EREVANI PETAKAN HAMALSARAN

Daryu� Qalvand

NEYMANI XNDIR� �ORRORD KARGI VERASERVO�

DIFERENCIAL-�PERATORAYIN HAVASARUMNERI HAMAR

A.01.02 { \Diferencial havasarumner" masnagitu�yamb

fizikama�ematikakan gitu�yunneri �ekna�ui gitakan

asti�ani haycman atenaxosu�yan

SE�MAGIR

EREVAN 2012

YEREVAN STATE UNIVERSITY

Daryush Kalvand

NEUMANN PROBLEM FOR DEGENERATEDIFFERENTIAL-OPERATOR EQUATIONS OF

FOURTH ORDER

SYNOPSIS

of dissertation for the degree of candidate of physical and

mathematical sciences specializing in

A.01.02 � �Di�erential equations�

Yerevan 2012

Atenaxosu�yan �eman hastatvel � EPH ma�ematikayi mexanikayi

fakulteti xorhrdi ko�mic:

Gitakan �ekavar` fiz-ma�. git. �ekna�u

L.P. Te�oyan

Pa�tonakan �nddimaxosner` fiz-ma�. git. doktor

A.H. Hovhannisyan

fiz-ma�. git. �ekna�u

A.H. Qamalyan

A�ajatar kazmakerpu�yun` X. Abovyani anvan Haykakan petakan

mankavar�akan hamalsaran

Pa�tpanu�yun� kkayana 2012�. noyember 27-in �. 1500-in Er ani petakan

hamalsaranum gor�o� BOH-i 050 masnagitakan xorhrdi nistum (0025, Er an, Aleq

Manukyan 1):

Atenaxosu�yan� kareli � �ano�anal EPH-i gradaranum:

Se�magirn a�aqvel � 2012�. hoktemberi 26-in:

Masnagitakan xorhrdi gitakan qartu�ar T.N. Haru�yunyan

Dissertation topic was approved at a meeting of academic council of the facultyof Mathematics and Mechanics of the Yerevan State University.

Supervisor: candidate of physical and mathematical

sciencesL.P. Tepoyan

O�cial opponents: doctor of physical and mathematical

sciencesA.H. Hovhannisyan

candidate of physical and mathematicalsciencesA.H. Kamalyan

Leading organization: Armenian State Pedagogical University

named after Khachatur Abovian

Defense of the thesis will be held at the meeting of the a specialized council 050 of HAC

of Armenia at Yerevan State University on November 27, 2012 at 1500 (0025, Yerevan,

A.Manoogian str. 1).

The thesis can be found in the library of the YSU.

Synopsis was sent on October 26, 2012.

Scienti�c secretary of specialized council T.N. Harutyunyan

General characteristics of the work

Relevance of the theme.

The study of di�erential equations whose coe�cients are unboundedoperators in Hilbert or Banach spaces, suitable not only because they containmany partial di�erential equations, but also because we are able to look froma uni�ed point of view of both on the ordinary di�erential operators and thepartial di�erential operators.

In the dissertation we consider the Neumann problem for degeneratedi�erential-operator equations of fourth order

Lu ≡ (tαu′′)′′ +Au = f, (1)

where t ∈ [0, b], 0 ≤ α ≤ 4, f ∈ L2((0, b),H), the operator A has complete systemof eigenfunctions {ϕk}k∈N, which form a Riesz basis in H, i.e. any x ∈ H hasthe representation

x =

∞∑k=1

xk(t)ϕk

and there are some positive constants c1 and c2 such that

c1

∞∑k=1

|xk|2 ≤ ‖x‖2 ≤ c2∞∑k=1

|xk|2.

The most important class of operator equations (1) are degenerate ellipticfourth-order equations. Degenerate elliptic equations encountered in solving ofmany important problems of applied character, such as the theory of smalldeformations surfaces of rotation, the membrane theory of shells, the bendingof plates of variable thickness with a sharp edge (see, for instance G. Jaiani[8]). Particularly, important are these equations in the gas dynamics. Startof numerous studies put the work of F. Tricomi [27], devoted to the second-order equation with non-characteristic degeneration. Fundamental role in thetheory of degenerate elliptic equations was played the article of M.V. Keldysh[9], where was �rst studied the �rst boundary value problem for the second-order elliptic equation with characteristic degeneration (see also the article ofG. Jaiani [7]). The next stage was the work of Bicadze [1], where was �rstformulated a weight problem. By G. Fichera [5] was created a uni�ed theoryof second-order equations with nonnegative characteristic form. S.G. Mikhlin[15], L.D. Kudryavtsev [11], [12] and others have investigated degenerate ellipticequations (both second and higher order) by variational methods. Fourth orderelliptic equations degenerating on the boundary of domain, for which we cannot apply variational methods, were �rst considered by V.K. Zakharov [30], [31].He extended the results of M.I. Vishik [28] on the fourth-order equation on theplane, provided that the coe�cient of the third-order derivative with respectto the y ful�ll to some condition near the line of degeneracy y = 0. It turnedout that for the fourth-order degenerate equation also the �lower terms� havein�uence for the statement of the boundary value problem. A similar fact has

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been studied by other methods by Narchaev [17], [18]. E.V. Makhover in [13], [14]obtained the conditions for the discreteness and non-discreteness of the spectrumfor the degenerate elliptic operator of fourth order (see also the book written byK. Mynbayev, M. Otelbaev [16]). Degenerate di�erential equations in abstractspaces have been studied by V.P. Glushko and S.G. Krein in [6], by A.A. Dezin[2], by V.V. Kornienko [10] and other authors.

This work is a direct continuation of [2], [22], [25] and is based on the studyof one-dimensional di�erential equation (1), i.e. in the case where the operatorA is the multiplication operator the number a ∈ C. Note that this approachwere applied in the book of A.A. Dezin [3], by V.K. Romanko [20], etc.. Thisapproach makes it possible complete to study a number of phenomena, whichwere not fully explored. At the same time it is easy to trace the connectionbetween ordinary di�erential equations and operator equations.

It is worth to note that the operator A : H→ H in general is an unboundedoperator.

Note that the operator equation (1) contains, in addition to ellipticequations, wide class of degenerate partial di�erential equations both classicaland nonclassical types.

We are interested in the nature of the boundary conditions with respect tot (at t = 0, b) being connected to the equation (1) and ensuring a unique solutionfor any right-hand sides f ∈ L2((0, b),H).

The aim of the thesis:

• to obtain estimates for the elements of the weighted Sobolev spaceW 2α(0, b)

near the point t = 0

• to prove that the one-dimensional operatorBu ≡ (tαu′′)′′+u is positive andself-adjoint as well as that the inverse operator B−1 : L2(0, b) → L2(0, b)is bounded for 0 ≤ α ≤ 4 and is for 0 ≤ α < 4 compact

• to give a description of the domain of de�nition of the operator Lu ≡(tαu′′)′′ + au depending on the order of degeneration α

• to give su�cient conditions for the uniquely solvability of the Neumannproblem for the di�erential-operator equation

• to give the description of the spectrum of the operator L when the operatorA is self-adjoint

Scienti�c innovation. All results are new.Practical and theoretical value. The results of the work have

theoretical character. The results of the thesis can be used in the study of theNeumann problem for the degenerate elliptic equations.

Approbation of the results. The obtained results were presented

• at the research seminar of the chair of Di�erential equations of the YerevanState University

4

• at the International Conference on Di�erential Equations and DynamicalSystems, 11-13 July, Iran, 2012

• at the 43th Annual Iranian Mathematics Conference, University of Tabriz,27-30 August, Iran, 2012.

The main results of the thesis

The thesis consists of an introduction, three chapters and a bibliography.Chapter 1 consists of two sections and is devoted to the one-dimensional

case.In Section 1.1 we de�ne the weighted Sobolev space W 2

α(0, b), α ≥ 0 as thecompletion of C2[0, b] in the norm

‖u‖2W2α(0,b) =

∫ b

0

tα |u′′(t)|2 dt, (2)

where C2[0, b] is the set of twice continuously di�erentiable functions u(t) de�nedon [0, b] and satisfying the conditions

u(0) = u′(0) = u(b) = u′(b) = 0. (3)

Proposition 1.1. For every u ∈ W 2α(0, b) close to t = 0 we have following

estimates

|u(t)|2 ≤ C1t3−α‖u‖2W2

α(0,b), for α 6= 1, 3,

|u′(t)|2 ≤ C2t1−α‖u‖2W2

α(0,b), for α 6= 1.

(4)

For α = 3 the factor t3−α should be replaced by | ln t|; for α = 1 the factor t1−α

by | ln t| and the factor t3−α by t2| ln t|.

Proposition 1.2. For every 0 ≤ α ≤ 4 we have a continuous embedding

W 2α(0, b) ↪→ L2(0, b), (5)

which for 0 ≤ α < 4 is compact.Let ψh(t) ≡ 0 for 0 ≤ t ≤ h and

ψh(t) =

{h−3(t− h)2(5h− 2t), h < t ≤ 2h,1, 2h < t ≤ b.

De�ne uh(t) = u(t)ψh(t). Clearly, the function uh(t) belongs to the spaceW 2α(0, b).

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Proposition 1.3. For every function u ∈ W 2α(0, b) and α 6= 1, α 6= 3 the

norm ‖uh − u‖W2α(0,b) tends to zero by h→ 0.

Denote by W 2α(0, b) the completion of C2[0, b] in the norm

‖u‖2W2α(0,b) =

∫ b

0

(tα |u′′(t)|2 + |u(t)|2

)dt. (6)

Proposition 1.4 For every 0 ≤ α ≤ 4 we have the embedding

W 2α(0, b) ⊂ L2(0, b). (7)

Proposition 1.5. For every u ∈W 2α(0, b) we have

|u(t)|2 ≤ (C1 + C2t3−α)‖u‖2W2

α(0,b), for α 6= 1, 3,

|u′(t)|2 ≤ (C3 + C4t1−α)‖u‖2W2

α(0,b), for α 6= 1.

(8)

For α = 3 the factor t3−α should be replaced by | ln t|; for α = 1 the factor t1−α

by | ln t| and the factor t3−α by t2| ln t|.

Proposition 1.6. For every 0 ≤ α ≤ 4 we have a continuous embedding

W 2α(0, b) ↪→ L2(0, b), (9)

which for 0 ≤ α < 4 is compact (see [22]).In Section 1.2 we consider the Neumann problem for the one-dimensional

equation (1) inLu ≡ (tαu′′)′′ + au = f, (10)

where t ∈ (0, b), 0 ≤ α ≤ 4, f ∈ L2(0, b) and a = const.

De�nition 1.7. The function u ∈W 2α(0, b) is called a generalized solution

of the Neumann problem for the equation (10) if for every v ∈W 2α(0, b) we have

(see [21])(tαu′′, v′′) + a(u, v) = (f, v). (11)

First we consider the following particular case of the equation (10) for a = 1

Bu ≡ (tαu′′)′′ + u = f, f ∈ L2(0, b). (12)

Proposition 1.8. The generalized solution of the Neumann problem forthe equation (12) in the space W 2

α(0) exists and is unique for every f ∈ L2(0, b).

De�nition 1.9. We say that the function u ∈ W 2α(0, b) belongs to the

domain of de�nition D(B) of the operator B, if there exists a function f ∈L2(0, b), such that for every v ∈ W 2

α(0, b) and a = 1 holds the equality (11). Inthis case we write Bu = f .

6

As a result we get an operator B : L2(0, b)→ L2(0, b) in L2(0, b) with densedomain of de�nition D(B) ⊂W 2

α(0, b).

Theorem 1.10. The operator B : L2(0, b) → L2(0, b) is for 0 ≤ α ≤ 4positive and self-adjoint. The inverse operator B−1 : L2(0, b) → L2(0, b) existsand is bounded for 0 ≤ α ≤ 4. Moreover, the inverse operator is for 0 ≤ α < 4is compact.

Consequently, the operator B : L2(0, b) → L2(0, b) for 0 ≤ α < 4 has adiscrete spectrum, and its eigenfunction system is complete in L2(0, b). Observethat for α = 4 the spectrum of the operator B is non-discrete.

Consider special case of the equation (10) for a = 0

(tαu′′)′′ = f. (13)

For the solvability of the equation (13) we get the following result.Proposition 1.11. The generalized solution of the Neumann problem for

the equation (13) exists if and only if (f, P1(t)) = 0 for any polynomial P1(t) oforder 1.

Observe also that the generalized solution of the Neumann problem for theequation (13) is unique up to an arbitrary additive polynomial of order 1.

Theorem 1.12. The domain of de�nition D(L) of the operator L consistsof the functions u(t) for which u(0) is �nite when 0 ≤ α < 7

2and u′(0) is �nite

for 0 ≤ α < 2. The values u(0) and u′(0) can not be speci�ed arbitrarily, but aredetermined by the right-hand side of the equation(10).

In Chapter 2, which consists from two sections, we concentrate on operatorequations.

In Section 2.1 we investigate so-called Π-operators. Let

L(−iD)u ≡∑|α|≤m

aαDαu

be the di�erential operation with constant coe�cients de�ned on the set P∞

of smooth functions in V = (0, 2π)n that are periodic in each variable, whereα = (α1, α2, . . . , αn) is a multi-index and

Dα = Dα11 Dα2

2 . . . Dαnn , Dk ≡

1

i

∂

∂xk, |α| = α1+α2+. . .+αn, k = 1, 2, . . . , n.

Let s ∈ Zn. To every di�erential operation L(−iD) we can associate a polynomialA(s) with constant coe�cients such that

A(−iD)eis·x = A(s)eis·x, s · x = s1x1 + s2x2 + . . .+ snxn.

We de�ne the corresponding operator A : L2(V ) → L2(V ) to be the closure inL2(V ) of the di�erential operation A(−iD) �rst de�ned on P∞.

Let S = Zn. The set of exponentials {eis·x, s ∈ S} forms an orthogonal basisin L2(V ). Observe that eis·x, s ∈ S are the eigenfunctions for each Π-operator A

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corresponding to the eigenvalues A(s), s ∈ S. Denote the set of eigenvalues by

{A(s), s ∈ S} def= A(S).Proposition 2.1. The spectrum of each Π-operator A : L2(V )→ L2(V ) is

the closure A(S) in the complex plane C of the set A(S), which forms the pointspectrum σp(A). The set σc(A) = σ(A)\σp(A) is the continuous spectrum of theoperator A.

In Section 2.2 we consider the operator equation (1).Since the system of the eigenfunctions {ϕk}∞k=1 forms a Riesz basis in H

we can write

u(t) =

∞∑k=1

uk(t)ϕk, f(t) =

∞∑k=1

fk(t)ϕk. (14)

Therefore, the operator equation (1) can be decomposed into an in�nite chainof degenerate ordinary di�erential equations (see [2], [3])

Lkuk ≡ (tαu′′k)′′ + akuk = fk, k ∈ N, (15)

where fk ∈ L2(0, b), k ∈ N. Let D(Lk), k ∈ N denote the domains of de�nitionfor the one-dimensional operators Lk, k ∈ N. First we de�ne the operator L onthe �nite sums

um =∑

uk(t)ϕk, uk ∈ D(Lk), Lkuk = fk, (16)

by the equality

Lum ≡∑

Lkuk(t)ϕk =∑

fk(t)ϕk ≡ fm.

De�nition 2.2. The function u ∈ L2((0, b),H) is called the generalizedsolution of the equation (1), if there is some sequence of �nite sums (16), suchthat are valid the equalities

limm→∞

‖u− um‖L2((0,b),H) = 0, limm→∞

‖f − fm‖L2((0,b),H) = 0.

Theorem 2.3. The operator equation (1) is uniquely solvable for everyf ∈ L2((0, b),H) if and only if the equations (15) are uniquely solvable for everyfk ∈ L2(0, b) and uniformly with respect to k ∈ N we have

‖uk‖L2(0,b) ≤ c‖fk‖L2(0,b). (17)

It follows from the inequalities (17) that for 0 ≤ α ≤ 4 the inverse operatorL−1 is bounded. In contrast to the one-dimensional case (see Theorem 1.10) theoperator L−1 for 0 ≤ α < 4 will be compact only in the case, when the space H

is �nite-dimensional.

Theorem 2.4. Let holds the conditions

ρ(1− ak;σ(B)) > ε, k ∈ N, (18)

8

where ε > 0, ρ is the distance in the complex plane C. Then the generalizedsolution of the equation (1) exists and is unique.

Note that Theorem 2.4 describes the resolvent set ρ(L) of the operator Land thus also the spectrum.

When the operator A is self-adjoint we can give the following descriptionof the spectrum of the operator L.

Theorem 2.5. The spectrum σ(L) of the operator L coincides with theclosure of the direct sum σ(B) and σ(A− IH), i.e.

σ(L) = σ(B) + σ(A− IH) ≡ {λ1 + λ2 − 1 : λ1 ∈ σ(B), λ2 ∈ σ(A)}.

In Chapter 3 we investigate numerical solution of the one-dimensionalequation (10) by quintic and non-polynomial as well as cubic spline functions.Let 4 = {0 = t0 < t1 < ... < tn = b} be a partition of the interval [0, b] andS4 ∈ Ck−1[0, b] is a piecewise polynomial function with the continuous (k − 1)-th derivative. We approximate the solution of the equation (10) by S4(t) andestimate the norm of the di�erence f − S4. In order to do this we represent thedi�erence in the form

‖f − S4‖2L2(0,b) =

∫ b

a

|f ′′(t)− S′′4(t)|2dt =

= ‖f‖2L2(0,b) − 2

∫ b

a

(f ′′(t)− S′′4(t)

)S′′4(t) dt− ‖S4‖2L2(0,b).

A quintic spline functions Si(t), i = 1, . . . , N interpolating a function u(t) oneach subinterval [ti, ti+1] is a polynomial of, at most, degree �ve. The splinefunctions Si(t) for t ∈ [ti, ti+1] have the form

Si(t) =

5∑k=0

a(i, k)(t− ti)k,

where the coe�cients a(i, k), k = 0, 1, . . . , 5 are constants to be determined.We consider also the rate of the convergence.

References

1. A.V. Bicadze, Equations of mixed type. M.: Izd. AN SSSR, 1959.

2. A.A. Dezin, Degenerate operator equations, Math. USSR Sbornik, vol.43(3), 1982, pp. 287-298.

3. A.A. Dezin, Partial Di�erential Equations (An Introduction to a GeneralTheory of Linear Boundary Value Problems), Springer, 1987.

4. P.A. Djarov, Compactness of embeddings in some spaces with power weight,Izvestya VUZ-ov, matematika, vol. 8, 1988, pp. 82-85 (Russian).

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5. G. Fichera, On a uni�ed theory of boundary value problems for elliptic-parabolic equations of second order, Boundary Problems of Di�erentialEquations, The Univ. of Wisconsin Press, 1960, pp. 97-120.

6. V.P. Glushko, S.G. Krein, On degenerate linear di�erential equations inBanach space, DAN SSSR, 1968, vol. 181, no. 4, pp. 784-787.

7. G. Jaiani, On a generalization of the Keldysh theorem, GeorgianMathematical Journal, vol. 2, no. 3, 1995, pp. 291-297.

8. G. Jaiani, Theory of cusped Euler-Bernoulli beams and Kircho�-Loveplates, Lecture Notes of TICMI, vol. 3, 2002.

9. M.V. Keldysh, On certain cases of degeneration of equations of elliptictype on the boundary of a domain, Dokl. Akad. Nauk. SSSR, 77, 1951, pp.181-183 (Russian).

10. V.V. Kornienko, On the spectrum of degenerate operator equations,Mathematical Notes, vol. 68(5), 2000, pp. 576-587.

11. L.D. Kudryavtzev, On a variational method of determination of generalizedsolution of di�erential equations in the function spaces with power weight,Di�er. Urav., vol. 19(10), 1983, pp. 1723-1740 (Russian).

12. L.D. Kudryavtzev, On equivalent norms in the weight spaces, Trudy Mat.Inst. AN SSSR, vol. 170, 1984, pp. 161-190 (Russian).

13. E.V. Makhover, Bending of a plate of variable thickness with a cusped edge.Scienti�c Notes of Leningrad State Ped. Institute, vol. 17, no. 2, 1957, pp.28-39 (Russian).

14. E.V. Makhover, On the spectrum of the fundamental frequency, Scienti�cNotes of Leningrad A.I. Hertzen State Ped. Institute, vol. 197, 1958, pp.113-118 (Russian).

15. S.G. Mikhlin, Degenerate elliptic equations, Vestnik LGU, vol. 3, no. 8,1954, pp. 19-48 (Russian).

16. K. Mynbayev, M. Otelbaev, Weighted Functional Spaces and Spectrum ofDi�erential Operator, Nauka, 1988.

17. A. Narchaev, First boundary value problem for the elliptic equationsdegenerating on the boundary of the domain, Dokl. AN SSSR, 1964, vol.156, no. 1, pp. 28-31.

18. A. Narchaev, On degenerate elliptic equation, Izvestia AN Turk. SSR, ser.phyz.-techn., chem. and geol. sciences, 1966, no. 2, pp. 3-7.

19. E. Poulsen, Boundary values in function spaces, Math. Scand., vol. 10,1962, pp. 45-52.

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20. V.K. Romanko, On the theory of the operators of the form dm

dtm− A,

Di�erential Equations, 1967, vol. 3, no. 11, pp. 1957-1970 (Russian).

21. R.E. Showalter, Hilbert Space Methods for Partial Di�erential Equations,Electronic Journal of Di�erential Equations, Monograph 01, 1994.

22. L. Tepoyan, Degenerate fourth-order di�eential-operator equations, Di�er.Urav., vol. 23(8), 1987, pp. 1366-1376, (Russian); English Transl. in Amer.Math. Soc., no. 8, 1988.

23. L. Tepoyan, On a degenerate di�erential-operator equation of higher order,Izvestiya Natsionalnoi Akademii Nauk Armenii. Matematika, vol. 34(5),1999, pp. 48-56.

24. L. Tepoyan, On the spectrum of a degenerate operator, IzvestiyaNatsionalnoi Akademii Nauk Armenii. Matematika, vol. 38, no. 5, 2003,pp. 53-57.

25. L. Tepoyan, The Neumann problem for a degenerate di�erential-operatorequation, Bulletin of TICMI (Tbilisi International Centre of Mathematicsand Informatics), vol. 14, 2010, pp. 1-9.

26. L. Tepoyan, Degenerate di�erential-operator equations of higher order andarbitrary weight, Asian-European Journal of Mathematics, vol. 05, no. 02,2012, pp. 1250030-1 - 1250030-8.

27. F. Tricomi, On linear partial di�erential equations of second orderequations of mixed type, M., Gostexizdat, 1947 (Russian).

28. M.I. Vi�sik, Boundary-value problems for elliptic equations degenerate onthe boundary of a region, Mat. Sb., 35(77), 1954, pp. 513-568; Englishtransl. in Amer. Math. Soc. Transl., vol. 35, no. 2, 1964.

29. J. Weidmann, Lineare Operatoren in Hilbertr�aumen, Teil 1, Grundlagen,Teubner Verlag, Stuttgart, 2000.

30. V.K. Zakharov, Embedding theorems for spaces with a metric, degeneratein a straightline part of the boundary, Dokl. AN SSSR, vol. 114, no. 3, 1957,pp. 468-471.

31. V.K. Zakharov, The �rst boundary value problem for elliptic equationsfourth order degenerate on the boundary, Dokl. AN SSSR, vol. 114, no.4, 1957, pp. 694-697.

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List of publications of the author

1. Daryoush Kalvand, L. Tepoyan, Neumann problem for the fourth orderdegenerate ordinary di�erential equation, Proceedings of the Yerevan StateUniversity, Physical and Mathematical Sciences, no. 1, 2010, pp. 22-26.

2. Daryoush Kalvand, Neumann problem for the degenerate di�erential-operator equations of the fourth order, Vestnik RAU, Physical-Mathematical and Natural Sciences, no. 2, 2010, pp. 34-41 (Russian).

3. Daryoush Kalvand, L. Tepoyan, J. Rashidinia, Existence and uniquenessof the fourth order boundary value problem and Quintic Spline solution,Proceeding of 9th Seminar on Di�erential Equations and DynamicalSystems, 11-13 July, Iran, 2012, pp. 137-140.

4. Daryush Kalvand, Esmaeil Youse�, L. Tepoyan, Numerical solution offourth order ordinary di�erential equation by Quintic Spline in theNeumann problem, The 43th Annual Iranian Mathematics Conference,University of Tabriz, 27-30 August, Iran, 2012.

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ԴԱՐՅՈՒՇ ՔԱԼՎԱՆԴ

Նեյմանի խնդիրը չորրորդ կարգի վերասերվող դիֆերենցիալ-օպերատորային հավասարումների համար

Ատենախոսությունը նվիրված է Նեյմանի խնդրի ուսումնասիրու-թյանը չորրորդ կարգի վերասերվող դիֆերենցիալ-օպերատորային հավասարումների մի դասի համար: Այդ դասը իր մեջ ներառում է ինչպես դասական, այնպես էլ ոչ դասական մասնակի ածանցյալներով դիֆերենցիալ հավասարումներ: Միաժամանակ հնարավորություն է ստեղծվում նայել սովորական և մասնակի ածանցյալներով դիֆերենցիալ հավասարումներին միասնական տեսանկյունից: Ներկայացվող ատենախոսության մեջ դիտարկվել է Նեյմանի խնդի-րը չորրորդ կարգի վերասերվող դիֆերենցիալ-օպերատորային հավասարման համար

𝐿𝐿𝐿𝐿 ≡ (𝑡𝑡𝛼𝛼𝐿𝐿′′ )′′ + 𝐴𝐴𝐿𝐿 = 𝑓𝑓(𝑡𝑡), 𝑡𝑡 ∈ (0, 𝑏𝑏),𝛼𝛼 ≥ 0,𝑓𝑓 ∈ 𝐿𝐿2((0, 𝑏𝑏),ℋ), (1)

որտեղ 𝐴𝐴:ℋ → ℋ գծային օպերատորը (ընդհանրապես ասած` անսահմանափակ) գործում է որևէ ℋ սեպարաբել հիլբերտյան տարածության մեջ: Ենթադրվում է, որ ℋ-ում գոյություն ունի Րիսի բազիս` {𝜑𝜑𝑘𝑘}𝑘𝑘=1

∞ , այնպիսին որ 𝜑𝜑𝑘𝑘 ,𝑘𝑘 ∈ ℕ ֆունկցիաները հանդիսանում են 𝐴𝐴 օպերատորի համար սեփական ֆունկցիաներ 𝐴𝐴𝜑𝜑𝑘𝑘 = 𝑎𝑎𝑘𝑘𝜑𝜑𝑘𝑘 ,𝑘𝑘 ∈ ℕ: Ատենախոսության մեջ նախ դիտարկվում է միաչափ դեպքը, այսինքն երբ 𝐴𝐴 օպերատորը 𝑎𝑎 թվով բազմապատկման օպերատոր է` 𝐴𝐴𝐿𝐿 =𝑎𝑎𝐿𝐿, 𝑎𝑎 ∈ ℂ: Այնուհետև օգտվելով

𝐿𝐿(𝑡𝑡) = ∑ 𝐿𝐿𝑘𝑘(𝑡𝑡)𝜑𝜑𝑘𝑘∞𝑘𝑘=1 , 𝑓𝑓(𝑡𝑡) = ∑ 𝑓𝑓𝑘𝑘(𝑡𝑡)𝜑𝜑𝑘𝑘∞

𝑘𝑘=1 ներկայացումներից (1) դիֆերենցիալ-օպերատորային հավասարումը բերվում է սովորական դիֆերենցիալ հավասարումների հետևյալ անվերջ շղթային 𝐿𝐿𝑘𝑘𝐿𝐿𝑘𝑘 ≡ (𝑡𝑡𝛼𝛼𝐿𝐿𝑘𝑘′′ )′′ + 𝑎𝑎𝑘𝑘𝐿𝐿𝑘𝑘 = 𝑓𝑓𝑘𝑘(𝑡𝑡), 𝑡𝑡 ∈ (0, 𝑏𝑏),𝛼𝛼 ≥ 0,𝑓𝑓𝑘𝑘 ∈ 𝐿𝐿2(0,𝑏𝑏),𝑘𝑘 ∈ ℕ: (2) Սկզբում սահմանվում է (2) տեսքի սովորական դիֆերենցիալ հավա-սարման համար Նեյմանի խնդրի ընդհանրացված լուծումը Սոբոլևի 𝑊𝑊𝛼𝛼

2(0, 𝑏𝑏) կշռային տարածության մեջ: Այնուհետև սահմանվում է (1) դիֆերենցիալ-օպերատորային հավասարման համար Նեյմանի ընդհանրացված խնդրի լուծումը:

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Ստացվել են հետևյալ հիմնական արդյունքները.

● Սոբոլևի կշռային 𝑊𝑊𝛼𝛼2(0, 𝑏𝑏) տարածության տարրերի և նրանց ածանց-

յալների համար ստացվել են գնահատականներ 𝑡𝑡 = 0 կետի շրջակայ-քում, ինչպես նաև ապացուցվել են ներդրման և կոմպակտության թեորեմներ: ● Ցույց է տրվել, որ միաչափ 𝐵𝐵𝐿𝐿 ≡ (𝑡𝑡𝛼𝛼𝐿𝐿′′ )′′ + 𝐿𝐿 օպերատորը 𝐿𝐿2(0,𝑏𝑏) տարածության մեջ դրական է և ինքնահամալուծ, երբ 0 ≤ 𝛼𝛼 ≤ 4, իսկ հակադարձ 𝐵𝐵−1: 𝐿𝐿2(0, 𝑏𝑏) → 𝐿𝐿2(0,𝑏𝑏) օպերատորը գոյություն ունի ու անընդհատ է, երբ 0 ≤ 𝛼𝛼 ≤ 4 : 0 ≤ 𝛼𝛼 < 4 դեպքում ապացուցվել է հակադարձ օպերատորի կոմպակտությունը: ● Տրվել է միաչափ 𝐿𝐿𝐿𝐿 ≡ (𝑡𝑡𝛼𝛼𝐿𝐿′′ )′′ + 𝑎𝑎𝐿𝐿 օպերատորի որոշման տիրույթի նկարագիրը կախված վերասերման 𝛼𝛼 ցուցիչից: ● Ստացվել են բավարար պայմաններ, որոնց դեպքում Նեյմանի խնդրի ընդհանրացված լուծումը դիֆերենցիալ-օպերատորային հավասար-ման համար գոյություն ունի և միակն է կամայական 𝑓𝑓 ∈ 𝐿𝐿2�(0,𝑏𝑏),ℋ� համար: ● 𝐴𝐴 օպերատորի ինքնահամալուծ լինելու դեպքում ցույց է տրվել, որ

𝐿𝐿: 𝐿𝐿2�(0,𝑏𝑏),ℋ� → 𝐿𝐿2�(0, 𝑏𝑏),ℋ�

օպերատորի սպեկտրը համընկնում է 𝐴𝐴 − 𝐼𝐼ℋ:ℋ →ℋ և 𝐵𝐵: 𝐿𝐿2(0, 𝑏𝑏) →𝐿𝐿2(0, 𝑏𝑏) օպերատորների սպեկտրների 𝜎𝜎(𝐿𝐿) = 𝜎𝜎(𝐴𝐴 − 𝐼𝐼ℋ) + 𝜎𝜎(𝐵𝐵) ուղիղ գումարի փակման հետ, որտեղ 𝐼𝐼ℋ -ը միավոր օպերատորն է ℋ տարածության մեջ:

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ЗАКЛЮЧЕНИЕ

ДАРЮШ КАЛВАНД

Задача Неймана для вырождающихся дифференциально-операторных уравнений четвертого порядка

Диссертация посвящена к исследованию смешанной задачи для одного класса вырождающихся дифференциально-операторных уравнений четвертого поряд-ка. Это довольно широкий класс, который содержит как классические, так и не-классические дифференциальные уравнения в частных производных.

В представленной диссертации рассматривается Задача Неймана для вырож-

дающихся дифференциально-операторных уравнений четвертого порядка

�� � ��������� � � ����, � � �0, ��, � � 0, � � ����0, ��, ��, (1)

где : � � � является линейным оператором (вообще говоря неограни-

ченный), действующим в некотором сепарабельном гильбертовом пространстве

�. Предполагается, что в � существует базис Рисса �������∞ , такой что

функции �� , � � � являются собственными функциями �� � ��� , � � �

для оператора .

В диссертационной работе сперва изучается одномерный случай, т.е. слу-

чай, когда является оператором умножения � � �, � ! на число

. Затем используя представления

���� � ∑ �������#��� , ���� � ∑ �������

#���

дифференциально-операторное уравнение (1) приводится к бесконечной

цепочке обыкновенных диффeренциальных уравнений

���� � ���������� ��� � �����, � � �0, ��, � � 0, �� � ���0, ��, � � �. (2)

Сперва для обыкновенного дифференциального уравнения вида (2)

определяется обобщенное решение задачи Неймана в весовом пространстве

Соболева $���0� и $�

����. Затем определяется соответствующее обобщенное

решение задачи Неймана для дифференциально-операторного уравнения (1).

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В диссертационной работе получены следующие основные результаты:

● Для элементов весового пространства Соболева $���0, �� и их производных

получены оценки в окрестности точки � � 0 .

● Доказано, что одномерный оператор %� � ��������� �, &�%� ' $���0, ��

является в пространстве ���0, �� положительным и самосопряженным при

0 ( � ( 4 , а обратный оператор %*�: ���0, �� � ���0, �� существует и

непрерывен при 0 ( � ( 4 . При 0 ( � , 4 обратный оператор является

компактным оператором.

● Дано описание области определения для одномерного оператора L в

зависимости от показателя вырождения � .

● Получены достаточные условия, при выполнении которых задача Неймана

для дифференциально-операторного уравнения однозначно разрешима.

● Когда оператор является самосопряженным, тогда спектр оператора

�: ��-�0, ��, �. � ��-�0, ��, �.

совпадает /��� � /� 0 1�� /�%� с замыканием прямой суммы спектров

операторов 0 1�: � � � и %: ���0, �� � ���0, �� , где 1� - единичный

оператор в пространстве.