Smoothness of generalized solutions of the Neumann problem for a strongly elliptic differential-difference equation on the boundary of adjacent subdomains

D. A. Neverova

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 66 (2020) no. 2, pp. 272-291

Résumé

This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of $Q$ were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can be violated near the boundary of these subdomains even for infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain $Q$ by throwing out all possible shifts of the boundary $\partial Q$ by vectors of a certain group generated by shifts occurring in the difference operators. For the one dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. Also there was obtained the smoothness (in Sobolev spaces $W^k_2$) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding $\varepsilon$-neighborhoods of certain points. However, the smoothness (in Hölder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Hölder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Hölder space.