On the continuity of the solutions of a class of non-local problems for an elliptic equation

This paper is devoted to a study of the connection between the notion of an $(n-1)$-dimensionally continuous (weak) solution for a non-local problem, which was earlier introduced by the authors, with the notion of a classical solution. Under natural suppositions on the operator entering the non-local condition, the continuity of the weak solution in the closure of the domain under consideration is proved for all arbitrary continuous boundary function. The notion of an $(n-1)$-dimensionally continuous solution is convenient when studying the Fredholm property of the problem. In the previous paper of the authors tl.e Fredholm property in such a setting was proved for a wide class of non-local problems. When studying the uniqueness it is easier to deal with a classical solution. The main result of this paper enables one, in particular, to use simultaneously the advantages of both approaches: to apply the classical maximum principle in the proof of the uniqueness (and hence, by the Fredholm property, the existence) of a weak solution.