1(K) (for a subsequence).

Now arguing as in Lemma 2.1.2, we get that M(t) = 0 or M(t) =M (0, O).

We can now take the limit in (2.2.10) for the sequences ek" and Anand we obtain

1 _0= wi] wi]2 Dn{:¡>o} ‘Dn{z¡>0} ’Dn{z¡<0}

So that,

o =/ (la? —M(0,0) —Mm) qbdx'dt.Dn{z¡=0} 2

Since w is arbitrary we get ¿a2 = M(0,0) —M(t). So that, inparticular, M(t) is constant and then w_chave that M(t) E 0 or M(t) EM(0,0). Since a > 0 we deduce that M(t) E Oand

a = ‘/2M(0,0).

The proof is finished. D

3. CONSEQUENCES AND APPLICATIONS 65

3. Consequences and applications

In this section, we study some consequences of the results collectedin Sections 1 and 2.

First, we prove a result that guarantees the nondegeneracy of a limitfunction u and then we combine this result with Theorem 2.2.5 andconstruct a family u‘ of solutions to (PE) such that the limit functionu = lim uEis a viscosity solution to (P) for rather general initial data.

Now we prove a proposition that says that, under suitable assumptions, u+ does not degenerate at the free boundary. The proof is similarto Theorem 6.3 in [10], where the nondegeneracy of u+ was proved inthe strictly two phase case. Here we assume, instead of (2.1.1) thesomewhat stronger condition that for every K C D compact, thereexist 0 < n < 1 and eo > Osuch that for every O < e 5 eo

E

(2.3.1) u? 5 n in K n {u E 0}°.

PROPOSITION2.3.2. Let u = lim u“, where (u‘k, Y”) are uniformlybounded solutions to (1.0.1) satisfying (1.0.4) with wo > —1, such thatY” 2 O and the functions ue" satisfy (2.3.1). Let (1:0,to) e 3{u > 0}.

Let us assume that there ezists l/ E IR”, with |11|= 1 such that

liminf H“ > 0} n {<1- zw) > 0} nat-(20,220)!"’°+ IQ: (zo, to)|

>01

and

liminf H“ = 0} n {I > a2H°+ ¡car-(xo,ton

with al + a2 > á, then there ezists a constant C > 0 depending onlyon N, f, a1+ a2 and r0 > 0 such that, ifO < r 5 ro,

sup u 2 Cr.ÜpQr-(IoJo)

PROOF. Without loss of generality, we may assume that (9:0,to) =(0,0) and that 1/= el = (1,0, ...,0).

We will note Q,- = Q,- (O,0) and

(una, t)=(met), (Worm)=(mrzt),1

u,(:c, t) = ;u(ra:, r2t).

STEP 1. Let us see that there exists r0 > 0 and a constant c suchthat if r < ro and ek < eo = 60(7‘),then

[Al-(Yskhfek/JWWM 2 c.

Without loss of generality we may assume that r0 is small so thatIu‘I S 1 in (2,70.

Let 0 < 4A < al + 02 — From our assumptions, it follows thatthere exists 1-0> 0 such that, for r 5 r0,

r>00 >00 _ r=0°ñ <00 _ 1¡{u } {ag } Q1|+I{u } {f1 } Q1|>_+2/\_

IQ] I IQ] I 2

We now fix 7' with this property. Then, there exists 'y > 0 small suchthat

¡{ur>7}0{I1 > amen +1{ur=o}°n{xi< 0mmIQÍI IQÍI

Let us now define

A,={u,>7}ñ{zl >0}ñQ¡’, Br: {u,=0}°ñ{z¡ <0}0Q1‘and —B, = {(zl,m',t)/ (—x¡,z',t) G Br}.

Then, we have

1>— /\._2+

IAr n (-Br)l 2 AIQÍI = Á

For 0 5 121< 1, let

9(I1) = |{(:v’,t) (Inaz'at) G Ar ñ(-13r)}l

Let 0 p;\}| > 0

In fact, if notl

As IArñ(—Br)I=/ g(zl)da:1=/ _ 9(I1)d1'1spx0 {95M}

which is a contradiction. Therefore, we have in particular that thereexists O < :cï < 1 such that

9th) = lArI= |{(I',t)/ (mLI'J) E Ar0 (-Br)}| > pÁ

Let n > 0 be the constant in (2.3.1) in Q1(0,0), let 0 < 6’ < ó,0<b<b’<1besuchthat

n < —wo(0,0) + 6 < b.

Let ¡e > O be such that

f(s)>n>0 forsgb'.Then, for (z’, t) e A, we have

1 7Ek r I

E—k/T(u ),(:z:1,z ,t) > —2(Ek/T)> b,1_ 6k _ 1' ' _

¿lc/rw ),( 11,1:,t) < wo(0,0) +6

if 6k < El = 61(7‘)is small. So that, for every (z’,t) e A, there exists:ïtï G (-1,1) such that —w0(0,0)+ 6 < 1 (u‘k),(íï,x’,t) 5 b.— Ek/r

Now, by the uniform Lipschitz regularity of (u‘k), and (YEk)raand(1.0.4), we have that for Ek 5 60(5 61) and r 5 r0,

(u )r r I (Yík)r— t <b dEIC/T (1:11:51 )— an Ek/T

where C depends on ó,6’,b,b’,on the Lipschitz constant of u“ and Y“in Q1“and ro depends only on wo.

5k . - E

(21,2],t) 2 6' 1f |:z:1—:1:1|< CT]:

Finally we haveiïfánïïív{(2,0e Qï/ 2 6'and¡(“MW2 n}

n

ek/r ek/r ¿“k/r

2 ¿’L|A,|2Cí'° 2 2Có’np5E c.Ek/T T

25'

STEP 2. Now we will prove that there exists a constant C > 0 suchthat, for every r > 0 small,

sup u 2 Cr.apor

We will proceed by contradiction. If the result were false, there wouldexist a sequence rn —>0 such that

1sup u 5 —rn.

6,70;," n

Since u“ —>u uniformly in Q_‘ there holds thatro?

2sup uEk S _Tm

ÜPQ;I'n n

fork 2 Thereforewehave

SIN

Sup (u‘k)rn Sapra;

In addition, by STEP 1,

//Qf(YEk)’"fek/rn((u€k)rn) Z C,

as long as rn < ro and 6k 5 eo.

Since (u5*)nl are solutions to

a E E E

¿(u ")rn = (Y *)rnfek/rn((u ")r..)

in Q; , there holds the representation formula

(mano) = f (mm —j/Q;(YEk)r"¡sk/rn((uik)rn)a,ano;

A(u6k)rn _

where

PZOonapQ5, / P=1,apQï

GZOÍHQQ, G2n>0ian.It follows that

(uek) (o 0) < 3 _,.1,c < —EcTn 1 _ n 2 1

if n is big enough and ek 5 min{eo, 6k(n)}. But this gives a contradictionsince (uík)rn(0, O) —>0 as k —)oo. Thus the proof is complete. El

REMARK 2.3.3. Proposition 2.3.2 remains true if we change thehypothesis that u‘k satisfies (2.3.1) by

u‘k

(2.3.4) €— —) —wo a.e. {u E 0}°.le

In fact, as in the proof of Proposition 2.3.2 we consider for each 0 <r < 1 the sets A, and Br. So that, for some 0 < A < 1

IArr-1(_Br)lZSince B, C {ur E 0}°, there holds that

(“6)1'6T (-zl,z',t) —>—w0(—r:1:1,ra:',r2t) a.e. A, n (-Br).

Let 0 < ,u < 1. There ezists C, C (A, ñ (-B,)) such that ICrl =uIA,ñ (-B,)| and

(“Elr

e/rLet ó > O. There ezists el = 61(r) such that

(—z¡,:t’,t) —>—wo(—r:cl,r:r',r2t) uniformly in Cr.

(u )r(—:1:¡,a:’,t) S —w0(—ra:1,r:t’,r2t) +é S —wo(0,0) + 6 in Cre/r 2

ife < 61 and r f r0 = ro(ó). Now, the prooffollows as in Proposition2.3.2 by taking /\ = uAIQfl and

AT:={(II1t)/(Ïlizlat)eREMARK2.3.5. Proposition 2.3.2 remains true if we change condi

tion (2.3.1) by condition (2.1.1). In fact, as in the proof of Theorem2.1.5 we see that condition (2.1.1) implies that

B¿k(u"‘,a:, t) —>0 Lloc({u E 0}°).

As in Theorem 2.1.5 we deduce that u‘k satisfies (2.3.4

Using Remark 2.3.3, Remark 2.3.5 and Theorem 2.2.5 we get thefollowing Corollaries.

COROLLARY2.3.6. Let u = lim u“ where (u‘k, Y“) are uniformlybounded solutions to (1.0.1) in a domain D C RN“ with Y“ 2 O,which verify (1.0.4) with wo > —1 and such that u“ satisfies (2.1.1).If the free boundary D ñ a{u > 0} is given by 3:1= g(:t',t) with g eLip(1, 1/2), then, u is a viscosity solution of the free boundary problem(P).

COROLLARY2.3.7. Let u = lim u“ where (u‘k, Y“) are uniformlybounded solutions to (1.0.1) in a domain D C RN“ with YE" Z 0,which verify (1.0.4) with wo > —1 and such that u‘k satisfies (2.3.4)and uÏ" S 0. If, for every (1:0,to) e D n 8{u > 0}, {3:e RN / (1:,to) EDñ {u > 0}} is given by x1 > (a:')with Ó, Lipschitz continuous then,u is a viscosity solution of the free boundary problem (P

PROOF. We only need to see that u does not degenerate at pointsof the free boundary which are regular from the zero side. Let (9:0,to)be any such point. We see that we can apply Remark 2.3.3 at thatpoint. In fact, since uf" 5 0, u is decreasing in time. Therefore,

{(I,t)/I¡ > ©(x), t 5 to} C {u > 0}and the parabolic density of this set is positive. El

In particular, Corollary 2.3.7 can bc applied to solutions of (1.0.1)with uf) constructed as in [13] and Yo‘a small perturbation of uf). Infact we have the following result.

COROLLARY2.3.8. Let uo e CGR”) n 02({u0 > 0}) be such that

||uo||c2(m) < oo, Aug 5 0 and (uo)Il —AIVuol 2 0 in {uo > O}with A > 0. Assume, moreouer that 0 < a2 5 IVuol 5 al < \/2M0in a neighborhood of the free boundary: e {U0> O}/dist(x, {uo =

0}) 5 7}, and Mo = [01sf(s). Then, there ezists a sequence (uf),Yo‘)G

(Cl(lRN))2 with uf, —)uo uniformly in RN (so that uf, are uniformlybounded) and, moreover, they satisfy(2.3.9)

(1) AUÉ - Yo‘fJUÉ) S 0

(2) (uan - ÁIVHÉIZ 0ye _ e

(3) OTUQ —)wo uniformly on compact sets, with wo > -1.

wo e IR is any constant such that wo 2 —1}with n > 0 small enough.

Let (uíY‘) be the solution to (1.0.1) with initial datum (u3,Y05)(so that, in particular, u‘ and Y‘ are uniformly bounded). For everysequence ej —)0 there ezists a subsequence ejk such that there ezists

u = lim ue“k-voo

and u is a viscosity solution to the free boundary problem (P

PROOF. Let uf, be the approximations constructed in [13]. Theapproximations are constructed in the following way. First we extenduo to a neighborhood of {uo > 0}: S := e RN/dist (I, {ug > 0}) 5'y} in such a. way that ||uo||02(s) < oo. For e small enough we define

u3(z) = EF( u°(x))) in {-Ce 5 U05 e}.¿(1 _

2Mo 5'

where F E C2(R) is such that

F” g (1+ 6)Ff(F) + aF’, F(0) = 1, F’(O)= —\/2M0.

Here ó > 0, a > 0 are such that F has a strict minimum at a finitepoint Esuch that s‘/2Mo > 1. (s —>+oo as ó —)0), and F is decreasingfor s < s.

The constant C is taken as C = s\/2MO —1.

We define

uf) = uo in {U0 > E}

uf, = €F(5) in RN \ {uo > —CE}.

As in [13], we see that us e CICR”).

Let wo e IR be such that wo 2 —n > —F(E) with 17> 0 to be fixedlater and let

Yo‘= ug + ewo.

Then, Yo‘2 O. It is immediate to verify that (2.3.9) 1) is satisfiedin {uo > E} and RN \ {uo > —Ce}. Let us see that it is satisfied in{—Ce 5 U0 5 e}. In fact,

1 321 1e _ E e = II 2 _ ’ _ _ _

Auo nfemo) 2MOEFIVuOI mFAuo EFf(F) E f(F)1+6 a a

< F F 2 FI 2 __ I_2M0€ +2MO€1 wo

—¿Fm —7m?)where a > 0 is such that IAuol g a.

Let 0 ae there holds that

Aus —mama) si [[(1+ ó)<1—mA? —11mmaa2 a6 ,

+ (TM: — r—2MO)F—wof(F)]

s ¿[—uF —wo]f(F) s ¿[—um) —wo]f(F) s oifn=[JF

Clearly, (2.3.9) 3) holds. Let us see that (2.3.9) 2) also holds. Weonly need to verify this property in the set {-CE < uo < e} and thisis clear from the fact that

1 1 'LLoVE=——F’ —1—— V.“0 2Mo ( 2Mo( 6)) u°

Now, by the results of Chapter 1, for every sequence ej —)0 thereexists a subsequence and a continuous function u such that u‘ik —>uuniformly on compact subsets of IR” x (0,00).

On the other hand, uf is a solution to the following equation

AU —Ut = fi;(u)U.

Here 65(3) = sf5(s). Since, for e small enough uï(:r, 0) S Owe concludethat

(2.3.10) u: 5 0 in RN x (0,00).

In a similar way we see that ui, —Aux, 2 0 for every i. So that

/\ y

(2.3.11) uf“ —ÑIVu‘| 2 o in IR" x (0,00).

Clearly (2.3.10) and (2.3.11) imply that

ut S 0 and uIl —%|V11| Z 0 in {u > 0}.

In particular, the free boundary is Lipschitz in space.

So that, in order to apply Corollary 2.3.7 we only need to verifythat u‘k satisfies (2.3.4). On one hand, given K C {no E 0}° compact,there exists 60 such that for e < eo

¿51(1) F(3)

35(us,z,o)=/ (8+wo)f(s)= (8+wo)f(s)- wo WO

On the other hand,6E(B¿(u‘,ar,t)) = YEf¿(uE)uÏ5 0.

Therefore,19(5)

B¿(u8,:c,t) 5/ (s + wo)f(s) for a: in K, t > 0._w0

As in the proof of Theorem 2.2.5 we see that, since uf 5 0, thereholds that B¿(u‘,z, t) —) M(:r, t) in Llloc({u E 0}°) and, for almost

every (z,t) we either have M(:r,t) = 0 or 1l71(a:,t) = M = f1w0(s +w0)f(s). Since

V(B¿(u5,9:, t)) = Y‘f¿(u€)Vu‘ —>O in Llloc({uE 0}°)

there holds that M(a:,t) = M(t) in {u E 0}°. Therefore,_ ¡”(5)

M(t) 5/ (s + w0)f(s) (1.6. {u E 0}°."¡0

Since F(5) < 1, there holds that 117I(t)E 0.


Thus, for every sequence ek —>0fif * (s + wo)f(s) —>o a.e. {u E 0}°_wo

and we deduce that u“ satisfies (2.3.4). El

Combining the regularity results for viscosity solutions of [17], Corollary 2.3.6 and Corollary 2.3.7 we have the following regularity resultfor limit functions.

COROLLARY2.3.12. Let u as in Corollary 2.3.6 or Corollary 2.3. 7.If, moreover, the free boundary D ñ 8{u > 0} is given by 1:1= g(:1:’,t)with g Lipschitz continuous, then, u is a classical solution of the freeboundaryproblem(P

CHAPTER 3

Uniqueness of limit solutions

The main point in this Chapter is to give a positive answer tothe question of whether the limit of a sequence of solutions to (PE)is determined once the limit of w‘k/e;c and of uff are fixed. So westudy the uniqueness character of the limit functions (or limit solutions)studied in the previous Chapters.

Some geometric assumptions are needed. In fact, uniqueness ofthe limit fails, in a general setting, even in the case wE E 0. Thesegeometric assumptions are similar to that used to prove uniqueness ofthe limit for the case ue = Ye in [30]. We state these assumptions inSection 1.

In fact, we follow here some of the ideas in [30] which are basedon the fact that any limit function is a supersolution to (P). This isstill true in our case. Unfortunately the simple construction in [30] ofsupersolutions of (PE) that approximate a strict supersolution of (P),when w‘ E 0, does not work in the general case unless one asks for alot of complementary conditions on the reaction function f.

Therefore, we follow here the construction done in [22]. The proofthat this construction works in based on blow up of the constructedfunctions.

In Section 2 we prove some technical lemmas needed in the proofof the uniqueness result.

In Section 3 we prove that, under the geometric assumptions in consideration, a semi-classical supersolution of (P) is the uniform limit ofsupersolutions of (PE),and restate an analogous result for subsolutions.

In Section 4 we arrive at the main point of the Chapter, we provethat, under suitable assumptions, there exists a unique limit.

We end this Chapter with a discussion of different geometries wherethese results hold.

76 3. UNIQUENESS OF LIMIT SOLUTIONS

1. Preliminaries

Following [30] we give the definition ofsupersolution of problem (P)that will be needed in this Chapter. Note that this definition differsfrom the one given in Definition 2.2.1 since we are not assuming thatthe function be Cl up to the free boundary or that the free boundarybe C1.

DEFINITION3.1.1. A continuous nonnegative function u in ÓT =RN x [0,T], T > 0, is called a semi-classical supersolution of (P) ifu E C1({u > 0}) and

(i) Au—u¿50 inQ={u>0};(ii) limsupnawlshh't) |Vu(y,s)I S \/2M(:r,t) for every (1:,t) e

39 n QT;(iii) u(-,0) 2 ug.

Respectively,u is a semi-classical subsolution of (P) if conditionsand are satisfiedwithreversedinequalitiesand liminf instead

oflirnsupinA function u is a classical solution of (P) if it is both a semi

classical subsolution and a semi-classical supersolution of (P), u EC1({u > 0}) and the free boundary 8{u > 0} ñ QT is a Cl surface.

Next, a semi-classical supersolution u of (P) is a strict semi-classicalsupersolution of (P) if there is a 6 > 0 such that the stronger inequalities

(ii’) limsup93(y_s)_,(1’t)|Vu(y,8)l S \/2M(:c,t) —¿for every (m,t) G39 n QT;u(-,0) Z uo+ 6 OTI00 = {U0>

hold. Analogously a strict semi-classical subsolution is defined.

As a consequence of the results in Chapter 1, one can check thatevery limit solution u = limJ-_,°°us} of (P) is a semi-classical supersolution in the sense of Definition 3.1.1. In fact,

PROPOSITION3.1.2. Let u‘i be solutions to (st) - with wei satisfying (0.2.4) and wo > —1 - such that uíi —>u uniformly on compact sets and e]- —)0. Assume that the initial datum uo is Lipschitzcontinuous and that the approximations of the initial datum verify|ufi(:c)|,|Vu3(:r)| 5 C and ug e C‘({u5 > 0}). Then u is a semiclassical supersolution of (P).

1. PRELIMINARIES 77

PROOF. We have to verify conditions (i)-(iii) of Definition 3.1.1.

By Proposition 1.5.1, and (ii) hold.

Now, from our assumptions on the initial datum uo, by Propoítion5.2.1 of [27], we have that uE —)u uniformly on compact sets of QT sothat u is continuous up to t = 0 and (iii) also holds. El

Let us suppose that the initial datum uo of problem (P) is starshaped with respect to a point 1:0,that we always assume to be O, inthe following sense: For every /\ e (0, 1) and a: E RN,

(3.1.3) u0(/\:r) 2 uo(z), A00 CC Qo,

where Qo = {U0 > O}.

Also, assume that

(3.1.4) wo(/\:r,0) S wo(x,0) ifa: e RN, 0 < /\ <1and

(3.1.5) wo > —1+ 61 for some 61 > 0.

Let u be a semi-classicalsupersolution of Let /\ and /\’ be tworeal numbers with 0 < A < /\’ < 1. Define

(3.1.6) u,\(a:,t) = ¿MAL/Vi)

in QT/p. The rescaling is taken so that u,\ satisfies the heat equationin

(3.1.7) QA= {(z,t) : (Az,/\2t) E

Moreover, the fact that 0 < A < /\’ < 1 makes u,\ a strict semi-classicalsupersolution of (P).

In fact, let us first see that

M(/\a:,/\2t)SM(a:,t) if0</\<1.This is a consequence of the fact that the function

1

a —>/ (s+a)f(s)ds—a

is nondecreasing and

(3.1.8) wo(A:c,/\2t) 5 w0(93,t) if 0 < /\ < 1.

In fact, the function w,\(z, t) = wo(/\a:,/\2t) is caloric and w,\(:c,0) 5w0(1:,0) if O< A < 1 by hypothesis. Thus, by the comparison principle,w,\(:v, t) 5 wo(:c, t) in QT.

Now, let (Io,to) E 3{u,\ > O}. Then,

. _ A

lim sup IVu,\(a:, t)| = hm sup —,Vu(/\a:, Azt)|9A9(1=,t)->(1=o.to) 99(A1,A2t)->(Axo,,\210) /\

g ¿Mm/¡(Am A2260)/\

S V 21VI(.'L'0,to) — — V 2M0,

where 0 < Mo < M(a:,t) in QT, by (3.1.5).

On the other hand, since A90 CC Qo, there holds that

uo(/\a:) 2 'y > 0 ifn; E (20.

Thus, for a: e Qo,

1 1

u,\(a:,0)= Ñuo(/\z) = u0(/\x)+ —1)u0(/\:v)

2 uo(z)+ —1)7.

The following comparison lemma for problem (P) can be proved asLemma 2.4 in [30].

LEMMA3.1.9. Let U0 satisfy (3.1.3) and wo satisfy (3.1.4)-(3.1.5).Then every semi-classical subsolution of (P) with bounded support, issmaller than every semi-classical supersolution of (P). i.e. if u’ isa semi-classical subsolution such that Q’ is bounded and u is a semiclassz'calsupersolution then

Q’CQ and u'Su,

where Q’ = {u’ > 0} and Q = {u > 0}.

PROOF. Let u’ be a subsolution and u be a supersolution of (P) inQT. We only need to show that Q’ C Q since the comparison betweenu’ and u will follow from this inclusion by the maximum principle.

Suppose first that u’ e CWW) and u e C‘Ñ). Let

A0 = sup{/\ e (0,1) : (2’ C QA},

where QAis defined in (3.1.7). We have to show that A0= 1. Supposenot, then A0 < 1 and Q’ C QAO,and there is a common point (2:0,to) G39’ 089M nQT. Let A0 < A},< 1 and uio be as in (3.1.6). Then

2. AUXILIARY RESULTS 79

u’ 5 u,\o in Q’. At (1:0,to), (as u’ and u are regular) by Hopf’s Lemmawe have

au’ au“ÑÜOJÜ < -5(lïo,to),

where 1/ is the outward spatial normal for Q’ at (1:0,to). Now since

Üu’ ,

—a—y(ï0vt0)= IVu(-Ï01t0)l Z 2M(xo,to)

and8

—a—‘;<zo,to)= IVu(-'vo,to)l < \/2M(3_o,to),

we arrive at a contradiction. Observe that here, we do not need thestrong inequality (ii’), so we only need the weaker assumption wo > —1in QT instead of (3.1.5) in this case.

The general case, can be reduced to the previous one as in [30]. Infact, let ïí be a supersolution. Choose O < /\ < X < 1 close to 1 andregularize ïí by

11(1)t) = (17A(1=,t+ h) - 77)+,

for small h,n > 0. Analogously regularize a subsolution Then wewill fall into the hypotheses of the previous case and then we can finishthe proof by letting first h, 77—>0+ and then /\ —)1-.

2. Auxiliary results

This section contains results on the following problem:

(Po) Au - ut = (u + wo)f(u),

where the function f is as in Section 1 and wo is a constant, wo > —1.The results will be used in the next sections where (Po) appears as ablow-up limit.

These results and their proofs are analogous to those in Section 4in [22] where the case wo = 0 was analyzed. We prove them here forthe sake of completeness.

LEMMA 3.2.1. Let a, b 2 O and let 1/)= 1,0“, be the classical solutionto

¡pss= +w0)f(w) for S > 0»

(3'22) M0) = a, 1/240)= 4%.

Let B(T) = f:,,,(p + wo)f(p) dp

(3.2.3) Ifb = 0 and a e {-wo} U [1,+00), then 1/1E a.

(3.2.4) Ifb = 0 and a e (-wo, 1), then lim,_,+°°1/1(s)= +00.

(3.2.5) Ifb e (o, B(a)), then limwoc 1/2(s)= +00

(3.2.6) IfO < b = B(a), then 1/),< 0 and llms_,+°°1,/)(8)= —wo.

(3,2,7) Ifb e (B(a), +00), then 1/),< 0 and lims_,+°°1,b(s)= —oo.

PROOF. We first recall that the function f is Lipschitz continuousand therefore, there is a unique classical solution to (3.2.2).

Let us multiply equation (3.2.2) by 11),.We getdwssws= + = fOI'S>

Then, if we integrate the expression above, we deduce that

(3.2.8) gate) - Bws» = ¿11130)- Bono» = b- Bm),for every s 2 0.

I. Assume b = 0 and a e {-wo} U [1,+oo). Then, (3.2.3) followseasily if we recall that (s + wo)f(s) = 0 for s e {-wo} U [1,+oo).

II. Assume b = 0 and a G (-wo, 1). Since 11),,> 0, then ws(s) 2 0.Moreover, 1/),(3) > 0 if s > 0 (otherwise 1/)E a in some interval, whichis not possible). In particular, given so > 0, we must have, for s > so,

M3) Z Úlso) + Ús(30)(3 —So)

and hence, (3.2.4) follows.

III. Assume b e (0, B(a)). From (3.2.8) we deduce

B(d)(3)) 2 B(a) - b > 0,

which implies, for some constant n,

(3.2.9) tb(s) 2 p > —w0.

Let us suppose a > 1. Then, 1,12,,= (1/)+ wo)f(1/1) = 0 near theorigin. Hence

1/)(5)= a —fis,

as long as 1,!)(3)> 1. In any case ( a > 1 or a S 1), there exists so 2 0such that 1/)(30)S 1 and ¡[23(30)= —\/%, and therefore, there existssl 2 0 such that

¿(31)<1, lil/ls(sl)<

If we had 11),5 0 for s 2 sl, then, from (3.2.9) and from equation(3.2.2), we would get, for s 2 sl,

-wo ó > 0,

for some constant ó. Thus,

0 2 1/23(s)2 ws(31) + ¿(s - 31),

for s 2 sl, which is not possible.

That is, we have shown that there exists 32 > Osuch that w3(32) >0. Then, 1/23,> 0 now gives, for s 2 32,

WS) 2 M82) + ws(82)(s - S2),

that is, (3.2.5) holds.

IV. Assume 0 < b = B(a). Now, (3.2.8) gives

(3.2.10) áwfls) = 30143)), for s z o.

If there existed so 2 O such that 1/),(30)= 0, then B(w(so)) = 0,implying M30) = —wo. The uniqueness of (3.2.2) would give ¡12(3)E1/)(30),a contradiction.

Hence, ¡03(5)< 0 and thus B(w(s)) > 0. This implies that WS) >—woand that there exists

lirn 1p(s) = 'y, —wo5 7 < +oo.3->+oo

If 7 > —wo,it follows from (3.2.10) that

SEE-1001,03“)= —\/2B('y) < 0,

and then «12(3)< —wofor s large. This gives a contradiction and thus,(3.2.6) holds.

V. Finally, assume b E (B(a), +00). Then, (3.2.8) gives

-1PÏ(S) Z b - B(a) > 0

In particular, ¡bsnever vanishes and we have, w,(s) 5 — 2(b —B(a)).It follows that

IMS) S 1/)(0)- \/2(b - 13(0)) 8,

for s > 0, then (3.2.7) holds and the proof is complete. El


LEMMA3.2.11. Let B(T) = frwo(p + wo)f(p) dp.

a) Let d)" Z —wo,symmetric with respect to s = g, be a solution to

¡bss = (1/)+ wo)f(t/)) i" (0, n),

¡12(0) = 1/1(n) = a G (——wo,1).

Then, 11:2(0): -—\/m with bn /‘ B(a) as n —>oo.

b) Let 1p" Z —wo be a solution to

1/)” = (w + wo)f(1/)) i" (0,71),

(3.2.13) WO) = a G (—w0,1],

1M") = ‘wo

Then, 111:(0)= —\/m with bn \, B(a) as n —)oo.

(3.2.12)

PROOF. Part a). Since 1/)"is symmetric, 1,1)?(3)= 0.

On the other hand, since1

yWV-BWÜ=%—BWLthere holds that

—B(1,b”(n/2)) = bn —B(a).

In particular, there holds that bn 5 B(a).

We claim that WK?) —>—woas n —>oo. In fact, if not there wouldexist a > —wosuch that, for a subsequence that we still call 1,1)",

112"(s)2 1,0"(n/2) 2 a, for 0 S s S n.

On the other hand, there holds that 11)"(s)5 a for 0 S s S n. Thus,(1,0"+ wo)f(1,/)"(s)) 2 fio > 0 for 0 5 s 5 n. Therefore, 1,11;;2 fio forOgsgnandthus

1/)"(s)2 a + %(s —n/2)2, for s e [n/2,n].

In particular,

a = 1/)"(n) 2 a + (Bo/8M2 —>oo, as n —>oo

which is a contradiction. Thus,

bn —B(a) = —B(w"(n/2)) —>0, as n —)oo.

Part b). Since

áflf-BWÜ=M—BWL

there holds that1

¿(wm)? = bn- B<a)2 o.We claim that tpfin) —)0 as n —)oo. In fact, if not, there would exista > 0 such that, for a subsequence that we still call u", 1/):(11)5 —a.Since ug; 2 0, there holds that

112?(n) 2 7,1)?(8),

for 0 5 s 5 n. Thus,

1p:(s) S —a for 0 5 s 5 n.

Therefore,

a +w0 = 1p"(0) —1,!)"(n) = —1/);‘(9)n 2 om —>oo, as n —) oo

which is a contradiction. Therefore, 11;;‘(n)—)0 as n —)oo and thereholds that

bn —>B(a),

¿sn->00. Ü

LEMMA3.2.14. Let B(T) be as in the previous Lemma and let 72., ={(x,t) E RN+1/a:¡ > 0,—oo < t 5 7}, 0 5 0 <1+wo and letU e C2+°'1+%('R_7) be such that

AU —Ut = (U+wo)f(U) in R7,

U=1—0 0n{a:1=0},—w05U51—9 ¿mz-7.

1) If0 = O, then |VU| 5 ‘/2B(1) on {1:1= 0}.

2) If0 <0< 1+w0 and0< a < B(1) is such that1-0

/ (p+ www) dp= B(1)—a,U0

then IVUI = \/2(_B(1)_—0_) on {1:1= 0}.

PROOF. For 0 e [0, 1 + wo), let Vn be the bounded solution to

AV- 14= (V+wo)f(V) in {o < 1:1< n, :r' e RN-1,t> o},V(0,:1:',t)=1— 0,

V(n,a:',t) = —wo,

V(:v, 0) = o,

and let Wn be the bounded solution to

AW —W, = (W+wo)f(W) in {0 < :L'l< n, z' e RN-1,t> o},

W(0,:r’,t) = 1- 6,W(n,:r’,t) = 1- 6,

W(z,0) = 1- 0.

Let us point out that Vn and W" are actually functions of (1:1,t).

For k E N, let V,Ï(:c,t) = Vn(:r,t + k) and W/f(a:,t) = Wn(a:,t +k). Since Vf, U and W]: are bounded solutions to equation P0 in thedomain {0 < 11:1< n, I’ e RN", —k < t 5 7}, and on the parabolicboundary of this domain, we have Vf 5 U 5 ij. It followsthat

V:(a:,t) 5 U(:L‘,t) S l/V,Ï(:r,t)

in {0 < 1:1 < n,:1:’ e RN‘1,—k < t 5 7}. On the other hand (see[19]), Vn(x,t) —) W1C“) uniformly as t —>oo, where 11222 0 is asolution to (3.2.13) with a = 1 —6.

Analogously, Wn(1:, t) -> wflzl uniformly as t —)oo, where 1,!)120, symmetric with respect to :cl = g, is a solution to (3.2.12) witha = 1 — 0.

Therefore, letting k —)oo we get

112Ï(1:¡)5U(a:,t) 5 1/2311) for 0 5 115 n, t 5 7.

In particular,

(MMO) S Uz¡(0,z',t) S (w:)5(0), for t g 7.

Let 0 = 0. We deduce from Lemma 3.2.11, b) that

-IVU(0, 2’,t)| = Unwnv',t) 2 711330(wwe) = MW.

Let 6 > O. We deduce from Lemma 3.2.11, a) and b) that

— 2(B(1) —a) = ¿i330(1/040) s Uma-ct)

s muito) = - 2(B(1)— ).Therefore,

—|VU(0,a:’,t)| = UI,(0,x’,t) = —\/2(B(1)— a).

LEMMA 3.2.15. Let Ej, 75). and reí be sequences such that Ej > 0,EJ- —>0, 7€]. > 0, 75]. —+7, with 0 5 'y S +00, re}. > 0, re]. —) r with0 5 r 5 +oo, and such that r < +00 implies that 'y = +oo. Let p > 0and

p P2 pz

A5,.= {(2,19/ Intl< —, mmm-2) < t < “limar-2)}EJ- Ej Ej

Assume that 0 5 0 < 1 + wo(:co,to) and let ü‘i be weak solutions to- 2

Aüej_ = (üej+ w51(6j1Ï+35,53 tEj))¿j

in {1:1> h¿j(:c', t)} ñ Agj,

17,51:1- 0 on {2:1= h¿j(:v',t)} ñAej,

w‘i (61'21:+ 251.,¿ft + tEJ.)

Ej

where (zeptej) —)(xo,to), with ü‘í e C({a:1 2 h¿j(:v’,t)} {WA-ej),andVüei e L2. Here ha]. are continuous functions such that h¿¡(0,0) =0 with ha}. —) O uniformly on compact subsets of IR"-l x (-7,7).

Moreouer, we assume that Ilhejllcnm) + IIszhE¡I|C,_g(K) are uniformlybounded, for every compact set K C IRN’1x (-r, 7).

5 a‘i 51- a in {xl 2 ñsj(z’,t)}WT,»

Then, there ezists a function ü such that, for a subsequence,

ü e CZ+°'1+%({21 2 O, 'y > t > —r}),

ü‘j —)ü uniformly on compact subsets of {1:1> O, 'y > t > —T},

Aa —a, = (a + w0(:vo,to))f(ü) in {1:1> 0, 7 > t > —'r},

ü=1—0 on{:cl=0,'y>t>—r},—wo(:co,to)5 ü 51-0 in {9:1Z 0, 'y > t > —r}.

If 7 < +00, we require, in addition, that

llïlíj (mi, t + 7€,-“ 7)“C‘(K) + “VIII-25.5(II) t + 76,‘_ 7)“c°-%(K)

be uniformly boundedfor euery compact set K C RN’l x (-oo, 7]. Andwe deduce that

u e C2+°'1+%({zl 2 0, t g 7}).

Ifr < +00, we let

BE].= {as/ ¡2| < 5, 2:1> fight-75)},J

and we require, in addition, that for every R > 0,

III-¿51'(1;, —T¿J.)“Cn (Eejnfinmn S CR,

and that there ezists r > 0 such that

“üEJ'(I, _TEj)IICl+a(Etjn'l-jr(0)) S Cr.

Moreover, we assume that ||h¿j(:c’,t —r5}.+ r)||Cn(K) + IIV¡Ih¿j(:r’,t —

T51.+ r)||Ca,g(K) are uniformly bounded for every compact set K CIR”-1 ><[-r, +00).

Then, there holds that

ü G C°’%({:cl Z O, t 2 —r}), Vü e C({0 S 171< r, t 2 —r}),

¡EJ-(¿5,_TEJ_)—>ü(a;, —7') uniformly on compact subsets of {:cl > 0}.

In any case (7,7 be infinite or finite)

IVü€¡(0,0)| —)|Vü(0,0)|.

PROOF. We will drop the subscript j when referring to the sequences defined in the statement and e —>0 will mean j —>oo.

Case I. r = +00, 'y = +oo.

In order to prove the result, we first apply suitable changes of variables to straighten up the boundaries 1:1= h¿(1:’,t). Namely, for everye, we let

y = ¡15(2) t)where

Hï=111 —ïlE(I’,t), I'll-5:17h i> 1,and we define

o‘(y, t) = aim, t).

Let R > 0 be fixed and let

Bi;= / yl> OBR(010)and let -5-¿e ewhere

F6H€ .6

ÜH‘ J Www-ag"(3-2-16) aij(yat)= k axk ask,


Note that there exists CR > 0 such that

(3.2.17) IIasJ-Ilcosfi, s Cn, ubïump s CR.

Moreover, there exists /\ > 0 such that, if e is small enough,

(3.2.18) zaïj(y,t)€¿€j 2 Ala? for (m) e BT;M

Here we have used the fact that |(DHE)“| are uniformly bounded onany compact set, if e is small enough.

Then the function 17‘ E C(B_;;), with Vï)‘ e L2(BÉ) is a weaksolution to

E 5 6223 t=(¡,5+ me) Ba,ÜE=1—0 0nB—Én{3/I=O}’

Sj _ _ ,if e is small enough.

By Theorem 10.1, Chapter III in [21], there exists CR > 0 suchthat

IIÜEIICa.g(fi) S CROn the other hand, by Theorem 1.4.3 in [12] we also have that

IIVÜEIILoqï) S CR'

McEwer, by Theorem 1.4.10 in [12], the functions V17Eare continuousin BE with a modulus of continuity independent of e.

2

Therefore, there exists a function ü e C°'°ï(B—‘g)and a subsequence2

that we still call 17‘such that 175—)ü and Vü‘ —)Vü uniformly in2

Clearly,

ü in {yl =O}OEÉ,1- 0

—wo(.’170,t0)Sí S 1- 6 in B_É.2

Since ÏLE—>0 and Vzlïle —>0 uniformly on compact sets, it is easyto see that we actually have that

ü":—)ü, and Vüe -> V‘ú uniformly on compact sets of BE.2

Clearly ü is a solution of AH —m = (ü + w0(a:0,to))f(ü) in BE._ 2Standard Schauder estimates imply that ü e C2+°’"+°ï(BÉ).

4

Since Ïz¿(0,0) = 0, VIIÍÉ(O,O) —>0 and V175(0,0) —)Vü(0,0), it iseasy to see that Vü5(0,0) —)Vü(0, 0).

Since R is arbitrary, a standard procedure gives the result in

{y1>0,—T<t<'y} forT=+ooand7=+oo.

Case II. 7' < +oo.

As in the previous case, we ap}_)lysuitable changes of variables tostraighten up the boundaries 1:1= ¡15(1",t). Namely, for every e, we let

y = H5(:c,t), s =t+TE -7',

where

Hf(:l:,t) = 11:1-Ïz¿(1:’,t), HÏ(:I;,t) = Ii, i> 1,

and we define

Ü‘(y,s) = 'ü5(.1:,t).

Let R > O be fixed and let

BE],={(y,s)/y1> 0, s > —7'}ñ BR(0,0)

and let, as before,

8 817€ 82'25 327€’5 — — ‘f'. _——— 5 —

LU- (“11(11,3) + (yas) asawhere aïj(y,s) and bï(31,3)are defined in BE), in a way analogous to(3.2.16) and moreover, they satisfy cstimates similar to those in (3.2.17)and (3.2.18) in BET.

Then the function 'Ü‘ e C(B,‘;'T), with Vü‘ e L2(B,*¡"T)is a weaksolution to

w552+z,€2t+t Las=(27€+ f(17‘5)mBET,{16:1-6 onBÉ,,ñ{yi=0}a

5- . -2t t

_1_”;(ïr'l':e—wga‘51—0 inBïïpj

'ÜE= gE(y) in BET ñ {s = -T},

if e is small enough, where we have called g‘(y) = ¡75(11,—'r). In addition,

“gellcfimnmzon 5 CR and “95“01+°(Tm)n{ynzo})5 C"

Moreover, g‘ = 1 —0 on {yl = 0}.

By Theorem 10.1, Chapter III in [21], there exists CR > 0 suchthat

Ilüellcmfiï) 5 CR.On the other hand, by Remark 1.4.11 in [12], applied to the functions175= 17€—ge, we also have that

IIVÜ‘II s CRL°°((B_5(0)x[-r.%])n{y120})

andthat the functionsV175arecontinuousin x [-T, n {yl 20} with a modulus of continuity independent of e.

Proceeding as in the case 'r = +oo and using that TE—>T we see

that there exists a function ïí e C“'aï(BE T)such that for a subsequence2 I

17‘-—)ü uniformly in a,

Vï)‘ —>VÜ uniformly on compact sets of B‘éfi,

ü‘ —>17, Vü‘ —>Vü uniformly on compact sets of BÉ'T,

ü""(y,—7'E)—)ü(y, -7') uniformly on compact sets of 2

{y1> 0} n Bg(0),

V175—)Vü uniformlyin x [-T, ñ {yl2 0}.This function ü satisfies

ü e C2+°'l+aï({y1 2 0, t > -T} ñ 3%(0, 0)),

Aü —üt = (ü + wo(a:o,t0))f(ü) in {yl > 0, t > —r} n 8%(0, 0),

ü =1—0 on {yl = 0, t 2 —7'}03%(0,0),

—wo(zo,to)5 ü 51- 0 in {yl 2 0, t Z -T} ñ Bg(0,0).Moreover, there holds that Vü€(0, 0) —>Vü(0, 0).

Since R is arbitrary, Case II is proved.

Case III. 7 < +00.

We proceed as in the previous cases. For every 5, we let

y = HE(I,t), s =t_7e +7,

where

Hf(:z:,t) = srl —h¿(1:',t), Hf(a;, t) = asi, i> 1,

and we define

ÜE(y,s) = ü‘(:z:,t).

Let R > 0 be fixed and let

“BI-2,7={(y18)/yl > 015 < 7} nBR(0aO)'

As in the previous cases, by using Theorem 10.1, Chapter III in [21],and Theorems 1.4.3 and 1.4.10 in [12] we deduce that there exists a

function ü e C°"%(BE 7) such that for a subsequence2 ’

175—)ü, Vï)‘ -—)Vü uniformly in B“; 7,2,

ff —)ü, VüE —)VE uniformly on compact sets of BE 7.2’

This function ü satisfies

ü e C2+°’l+%({y¡2 0, t 5 7} n 3%(0,0)),

A’ü —Üt = (Ü + wo(Io,t0))f(Ü) in {yl > 0, t < 7’}ñ 3% (0,0),

ü=1—0 on {yl=0,t57}ñB%(0,0),—w0(a:0,to)S ü 51- 0 in {y} 2 0, t S 'y}ñ B;(0,0).

Moreover, there holds that Vü¿(0, 0) —>Vü(0,0).

Since R is arbitrary, the lemma is proved. III

3. Approximation results

In this section we prove that, under certain assumptions, a strictsemi-classical supersolution to problem (P) is the uniform limit of afamily of supersolutions to problem (PE) (Theorem 3.3.1), and we statean analogous result for subsolutions (Theorem 3.3.7). Also, we provethat for compactly supported initial data, limit solutions have boundedsupport (Proposition 3.3.8).

The following construction follows the lines of Theorem 5.2 in [22].In our case we have to be more careful with the construction of theinitial data.

THEOREM 3.3.1. Let ïi be a semi-classical supersolution to (P) inQT with 17e Cl({ïi > 0}) and such that {ü > 0} is bounded. Assume,

3. APPROXIMATION RESULTS 91

in addition, that there exist 60,so > 0 such that

|Vïí+| g v/2M(:c, t) —¿o on o n a{ïz > 0},

IVïZI>óo inQn{0<ü<So}.

Let w‘ be a solution of the heat equation in QT such that —)wo(a:,t) uniformly in G with wo e C(G) n L°°(QT) and uerifies(3.1.5).

Then, there ezists a familyue E C with Vue G LMG), a.of weak supeilolutions to (P5) in QT, such that, as e —) 0, u‘ —) uuniformly in QT.

PROOF. Step I. Construction of the family u‘. Let 0 < 0 < 61 besuch that

l 60

/l (3+ W)f(s)ds= í,—o

where W is a suitable uniform bound of IIwELoo({ü>o}). For everye > 0 small, we define the domain DE = {a < (1 - 0)e} C QT.

Let zE be the bounded solution to

Az‘ —zf = (z‘ + w5)f¿(z‘) in DE,

with boundary data

¿(La = (1 —0)e on 8D nt > 0,23(2) 1n DE ñ {t = 0}.

In order to give the initial data 23, we let uf (s, 1:) be the solution to(3.2.2) with

1-0 e

a=1—0’ (8+w(z,0))f(s)ds’wo:—w‘(a:,0)/e E e

Assume first that IVÏZIis smooth. Then we let

E _ e 1 — 9 — E

(p(EPT)_ 1p Unam)!and we define

E E 1"20(z) = Ego(Eu(:v,0),:c).

If a is not regular enough, we can replace |ViZ(a:,0)| by a smoothapproximation Fs(z) so that the initial datum 23 is 01+“. We leavethe details to the reader.

92 3. UNIQUENBSS OF LIMIT SOLUTIONS

Finally, we define the family uE as follows:

us: {a in {az (1 —0)5},25 in D5.

Step II. Passage to the limit. If (1,0) G Ñ, we have O S ¿171230)51 —0. Since, from Lemma 3.1, we know that —w5(z,0)/€ 5 M3413) 51 —0 for s 2 O, it follows that —wE(.7:,0)5 25(x,0) 5 (1 —0)s. Sincef¿(s) 2 0, constant functions larger than —w5(:c,t) are supersolutionsto (PE). Therefore, (1 —0).»:is a supersolution if e < el and we mayapply the comparison principle for bounded super and subsolutions of(PE) to conclude that —w5 5 zE 5 (1 —0)5.

Hence,sup Iu‘ —El = sup Izs —El 5 CeQ_-r D‘

and therefore, the convergence of the family ue follows.

Step III. Let us show that there exists 50 > 0 such that the functionsuE are supersolutions to (P5) for 5 < 50.

If uE > (1 —0)z-:,then ue = ïi, which by hypothesis is supercaloric.Since f¿(s) 2 0 and (1 —0)€ 2 —w‘ if e < 61, it follows that u‘ aresupersolutions to (PE) here.

If u‘ < (1 - 0)E, then we are in DE and therefore, by construction,u‘ are solutions to (PE).

That is, the us’s are continuous functions, and they are piecewisesupersolutions to (PE). In order to see that u‘ are globally supersolutions to (PE), it suffices to see that the jumps of the gradients (whichoccur at smooth surfaces), have the right sign.

To this effect, we will show that there exists 50 > 0 such that

(3.3.2) IVu‘I 2 2M(:r, t) —60/2 on {ïZ=(1— 0)s}, for e < 50.

Assume that (3.3.2) does not hold. Then, for every j e N, thereexist ej > O and (335).,t51.)e Q, with

Ej —)0 and (IEj,t¿j) —)(11:0,t0)G > n = 0},such that(3.3.3)

qu(ÏEj1tEj)= and IVqu(IEj=tEJ')I< 2M(I6j’t6j)_

From now on we will drop the subscript j when referring to thesequences defined above and e —)0 will mean j —)oo.

We can assume (performing a rotation in the space variables ifnecessary) that there exists a family gs of smooth functions such that,in a neighborhood of (3:5,te),

Üf=(1-9k}={üfih%y-%1=ng-x¿t—hfl,{ue < (1- 6)6} = {(93,t) /z1 —3:51> g¿(:r’ —ze',t —t¿)},

where there holds that

gE(O,O) = 0, |V1:gE(O, 0)| —) 0, e —>0.

We can assume that (3.3.4) holds in (Bp(a:e)x (te - pz, tE+p2)) n {0 5t 5 T} for some p > 0.

(3.3.4)

Let us now define1 1

ü‘(z, t) = -E—u‘(:¡:E+ 6:17,te + 62t), ge(z’,t) = Eg¿(6a:',e2t),and let

_ tE _ T —teTe — 6-2 a '76 — E2

We have, for a subsequence,

Te —) T a 75 _* 7

where 0 S 'r,'y 5 +00 and 7' and 'y cannot be both finite.We now let

A5= (cc,t) / |12|< e, —min(7'5, < t < minha .e 62 52

Then, the functions ü‘ are weak solutions toe t 2t

Aü‘—ü:= (¡f+—w(IE+6? E+6 )) f(ü‘)in {1:1> 55(2’,t)} 0.14€,

üe = 1- 0 on {21 = ¿76(1',t)} DAS,e t 2t _5üe51-9 in{El2¿Eu/mms,

Note that we are under the hypotheses of Lemma 3.2.15. Then,there exists a function ü such that, for a subsequence,

ü G C2+°’1+°ï({z¡ 2 0, -T < t < 'y}),

ü‘E—>ü uniformly on compact subsets of {21 > 0, —7'< t < 7},

A’Ü,—Ü; = (Ü + wo(170,t0))f(ü) in {.171> 0, -7' < t < 7},ü: —0 on{:c¡=0,—'r<t<'y},

-w0(Io,to)S'ÜSl-0 in {13120,-T<t<')'}.

We will divide the remainder of the proof into two cases, dependingon whether T = +oo or T < +oo.

Case I. Assume T = +oo.

In this case, Lemma 3.2.15 also gives

|Vü‘(0,0)| —>|Vü(0,0)|.

On the other hand, ü satisfies the hypotheses of Lemma 3.2.14 andtherefore,

IV’ÜIZ V 2M(I0,t0) - on {131= 0},

which yields2 V2A/I(I0=tO)_ 360/8)

for e small. But this gives

[Vu5(z¿,t¿)| 2 21l/Í(Zï5,t¿)—60/2,

for e small. This contradicts (3.3.3) and completes the proof in caseT = +00.

Case II. Assume T < +oo. (In this case 'y = +00.)

There holds that ü‘(:v, —TE)- lu°(z¿ + 613,0), then_ É

_ 1..(3.3.5) u5(a:, —Te)= <pE(Eu(:c¿ + 61,0),175 + 69:).

Here we want to apply the result of Lemma 3.2.15 correspondingto T < +oo. In fact, we can see that there exist C,T > 0 such that

lll-¿Eh_T€)llcl+a('fir(0)) S C

Now Lemma 3.2.15 gives, for a subsequence,

a e C°’°ï({1:1 2 0, t 2 —r}),

ü5(a:, —7'8)—>ü(1:, —T) uniformly on compact subsets of {1:1 > O}.

Therefore, we get that (recall that in the case we are considering to =0),

au, —T)= ¿(1- e —|Vü+(:co,t0)|11,10).

where¿(3,3) = ¡KLM z) and z/¡(s,a:)is the solutionof (3.2.2)_ IVÜÜJ, )|’

With

a = 1- 0, b = /_ — (s + wo(z,0))f(s) ds, wo= wo(:c,0).wo(:I:,0)

Thus,ü(a:, —r) = ib(zl,1:o).

Since the function 112051,10)is a stationary solution to equation(P0), bounded for 2:12 0, and ü = 1/)on the parabolic boundary of thedomain {1:1> O, t > —T}, we conclude that

ü(a=,t) = 1Mïïlfilïo) in {11:1Z 0, t Z -T}

It follows from Lemma 3.2.1 and the choice of 0 that, on {1:1= 0, t 2_T},

2 1-0

(mmm) = (3+wo(ïo,to))f(3)d3wo(:I-'o,to)

¿wz-42 = á

2 M(a:o,t0) - 68-0.

This is,

IVüI 2 2M(a:o,t0) —60/4 on {1:1= 0,t 2 —r}.

But Lemma 3.2.15 gives

|Vü‘(0,0)| —)|Vü(0,0)|,

which yields¡Vi-¡(CHW Z 2M(130,to) —350/8,

for e small. Then,

|Vu‘(z¿,t¿)l Z 2M(:z:¿,t¿) —60 2

for e small. This contradicts (3.3.3) and completes the proof in caser < +00. El

REMARK 3.3.6. Observe that from the construction of u": done inthe previous proof, it follows that

uEE ü in {Ü> (1- 0)e}.

We state without proof the followingTheorem.

THEOREM 3.3.7. Let ïi be a semi-classical subsolution to (P) inQT with 17e C1({ïí > 0}) such that {ii > 0} is bounded. Assume, inaddition, that there exist 60 > O such that

IVzT'IZ 2M(1:,t)+60 onQ08{ïí>0}.

Let wEbe a solution of the heat equation in QT such that —>wo(a:,t) unifomlyin Andassume,moreoverthatwoe ñL°°(QT) and verifies (3.1.5).

96 3. UNIQUENESS OF LI.\IIT SOLUTIONS

Then, there ezistsa familyu‘ G C withVu"re Liga), ofwecïsubsolutions to (PE) in QT, such that, as e —)0, u‘ —>ü uniformlyin QT.

PROOF. The proof is analogous to Theorem 3.3.1. See [22] for asimilar result in the case ws = 0. El

Finally, we end this Section by showing that, for compactly supported initial data, the support of a limit solution of problem (P) isbounded.

PROPOSITION 3.3.8. Let uo e CGR”) with compact support. Letuf, conoerge uniformly to U0 with supports conoerging to the supportof ug and let we be a solution of the heat equation in QT such that

w —>w0(x,t) uniformlyin And assume, moreouerthat wo GñL°°(QT) and verifies(3.1.5). Finalli, let u‘ bethe solutionto

(PE) with function we and initial condition ug.

Let u = lim uEJ'. Then {u > 0} is bounded. Moreover, u uanishesin finite time.

PROOF. Let —1< wo < w‘(x, t)/e. Then it is easy to check that(3.3.9)

l

M“,o=/_ (s +wo)f(s) ds < M(x,t) z/ (s + wo(a:,t))f(s)ds.“’° -wo(:1:,t)

Let us now consider the following self-similar function

V(:r,t;T) = (T _ t)l/2¡¿(I_,L.I(T_t)—1/2),

where h = h(r) is a solution of

h”+(N_1+%r)h/+%h=0, 0<r<R,T

(3-310) h’(0) = o, h(r) > 0, 0 5 r < R,

h(R) = o, h’(R) = —‘/2.x\/1w0.

It is proved in [13], Proposition 1.1, that there exists a unique R > 0and a unique h solution of (3.3.10).

Moreover, it can be checked that if one picks T sufficiently large,then

V(x,0; T) 2 uo +1 in {uo > 0},and so V(a:,t;T) is a strict semi-classical supersolution of (P) withbounded support and positive gradient near its free boundary.

4. UNIQUENESS OF THE LIMIT SOLUTION 97

Now, let uEJ'be solutions to (Pej) —with initial data uff convergingunifomly to uo such that support uff —)support uo —such that u =lim uei.

By Theorem 3.3.1, there exists a family v‘i of supersolutions of (Pci)such that u‘i —)V uniformly on compact sets, and u‘i (a),0) 2 u‘i (rc,0).Therefore, by the comparison principle, we obtain uei 5 vai and passingto the limit u(a:, t) 5 V(a:, t; T), and the result follows.

4. Uniqueness of the limit solution

In this section we arrive at the main point of the Chapter: we provethat, under certain assumptions, there exists a unique limit solution tothe initial and boundary value problem associated to (P) as long ascondition (0.2.3) is satisfied.

Let us begin with the following Proposition that is the key ingredient in the proof of our main result.

PROPOSITION 3.4.1. Let 17be a strict semi-classical supersolutionto (P) with bounded support in QT such that there exists so > 0 sothat IVÜI > 0 in {0 < ïi < so} and let wE/s be solutioE to the heatequation in QT converging to wo uniformly with wo G C (QT) ñL°°(QT)and uerifies (3.1.5).

Let uE be solutions to (P5) with function w‘ and initial conditionug, where uf, are uniform approximations of uo with support ug —)support uo. Then

lirn sup u‘(1:, t) 5 ïZ(1:,t)540+

for every (1:,t) e QT.

PROOF.Let ïZbe a strict semi-classicalsupersolution of Letus first, define the following regularization

u(z, t) = (Ü(:r:,t + h) —n)+,

for h,r] > 0 small. So that u is a strict semi-classical supersolutionof (P) with Cl free boundary, C‘({u > 0}) and IVuI > 60 > 0 in aneighborhood of its free boundary. So, by Theorem 3.3.1, there existsv‘ supersolution of (PE) such that v‘ —)u uniformly in QT_h.

Now, using the comparison principle, we conclude that uE 5 v‘ inQT_h, and the Proposition now follows letting first e —>0+ and thenh, 77—) 0+. El

Finally, we arrive at the main point of the paper: The uniquenessof limit solutions of (P).

THEOREM 3.4.2. Let the initial datum 11.0be Lipschitz, with compactsupport and satisfy the condition (3.1.3). Then there ezists at mostone limit solution such that its gradient does not vanish near its freeboundary as long as the function w”:in problem (PE) satisfies condition(0.2.4

More precisely, let uff , 113"be uniformly Lipschitz continuous in RNwith uniformly bounded Lipschitz norms and ej, 5k —)0. Assume that

uff e Cl({uáj > 0}), úák e Cl({u3* > O}), uÉj,üÏ)" —) U0 uniformlyand support uáj, support üá"—)support U0. Let wei/ej and ask/sk besolutionsi the heat equation converginguniformly to the same functionwo G C'(QT) ñ L°°(QT), that uerifies (3.1.5). Also, assume that wosatisfies the monotonicity condition (3.1.4

Let uei (resp. ü“) be the solution to (st) with function w‘i andinitial datum uff (resp. solution to (Pak) with function w“ and initialdatum uff). Let u = lim u‘J' and ü = lim ü“. If there ezists so > Osuch that IVüI > 0 in {0 < ü < so}.

Then, u 5 a.

PROOF. Since a is asemi-classical supcrsolution of(P), ü e C1({ü >O}) and, by Propositon 3.3.8, its support is bounded, the function ü,\as defined in (3.1.6) satisfies the hypothcses of Proposition 3.4.1 inQT/¿z D QT. So by letting /\ —)1- we arrive at

(3.4.3) u(:c, t) 5 ü(.1:,t).

This finishes the proof. Ü

THEOREM 3.4.4. Let the initial datum uo be as in Theorem 3.4.2.Assume that there ezists a semi-classical solution u to (P) with initialdatum uo and let uÉj be uniformly Lipschitz continuous in RN with

e,- —+ 0, such that uáj e C‘({u3’ > 0}), uff —->uo uniformly andsupport u8’—)support uo. Assume wei/e]- is a solution of the heatequation converging to wo uniformly with wo e C(G) ñ L°°(QT) andverifying (3.1.5 Also, assume that wo satisfies the monotonicity condition (3.1.4).

Let uEJ'be the solution to (Pai) with function w‘i and initial datumugj and let u = lirn u‘i. Then, u = v.

In particular, there ezists at most one classical solution to (P

5. CONCLUSIONS 99

PROOF. Since u is a semi-classical supersolution to (P) and v is asemi-classical subsolution, Lemma 2.1 applies and we get that v 5 u.

On the other hand, if we define v,\ as in (3.1.6), with 0 < /\ < /\’ < 1,we have that v,\ satisfies the hypotheses of Proposition 3.4.1. Thus,there exists a family v‘i of supersolutions to (st) with function w‘isuch that, for a subsequence, vai —) v with initial data converginguniformly to uo. So by the comparison principle

u = lim us!"S lim v‘j = v.

This finishes the proof. El

5. Conclusions

We have proved that the limits of sequences of solutions to (PE)with different constitutive functions ws and initial data uf, coincide —as long as certain monotonicity assumptions are made —if the limit ofwE/e and of uf, are prescribed.

The monotonicity assumptions are necessary to provide strict semiclassical supersolutions as close as we want to any semi-classical supersolution. This kind of condition was also used with the same purpose—in the case in which we = 0 —in [30, 22]. In the latter, a differentgeometry was considered namely, the domain was a cylinder, Neumannboundary conditions were given on the boundary of the cylinder andmonotonicity in the direction of the cylinder axis was assumed. In [22]it was proved that, if a classical solution exists and w‘ = 0, then it isequal to any limit of solutions to (PE).

In our case, this is with w":7€O satisfying (0.2.4) and nondecreasing in the direction of the cylinder axis, the uniqueness result in thepresence of a classical solution still holds.

The cylindrical geometry has the advantage of giving the condition of nonvanishing gradient in the positivity set of any limit solution. Since in dimension 2 one can prove that limit solutions are semiclassical supersolutions up to the fixed boundary, the uniqueness oflimit solutions follows in this case without further assumptions.

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un problema de frontera libre en teoría de combustión · teoría de combustión fernández...

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