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On general boundary value problems for elliptic equations. (English. Russian original) Zbl 0962.58009
Dokl. Math. 60, No. 1, 19-21 (1999); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 367, No. 1, 23-25 (1999).
From the text: It is well known that necessary and sufficient conditions for an elliptic boundary value problem in a Sobolev space to the Fredholm are conditions known in the literature as the Shapiro-Lopatinskii conditions. It is well known that there are elliptic operators on a manifold with a boundary for which boundary value problems satisfying these conditions are absent. Examples are the Dirac and Hirzebruch operators (signatures). At the same time, the last operators play an important role in differential-geometric and topological problems. For this reason, it seems important to find a generalization of boundary value problems such that any elliptic operator on a manifold with a boundary admits Fredholm boundary value problems.
In this paper, we solve this problem in the general case. Specifically, for any elliptic operator, we indicate boundary value problems that define a Fredholm operator in corresponding (in general, non-Sobolev) spaces.
58J32 Boundary value problems on manifolds
35J55 Systems of elliptic equations, boundary value problems (MSC2000)