On solvability of nonlocal problems for a second-order elliptic equation

A. K. Gushchin   ; V. P. Mikhailov

Sbornik. Mathematics, Tome 81 (1995) no. 1, pp. 101-136

Résumé

This article is an investigation of the solvability of nonlocal problems for an elliptic equation, in which the values of the solution on the boundary of the domain $\mathcal{Q}$ under consideration are expressed in terms of its values at interior points and other points of the boundary. A new concept of solution (in the space of $(n-1)$-dimensionally continuous functions) is introduced, broader than concepts considered previously, and sufficient conditions are established for the problem to be Fredholm with index zero. The connection between solvability of the problem in this formulation and in the classical formulation is studied. In particular, there is a class of nonlocal problems (including some problems studied previously) that are Fredholm with index zero in the formulation introduced but not in the classical formulation (sometimes not even Fredholm). For a certain class of problems a theorem on unique solvability is proved.

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