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On nonlocal Sobolev problems. (English. Russian original) Zbl 1280.58015
Dokl. Math. 88, No. 1, 421-424 (2013); translation from Dokl. Akad. Nauk, 451, No. 3, 259-263 (2013).
From the text: An elliptic theory of problems in which boundary conditions are specified on submanifolds of arbitrary dimension was constructed by Sternin, who gave the name of Sobolev to problems of this type. The key point in the theory of such boundary value problems is reducing the problem under consideration to some operator on the boundary. It turns out that the type of the resulting operator depends essentially on the type of operators involved in the Sobolev problem. Thus, in the classical case, i.e., in the case where these operators are pseudodifferential, the operator arising on the submanifold is pseudodifferential.
On the other hand, in recent years, the theory of nonlocal elliptic problems has been extensively studied. In this connection, the question about the nature and properties of an operator obtained on a submanifold naturally arises. This problem is the subject of this paper; on the basis of its solution, we obtain, in the elliptic case, a finiteness (Fredholm) theorem for a nonlocal Sobolev problem and give an expression for its index.

MSC:
58J32 Boundary value problems on manifolds
58J05 Elliptic equations on manifolds, general theory
58J40 Pseudodifferential and Fourier integral operators on manifolds
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References:
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