ELLIPTIC EQUATIONS WITH ARBITRARILY DIRECTED TRANSLATIONS IN HALF-SPACES Текст научной статьи по специальности «Математика»

V- l™|■■■■ О

Серия «Математика»

2023. Т. 43. С. 64—77

Онлайн-доступ к журналу: http://mathizv.isu.ru

Research article

УДК 517.9 MSC 35R10, 35J25

DOI https://doi.org/10.26516/1997-7670.2023.43.64

Elliptic Equations with Arbitrarily Directed Translations in Half-Spaces

Viktoriia V. Liiko1, Andrey B. Muravnik1^

1 Peoples' Friendship University of Russia (RUDN University), Moscow, Russian Federation

И [email protected]

Abstract. In this paper, we investigate the half-space Dirichlet problem for elliptic differential-difference equations with superpositions of differential operators and translation operators acting in arbitrary directions parallel to the boundary hyperplane. The summability assumption is imposed on the boundary-value function of the problem. The specified equations, substantially generalizing classical elliptic partial differential equations, arise in various models of mathematical physics with nonlocal and (or) heterogeneous properties or the process or medium: multi-layer plates and envelopes theory, theory of diffusion processes, biomathematical applications, models of nonlinear optics, etc. The theoretical interest to such equations is caused by their nonlocal nature: they connect values of the desired function (and its derivatives) at different points (instead of the same one), which makes many classical methods unapplicable.

For the considered problem, we establish the solvability in the sense of generalized functions, construct Poisson-like integral representations of solutions, and prove the infinite smoothness of the solution outside the boundary hyperplane and its uniform convergence to zero (together with all its derivatives) as the timelike variable tends to infinity. We find a power estimate of the velocity of the specified extinction of the solution and each its derivative.

Keywords: differential-difference equations, elliptic equations, half-space Dirichlet problems, summable boundary-value functions

Acknowledgements: The research was supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-03-2020-223/3, FSSF-2020-0018).

For citation: Liiko V. V., MuravnikA.B. Elliptic Equations with Arbitrarily Directed Translations in Half-Spaces. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 43, pp. 64-77. https://doi.org/10.26516/1997-7670.2023.43.64

Научная статья

Эллиптические уравнения со сдвигами произвольных направлений в полупространстве

В.В.Лийко1, А.Б.Муравник1и

1 Российский университет дружбы народов, Москва, Российская Федерация И [email protected]

Аннотация. Исследуется задача Дирихле в полупространстве для эллиптических дифференциально-разностных уравнений с операторами, представляющими собой суперпозиции дифференциальных операторов и операторов сдвига в произвольных направлениях, параллельных краевой гиперплоскости. На краевую функцию задачи накладывается условие суммируемости. Указанные уравнения, существенно обобщающие классические эллиптические уравнения в частных производных, возникают в различных моделях математической физики, для которых имеют место нелокальные и (или) неоднородные свойства процесса или среды: теория многослойных пластин и оболочек, теория диффузионных процессов, биоматематические приложения, модели нелинейной оптики и др. В теоретическом плане интерес к таким уравнениям обусловлен их нелокальной природой — они связывают между собой значения неизвестной функции (и ее производных) не в одной точке, а в разных, что делает неприменимыми многие классические методы.

Для рассматриваемой задачи устанавливается разрешимость в смысле обобщенных функций, строится интегральное представление решения формулой пуассоновского типа, доказывается его бесконечная гладкость вне краевой гиперплоскости и его равномерное стремление к нулю (вместе со всеми его производными) при стремлении времениподобной независимой переменной к бесконечности. Доказывается степенная оценка скорости указанного равномерного затухания решения и каждой его производной.

Ключевые слова: дифференциально-разностные уравнения, эллиптические уравнения, задача Дирихле в полупространстве, суммируемые краевые функции

Благодарности: Работа выполнена при поддержке Министерства образования и науки РФ в рамках государственного задания (соглашение № 075-03-2020-223/3, FSSF-2020-0018).

Ссылка для цитирования: Liiko V. V., Muravnik A. B. Elliptic Equations with Arbitrarily Directed Translations in Half-Spaces // Известия Иркутского государственного университета. Серия Математика. 2023. Т. 43. C. 64-77. https://doi.org/10.26516/1997-7670.2023.43.64

1. Introduction

We consider the Dirichlet problem (with summable boundary-value functions) for functional-differential equations containing superpositions of differential operators and arbitrarily directed translation operations. Namely,

in the half-space |(ж, y) x G Rra, y > 0 j, we consider the problem

n n

x. (x,y) + Uyy (x,y) + Y_1 аз uXjXj (X + hj ,y) = 0, (1.1) 3=1 3 = 1

и = uo(x), x G Rra, (1.2)

y=0

where

a0 := max | < 1, (1.3)

j=1,n

hj := (hj\,...,hjn),j = 1,n, are arbitrary vectors from Rra, and u0 G Li(Rn).

This substantially generalizes the problem investigated in [9], where translation operators act only with respect to coordinate directions. In the present paper, this restriction is taken off, which means that translations of the independent (vector) variable x are not supposed to be either collinear or orthogonal to each other; instead, each angle between the directions of those translations is admitted.

The current interest to differential-difference equations, i.e., to equations where translation operators act on the desired function apart from differential operators, is caused both by their numerous applications not covered by classical models of mathematical physics (see, e.g., [13-15;21] and references therein) and by purely theoretical reasonings: the nonlocal nature of such equations generates qualitatively new phenomena not observed in the classical case of differential equations, while various research methods proved their efficiency in the theory of differential equations (e. g., such as the maximum principle) turn to be inapplicable. A comprehensive and profound explanation of the theory of bounded-region problems for elliptic differential-difference equations (as well as for the very close theory of nonlocal problems for differential elliptic equations) can be found in [3; 13; 16-18] (see references therein as well). Problems in unbounded regions are investigated not so profoundly. Moreover, the case of bounded, boundary-value functions is mainly investigated (see [8] and references therein). In the present paper, the case of summable boundary-value functions is studied. This decomposition of half-space problems into two classes (problems with bounded boundary-value functions and problems with summable boundary-value functions) takes place for the classical case of differential equations (both elliptic and parabolic ones) as well. Such a separation is reasonable because the specified difference in the problem settings generates solutions with qualitatively different sets of properties: for example, constant solutions are possible only for problems from the first of the above classes, while only finite-energy solutions are possible for problems from the second class. In other words, the Repnikov-Eidel'man

stabilization criterion (see [10; 11]), according to which solutions with and without limits (in general, different from zero) are possible, is valid only for problems with bounded boundary-value (initial-value in the parabolic case) functions. For problems with summable boundary-value (initial-value) functions, the solution always has a limit and it is always equal to zero, while the main research interest in such problems is concentrated on the estimate of the decay rate for the solution.

2. Operational scheme

We use the classical Gel'fand-Shilov operational scheme (see, e.g., [1, Sec. 10]): apply (formally) the Fourier transformation with respect to the (vector) variable x to problem (1.1)-(1.2). This leads to the following initial-value problem to an ordinary differential equation:

^ = ||£|2 + E a3£2e*h* j u, y e (0, +«>), (2.1)

«(0; 0= «o(0. (2.2)

Note that this is not a Cauchy problem because the order of the equation is equal to two, while the initial-value condition is unique. Thus, (2.1) is a second-order linear differential equation (depending on the n-dimensional parameter £) with constant coefficients, such that its characteristic equation has two roots

±

\

Id2 + E %eihj ? = ±

i=1

\

|C|2 + E a3¿2 cos h3 • £ + ^ a3^ sin h3 • £

i=t

=: ±p(cos d + i sin(

i=t

where

p(0 =

2

|C|2 + E ai^ cos h • + I E aik^ sin h • £

3=1

0(0 = 1 arctg ——1

Vi,k=1

sin hj

£ aj & sin h j • £

|£|2 + £ a3g cos h • £

=1

Arguing as in [6, Sec. 1], we solve problem (2.1)-(2.2), suitably select the "free" arbitrary constant (it arises because the amount of initial-value

2

4

conditions is less than the order of the equation), and (formally) apply the inverse Fourier transformation to the obtained solution. This yields the convolution of the boundary-value function with the integral (with respect to 0 of the function e-yp(^)cosm cos[^ ■ £ - yp(£) sin 0(£)]:

where

u(x,y) = J £(x - £,y)uo(Od£,

R"

£(X,y) = J e-yGl(C) cos [* ■ ^ - yG*(^ ^

3=1

Gi(A)= P(Q cos 9(0, and G2(0 = p(0 sin d($.

■ e ь Kt )

(2.3)

(2.4)

(2.5)

Denote ^ ajcos hj ■ £ by a(£). Denote ^ ajsin hj ■ £ by b(£). Then

3 = 1

P(0 = ([Id2 + a(0]2 + b2(()) 4 and 0(0 = 2 arctg

ICI2 + a(t)

This im-

plies that |0(OI < ^, i.e., cos0(0 > ^ and cos 20(0 > 0. Hence, cos0(0

can be represented in the form

/

1 + cos 20(£)

2

. Now, apply the relation

cos2 20(0 =

1

1 + tg2 20(0

and take into account the nonnegativity of cos 20(0 again. We obtain that

1

cos20(£) =

Further, since

it follows that

and, therefore,

20(e) = arctg tg 20(^) =

^1+tg2 20(e ) '

ко

ICI2 + a(£ y

ко

[

cos 20(e) = ( 1 +

[ICI2 + a(0]2

ICI2 + a(t) b2(0

[I£ I2 + a(0]2,

ICI2 + a(t)

I2 + й(^)]2 + ^ ^2 + *(0]2 + ^(0

because Condition (1.3) ensures the nonnegativity of the function |{|2+a(£) for each £ G Rn and its positivity everywhere apart from the origin:

n n

ici2 + E ai$ cos hj ■ e > ici2 - E K-$

3 = 1

3=1

>|e|2 - max K| E^2 = (1 - ac)|e|2 > 0

, -,3=1

3 1 ,n

provided that £ = 0. Then

cos 0(0 =

1 +

|£|2 + )

V[|C|2 + a(£)]2 + b2(0 _

In Rn, introduce the function

:= + fl(^)]2 + ^ = V|C|4 + a2(0 + &2(0 + 2a(£)|£|2

Then

|£|2 +

V^2 + a(£)]2 + b2(0

= ^ * F+S^ = ^12 ^

= —V p(0 + Id2 + a(0 > — Viei2 + a(e)

(by virtue of the nonnegativity of the function Therefore, the function G\ (£) is bounded from below by the expression

2

2

|£|2 - E K n

=

=1

|^|2 - E ^2 E K1>"72 =1 =1 2

|£|2 - e2 =1

/

1 - ao

which guarantees that function (2.4) is well defined in Rra x (0, +rc>).

Note that, applying the direct and inverse Fourier transformations in the present section, we do not care about the justification of the convergence of the integrals and the legibility of the change of the order of integrating. This entirely corresponds to the general scheme of [1, Sec. 10] because solutions

2

2

in the sense of generalized functions are meant. Lemma 1 (see the next section) deals with smooth functions, but it is proved independently.

3. Constructing of Poisson kernels

The following assertion is valid.

Lemma 1. The function £(x, y) defined by relation (2.4) is well defined in the half-space Rra x (0, and satisfies (in the classical sense) Eq. (1.1).

Proof. It is shown in the previous section that the function G1(() is estimated from below by the function ^JЮ Then the function £(x,y) is well defined in Rra x (0, +ro). Really, the modulus of the integrand function in (2.4) is majorized by the integrable function const exp д/l — |ao| . Now, compute the Laplacian of function (2.4). We have the relations

£XjXj (x,y) = —J e3e~yGm cos [ж ■ £ — yG2(0] d£, j = M,

Rn

£y (x,y) = — j Gi(£)e-^Gl(« cos [ж ■ £ — yG2(0] d£

Rn

+ J G2(0e~yGl^ sin [ж ■ £ — yG2(0] d£,

R"

and

£yy(X, y) = j Gl(£)e-^1(« cos [ж ■ £ — yG2(0] d^

R"

— j Gi(OG2(Oe-yGl{0 sin [ж ■ £ — yG2(0] d^

R"

— j Gi(0G2(0e--yGl{° sin [ж ■ £ — yG2(0] d£

R"

— j G2(0e-yGl^ cos [ж ■ £ — yG2(0] d£ =

Rn

= / [Gi(0 — G2(0] e-yGl(?) cos [ж ■ £ — yG2(C)] d^

R"

—2j Gi(£)G2(C)e-yGl(^) sin [ж ■ £ — yG2(0] d£

R"

(note that the differentiating inside the integral is legible in all these cases because the arising integrand factors have no singularities and their growth with respect to £ is at most polynomial). Further, it follows from (2.5) that

2Gi(£)G2(0 = 2p(0 cos0(Op(Osin0(0 = p2(0 sin20(0.

It is proved above that cos 20(0 > 0. Therefore,

tg20(£ )

sin 20(0 =

^1+tg2 20(0

a, g. sin hj • £

j=1

ICI2 + E a,g cos h, • e

3=1

1 +

(n

S'

E aj$ sin hj • £

(

ICI2 + E ajC2 cos hj • £

j=1

Hence, the following relation is valid:

E a,g sin hj • £

sin 20(0 =

j=1

ICI2 + E aj g cos hj • £

j=1

(

ICI2 + E aj g cos hj • £

j=1

)

\

rj + (E aj £2 sin hj • ^

ICI2 + E aj£2 cos hj • £ ) + ( E aj£2 sin hj • £

Since the sum I£I2 + ^ aj g cos hj • £ is nonnegative by virtue of Condition

j=i

(1.3), we eventually obtain the relation

E aj g sin hj • £

sin 20(0 =

j=1

\

(

^ + (jg aj £2 sin hj • ^

ICI2 + E aj£j2 cos hj • £ ] + ( E aj£j2 sin hj • £

Therefore, the following relation holds:

2

x

2Gi(C)G2 (0 = = p2(0-

£ aj0 sin hj ■ £

j=i

\

(

j=i

Id2 + £ ^ cos h3 ■ £ ) + ( £ a3$ sin h3 ■ £

j=i

n

$sin h ■ £ = ко.

3 = i

Now, using (2.5), we obtain that

G\(i) — G2(0 = P2(0 cos2 0(0 — sin2 0(0 p2(0 cos 20(0 =

= P2(0l 1 —

1

tg2 20(0 _ о

= P2(0

1

1 + tg2 20(0 ' V V1+tg2 20(0'

which is equal to

p2(0

\

Id2 + £ a3о cos h3 ■ e

j=i

)

<

i=i

Id2 + £ ^ cos hj ■ ^ + ^ £ a,£2 sin hj ■ ^

|£|2 + Ecos hj ■ e = icI2+a®.

3 = i

This means that

£yy(x,y) = J [ICI2 + a(0] e-yGl(^) cos [ж ■ £ — ^(0] d(—

R"

— j b(0e-yGl(?) sin [ж ■ £ — yG2(0] dO

Rn

2

Now, substitute the function Sin Eq. (1.1):

E Sxiy) + SvvV) + E a3Sxixi+ h3 ' ^ = 3=1 3=1

J e-yGl(a(£) cos [x ■ £ - yG2(C)j - &(£) sin [z ■ £ - yG^)] )

n „

E a3 J &-yGm cos [(x + h3) ■ £ - yG2(£)j

J 6

3 = 1 R™

This is equal to

f cos [x ■ e - yG2(e)j E «3 d cos h, ■ e-

Rn 3=1

n

- sin ■ £ - yG2(£)] E d]£2 sin h3 ■ C-

3=1

n \

- E a3 el cos [(x+hi ) ■ e - yG2(e)]i =

3=1

= J E a3^ ( cos h3 ■ £ cos [x ■ £ - yG^)] -

Rn 3 = 1

- sin h ■ £ sin [a ■ £ - yG2(0] - cos [(a + h) ■ £ - d£,

which is equal to

n

2

J e-yG1(C) £ a .^ cos [(X + h3) ■ £ - yG2(£)] -

e Z^ ^

Rn 3 = 1

- cos [(^ + hj) ■ £ - yG^)]) d£.

This is equal to zero, i.e., the function £satisfies (in the classical sense) Eq. (1.1) in the half-space Rra x (0, +rc>). □

4. Convolutions with summable functions

Now, majorize the function £(x,y) itself as well as its derivatives of arbitrary orders:

f IC|me-^ « 1 = -- 1m+n + f Wme-1 dv =

/ (1 -M) "+" Ут+п £

oo

= const r +n-le-pd = const

ym+n J ' ' ym+n'

0

Taking into account Lemma 1, we obtain the following assertion.

Theorem 1. If u0 e Li(Rn), then the function u(x,y) defined by relation (2.3) is an infinitely differentiable solution of Eq. (1.1) in Rn x (0, This function and each its partial derivative tend to zero as у ^ uniformly with respect to x e Rn.

Note that, similarly to less general cases investigated earlier (see [9] and references therein), a greater smoothness is established for the solution of the equation (not of the boundary-value problem). At the moment, the solvability of problem (1.1)-(1.2) is proved only in the Gel'fand-Shilov sense (see [1, Sec. 10] and cf. [7, Remark 2]): (2.3) is a generalized function of the n-dimensional variable x, depending on the real parameter у and differentiable with respect to this parameter on the positive semiaxis (see, e.g., [12, §9, Sec. 5]), Eq. (1.1) is treated as the equality of generalized functions of variable x, satisfied for each positive value of the parameter y, and Condition (1.2) is treated as a limit relation in the topology of generalized functions of variable x as the real parameter у tends to zero from the right (see, e.g., [12, §9, Sec. 4]).

However, it is established by Theorem 1 that this solution possesses a greater smoothness (more exactly, the infinite smoothness) outside the boundary-value hyperplane, i.e., in the open half-space Rn x (0, +ro).

Remark 1. Though the problem is set in an unbounded region, it is not a Cauchy problem (unlike the parabolic case, see, e.g., [4; 5]). This phenomenon (an elliptic equation considered in an anisotropic region acquires qualitative properties typical for nonstationary equations) is well known for the classical case of differential equations (see, e.g., [19; 20]). In the nonlocal case, this phenomenon is still actively investigated.

Remark 2. The solvability result obtained above complements the results of [2] where the solvability of the investigated problem is proved for various classes of boundary-value functions (in particular, it is known for classes of functions belonging to Lp(Rn), 1 < p < те, and such that their Fourier

transforms are compactly supported in a domain of Rra). As far as the authors are aware, the case where p = 1 was not considered earlier.

The second author is grateful to the participants of the 7th International Conference on Nonlinear Analysis and Extremal Problems (Irkutsk, July 2022) for useful discussions encouraging a better understanding of the obtained results and improving of their presentation.

The authors express their profound gratitude to A. L. Skubachevskii for the permanent attention to this work.

References

1. Gel'fand I.M., Silov G.E. Fourier transforms of rapidly increasing functions and questions of uniqueness of the solution of Cauchy's problem. Uspehi Matem. Nauk (N.S.), 1953, vol. 8, pp. 3-54. (in Russian)

2. Gorenflo R., Luchko Yu.F., Umarov S.R. On some boundary value problems for pseudo-differential equations with boundary operators of fractional order. Fract. Calc. Appl. Anal., 2000, vol. 3, pp. 453-468.

3. Gurevich P.L. Elliptic problems with nonlocal boundary conditions and Feller semigroups. Journal of Mathematical Sciences, 2012, vol. 182, pp. 255-440. https://doi.org/10.1007/s10958-012-0746-y

4. Muravnik A.B. On the Cauchy problem for differential-difference equations of the parabolic type. Dokl. Akad. Nauk, 2002, vol. 66, pp. 107-110.

5. Muravnik A.B. On Cauchy problem for parabolic differential-difference equations. Nonlinear Analysis, 2002, vol. 51, pp. 215-238. https://doi.org/10.1016/S0362-546X(01)00821-5

6. Muravnik A.B. On the Dirichlet problem for differential-difference elliptic equations in a half-plane. Journal of Mathematical Sciences, 2018, vol. 235, pp. 473-483. https://doi.org/10.1007/s10958-018-4082-8

7. Muravnik A.B. Elliptic differential-difference equations of general form in a half-space. Mathematical Notes, 2021, vol. 110, pp. 92-99. https://doi.org/10.1134/S0001434621070099

8. Muravnik A.B. Half-plane differential-difference elliptic problems with general-kind nonlocal potentials. Complex Variables and Elliptic Equations, 2022, vol. 67, pp. 1101-1120. https://doi.org/10.1080/17476933.2020.1857372

9. Muravnik A.B. Elliptic equations with translations of general form in a half-space. Mathematical Notes, 2022, vol. 111, pp. 587-594. https://doi.org/10.1134/S0001434622030270

10. Repnikov V.D., Eidel'man S.D. Necessary and sufficient conditions for establishing a solution to the Cauchy problem. Soviet Math. Dokl., 1966, vol. 7, pp. 388-391.

11. Repnikov V.D., Eidel'man S.D. A new proof of the theorem on the stabilization of the solution of the Cauchy problem for the heat equation. Sbornik: Mathematics, 1967, vol. 2, pp. 135-139. https://doi.org/10.1070/SM1967v002n01ABEH002328

12. Shilov G.E. Matematicheskii analiz. Vtoroj spetsial'nyi kurs [Mathematical analysis. The second special course]. Moscow, Moskow Univ. Publ., 1984. (in Russian)

13. Skubachevskii A.L. Elliptic functional differential equations and applications. Basel-Boston-Berlin, Birkhauser, 1997.

14. Skubachevskii A.L. On the Hopf bifurcation for a quasilinear parabolic functional-differential equation. Differential Equations, 1998, vol. 34, pp. 1395-1402.

76

15

16

17

18

19

20

21

1

2

3

4.

5.

6

7

8

9

Skubachevskii A.L. Bifurcation of periodic solutions for nonlinear parabolic functional differential equations arising in optoelectronics. Nonlinear Analysis, 1998, vol. 32, pp. 261-278.

Skubachevskii A.L. Nonclassical boundary-value problems. I. Journal of Mathematical Sciences, 2008, vol. 155, pp. 199-334. https://doi.org/10.1007/s10958-008-9218-9

Skubachevskii A.L. Nonclassical boundary-value problems. II. Journal of Mathematical Sciences, 2010, vol. 166, pp. 377-561. https://doi.org/10.1007/s10958-010-9873-5

Skubachevskii A.L. Boundary-value problems for elliptic functional-differential equations and their applications. Russian Mathematical Surveys, 2016, vol. 71, pp. 801-906. https://doi.org/10.1070/RM9739

Stein E.M., Weiss G. On the theory of harmonic functions of several variables. I: The theory of Hp spaces. Acta Mathematica, 1960, vol. 103, pp. 25-62. Stein E.M., Weiss G. On the theory of harmonic functions of several variables. II: Behavior near the boundary. Acta Mathematica, 1961, vol. 106, pp. 137-174. Vorontsov M.A., Iroshnikov N.G., Abernathy R.L. Diffractive patterns in a nonlinear optical two-dimensional feedback system with field rotation. Chaos, Solitons, and Fractals, 1994, vol. 4, pp. 1701-1716.

Список источников

Гельфанд И.М., Шилов Г. Е. Преобразования Фурье быстро растущих функций и вопросы единственности решения задачи Коши // Успехи математических наук. 1953. Т. 8. С. 3-54.

Gorenflo R., Luchko Yu. F., Umarov S. R. On some boundary value problems for pseudo-differential equations with boundary operators of fractional order // Fract. Calc. Appl. Anal. 2000. Vol. 3. P. 453-468.

Гуревич П. Л. Эллиптические задачи с нелокальными краевыми условиями и полугруппы Феллера // Современная математика. Фундаментальные направления. 2010. Т. 38. С. 3-173.

Муравник А. Б. О задаче Коши для некоторых дифференциально-разностных уравнений параболического типа // Доклады РАН. 2002. Т. 385. С. 604-607. Muravnik A.B. On Cauchy problem for parabolic differential-difference equations // Nonlinear Analysis. 2002. Vol. 51. P. 215-238. https://doi.org/10.1016/S0362-546X(01)00821-5

Муравник А. Б. О задаче Дирихле в полуплоскости для дифференциально-разностных эллиптических уравнений // Современная математика. Фундаментальные направления. 2016. Т. 60. С. 102-113.

Муравник А. Б. Эллиптические дифференциально-разностные уравнения общего вида в полупространстве // Математические заметки. 2021. Т. 110. С. 90-98. https://doi.org/10.4213/mzm13009

Muravnik A. B. Half-plane differential-difference elliptic problems with generalkind nonlocal potentials // Complex Variables and Elliptic Equations. 2021. Vol. 67. P. 1101-1120. https://doi.org/10.1080/17476933.2020.1857372 Муравник А. Б. Эллиптические уравнения со сдвигами общего вида в полупространстве // Математические заметки. 2022. Т. 111. С. 571-580. https://doi.org/10.4213/mzm13369

10. Репников В. Д., Эйдельман С. Д. Необходимые и достаточные условия установления решения задачи Коши // Доклады АН СССР. 1966. Т. 167. С. 298-301.

11. Репников В. Д., Эйдельман С. Д. Новое доказательство теоремы о стабилизации решения задачи Коши для уравнения теплопроводности // Математический сборник. 1967. Т. 73 (115). С. 155-159.

12. Шилов Г. Е. Математический анализ. Второй специальный курс. М. : Изд-во МГУ, 1984.

13. Skubachevskii A. L. Elliptic functional differential equations and applications. Basel-Boston-Berlin: Birkhauser, 1997.

14. Скубачевский А. Л. О бифуркации Хопфа для квазилинейного параболического функционально-дифференциального уравнения // Дифференциальные уравнения. 1998. Т. 34. С. 1394-1401.

15. Skubachevskii A. L. Bifurcation of periodic solutions for nonlinear parabolic functional differential equations arising in optoelectronics // Nonlinear Analysis. 1998. Vol. 32. P. 261-278.

16. Скубачевский А. Л. Неклассические краевые задачи. I // Современная математика. Фундаментальные направления. 2007. Т. 26. C. 3-132.

17. Скубачевский А. Л. Неклассические краевые задачи. II // Современная математика. Фундаментальные направления. 2009. Т. 33. C. 3-179.

18. Скубачевский А. Л. Краевые задачи для эллиптических функционально-дифференциальных уравнений и их приложения // Успехи математических наук. 2016. Т. 26. C. 3-112. https://doi.org/10.1070/RM9739

19. Stein E. M., Weiss G. On the theory of harmonic functions of several variables. I: The theory of Hp spaces // Acta Mathematica. 1960. Vol. 103. P. 25-62.

20. Stein E. M., Weiss G. On the theory of harmonic functions of several variables. II: Behavior near the boundary // Acta Mathematica. 1961. Vol. 106. P. 137-174.

21. Vorontsov M. A., Iroshnikov N. G., Abernathy R. L. Diffractive patterns in a nonlinear optical two-dimensional feedback system with field rotation // Chaos, Solitons, and Fractals. 1994. Vol. 4. P. 1701-1716.

Об авторах

Лийко Виктория Владимировна, канд. физ.-мат. наук, Российский университет дружбы народов, Российская Федерация, 117198, г. Москва, [email protected], https://orcid.org/0000-0003-1789-9772

Муравник Андрей Борисович,

д-р физ.-мат. наук, Российский университет дружбы народов, Российская Федерация, 117198, г. Москва, [email protected], https://orcid.org/0000-0001-8670-6589

About the authors Viktoriia V. Liiko, Cand. Sci. (Phys.-Math.), Peoples' Friendship University of Russia (RUDN University), Moscow, 117198, Russian Federation, [email protected], https://orcid.org/0000-0003-1789-9772

Andrey B. Muravnik, Dr. Sci.

(Phys.-Math.), Peoples' Friendship University of Russia (RUDN University), Moscow, 117198, Russian Federation, [email protected], https://orcid.org/0000-0001-8670-6589

Поступила в 'редакцию / Received 25.09.2022 Поступила после рецензирования / Revised 20.01.2023 Принята к публикации / Accepted 23.01.2023