Smoothness of generalized solutions of the second and third boundary-value problems for strongly elliptic differential-difference equations

D. A. Neverova

Contemporary Mathematics. Fundamental Directions, Proceedings of the S.M. Nikolskii Mathematical Institute of RUDN University, Tome 65 (2019) no. 4, pp. 655-671.

Résumé

In this paper, we investigate qualitative properties of solutions of boundary-value problems for strongly elliptic differential-difference equations. Earlier results establish the existence of generalized solutions of these problems. It was proved that smoothness of such solutions is preserved in some subdomains but can be violated on their boundaries even for infinitely smooth function on the right-hand side. For differential-difference equations on a segment with continuous right-hand sides and boundary conditions of the first, second, or the third kind, earlier we had obtained conditions on the coefficients of difference operators under which there is a classical solution of the problem that coincides with its generalized solution. Also, for the Dirichlet problem for strongly elliptic differential-difference equations, the necessary and sufficient conditions for smoothness of the generalized solution in Hölder spaces on the boundaries between subdomains were obtained. The smoothness of solutions inside some subdomains except for $\varepsilon$-neighborhoods of angular points was established earlier as well. However, the problem of smoothness of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations remained uninvestigated. In this paper, we use approximation of the differential operator by finite-difference operators in order to increase the smoothness of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in the scale of Sobolev spaces inside subdomains. We prove the corresponding theorem.

Bibliographie

[1] A. B. Antonevich, “On the index and normal solvability of general elliptic boundary-value problem with a finite group of shifts on the boundary”, Differ. Equ., 8:2 (1972), 309–317 (in Russian) | Zbl

[2] R. Bellman, K. Cooke, Differential-Difference Equations, Russian translation, Mir, M., 1967

[3] A. V. Bitsadze, A. A. Samarskii, “On some simplest generalizations of linear elliptic boundary-value problems”, Rep. Acad. Sci. USSR, 185:4 (1969), 739–740 (in Russian) | Zbl

[4] E. M. Varfolomeev, “On some properties of elliptic and parabolic functional differential operators arising in nonlinear optics”, Contemp. Probl. Math. Fundam. Directions, 21 (2007), 5–36 (in Russian)

[5] V. V. Vlasov, R. Perez Ortiz, N. A. Rautian, “Investigation of Volterra integrodifferential equations with kernels depending of a parameter”, Differ. Equ., 54:3 (2018), 369–386 (in Russian) | DOI | Zbl

[6] V. V. Vlasov, N. A. Rautian, Spectral Analysis of Functional Differential Equations, MAKS Press, M., 2016 (in Russian)

[7] V. V. Vlasov, N. A. Rautian, “Investigation of functional differential equations with unbounded operator coefficients”, Rep. Russ. Acad. Sci., 477:6 (2017), 641–645 (in Russian) | DOI | Zbl

[8] G. A. Kamenskiy, A. L. Skubachevskii, Linear Boundary-Value Problems for Differential-Difference Equations, MAI, M., 1992 (in Russian)

[9] T. Kato, Perturbation Theory for Linear Operators, Russian translation, Mir, M., 1972

[10] V. P. Mikhaylov, Partial Differential Equations, Nauka, M., 1976 (in Russian)

[11] A. D. Myshkis, Linear Differential Equations with Delayed Argument, Nauka, M., 1972 (in Russian) | MR

[12] D. A. Neverova, A. L. Skubachevskii, “On classical and generalized solutions of boundary-value problems for differential-difference equations with variable coefficients”, Math. Notes, 94:5 (2013), 702–719 (in Russian) | DOI | MR | Zbl

[13] G. G. Onanov, A. L. Skubachevskii, “Differential equations with deviating arguments in stationary problems of mechanics of deformable bodies”, Appl. Mech., 15:5 (1979), 39–47 (in Russian) | MR | Zbl

[14] V. V. Podyapol'skii, A. L. Skubachevskii, “Spectral asymptotics of strongly elliptic differential-difference operators”, Differ. Equ., 35 (1999), 393–800 (in Russian) | MR

[15] V. A. Popov, A. L. Skubachevskii, “A priori estimates for elliptic differential-difference operators with degeneration”, Contemp. Math. Fundam. Directions, 36, 2010, 125–142 (in Russian) | Zbl

[16] V. A. Popov, A. L. Skubachevskii, “Smoothness of generalized solutions of elliptic difference differential equations with degenerations”, Contemp. Math. Fundam. Directions, 39, 2011, 130–140 (in Russian)

[17] V. S. Rabinovich, “On solvability of differential-difference equations in $\mathbb R^n$ and in the half-space”, Rep. Acad. Sci. USSR, 243:5 (1978), 1134–1137 (in Russian) | MR | Zbl

[18] L. E. Rossovskii, “Elliptic functional differential equations with contractions and extensions of independent variables of the unknown function”, Contemp. Math. Fundam. Directions, 54, 2014, 31–38 (in Russian)

[19] A. M. Selitskii, “The third boundary-value problem for parabolic functional differential equation”, Contemp. Math. Fundam. Directions, 21, 2007, 114–132 (in Russian)

[20] A. M. Selitskii, A. L. Skubachevskii, “The second boundary-value problem for parabolic functional differential equation”, Proc. I. G. Petrovskii Semin., 26, 2007, 324–347 (in Russian) | MR

[21] A. L. Skubachevskii, “On eigenvalues and eigenfunctions of some nonlocal boundary-value problems”, Differ. Equ., 25:1 (1989), 127–136 (in Russian) | MR

[22] A. L. Skubachevskii, “Boundary-value problems for elliptic functional differential equations and their applications”, Progr. Math. Sci., 71:5 (2016), 3–112 (in Russian) | DOI | MR | Zbl

[23] A. L. Skubachevskii, E. L. Tsvetkov, “The second boundary-value problem for elliptic differential-difference equations”, Differ. Equ., 25:10 (1989), 1766–1776 (in Russian) | MR

[24] J. Hale, Theory of Functional Differential Equations, Russian translation, Mir, M., 1984

[25] E. L. Tsvetkov, “On smoothness of generalized solutions of the third boundary-value problem for elliptic differential-difference equation”, Ukr. Math. J., 45:8 (1993), 1140–1150 (in Russian) | MR | Zbl

[26] F. Hartman, G. Stampacchia, “On some nonlinear elliptic differential-functional equations”, Acta Math., 115 (1966), 271–310 | DOI | MR | Zbl

[27] D. A. Neverova, “Generalized and classical solutions to the Second and third boundary value problem for difference-differential equations”, Funct. Differ. Equ., 21 (2014), 47–65 | MR | Zbl

[28] D. A. Neverova, A. L. Skubachevskii, “On the smoothness of generalized solutions to boundary value problems for strongly elliptic differential-difference equations on a boundary of neighboring subdomains”, Russ. J. Math. Phys., 22:4 (2015), 504–517 | DOI | MR | Zbl

[29] G. G. Onanov, A. L. Skubachevskii, “Nonlocal Problems in the Mechanics of Three-Layer Shells”, Math. Model. Nat. Phenom., 12 (2017), 192–207 | DOI | MR | Zbl

[30] G. G. Onanov, E. L. Tsvetkov, “On the minimum of the energy functional with respect to functions with deviating argument in a stationary problem of elasticity theory”, Russ. J. Math. Phys., 3:4 (1996), 491–500 | MR

[31] A. L. Skubachevskii, “The first boundary value problem for strongly elliptic differential-difference equations”, J. Differ. Equ., 63:3 (1986), 332–361 | DOI | MR

[32] A. L. Skubachevskii, Elliptic functional differential equations and applications, Birkhauser, Basel—Boston—Berlin, 1997 | MR | Zbl