Regularity of solutions to the Robin problem for differential-difference equations

Abstract

This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. In contrast to elliptic differential equation, the smoothness of generalized solutions of boundary-value problems for differential-difference equations can be violated near the boundary of these subdomains even for infinitely differentiable right-hand side. Here subdomains are defined as connected components of the set, which is obtained from the domain Q by throwing out all possible translations of the boundary $\mathrm{\partial }Q$ by vectors of a certain group generated by translations occurring in the difference operators. We obtain necessary and sufficient conditions of smoothness of generalized solutions to the Robin problem for such equations on a boundary of neighbouring subdomains.

Keywords:

• Functional-differential equations
• smoothness of generalized solutions
• boundary-value problems

AMS Subject Classifications:

• 39A70
• 39A14

Acknowledgements

The author expresses her gratitude to A. Skubachevskii for valuable discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the Ministry of Science and Higher Education of the Russian Federation: agreement no. 075-03-2020-223/3 (FSSF-2020-0018).