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Elliptic expansion-contraction problems on manifolds with boundary. (English. Russian original) Zbl 1354.58020
Dokl. Math. 94, No. 1, 393-395 (2016); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 469, No. 2, 154-156 (2016).
The interesting note under review deals with a class of expansion-contraction boundary value problems on manifolds with boundary. Precisely, problems of the type \[ (D,B): \quad H^s(M) \to H^{s-m}(M)\oplus H^{s-b}(X), \] \(m=\text{ord} D,\) \(b=\text{ord}B,\) are considered on a smooth compact manifold \(M\) with boundary \(X,\) where the main operator \(D\) and the boundary operator \(B\) are nonlocal and associated with smooth self-mappings of the manifold \(M.\)
The authors calculate the trajectory symbols of such problems and, by an analogue of the Shapiro–Lopatinskii condition, derive the corresponding finiteness theorem.
MSC:
58J32 Boundary value problems on manifolds
47A53 (Semi-) Fredholm operators; index theories
35J30 Higher-order elliptic equations
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References:
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