Geodesic
On the continuity of the solutions of a class of non-local problems for an elliptic equation
Sbornik. Mathematics, Tome 186 (1995) no. 2, pp. 197-219 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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A. K. Gushchin; V. P. Mikhailov. On the continuity of the solutions of a class of non-local problems for an elliptic equation. Sbornik. Mathematics, Tome 186 (1995) no. 2, pp. 197-219. http://geodesic.mathdoc.fr/item/SM_1995_186_2_a2/

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