zbMATH — the first resource for mathematics

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
Full Text:
References:
 [1] A. Antonevich and A. Lebedev, Functional-Differential Equations, 1:C*-Theory (Longman, Harlow, 1994). · Zbl 0799.34001 [2] Rossovskii, L. E., No article title, Tr. Mosk. Mat. O-va, 62, 199-228, (2001) [3] Bitsadze, A. V.; Samarskii, A. A., No article title, Dokl. Akad. Nauk SSSR, 185, 739-740, (1969) [4] Savin, A. Yu., No article title, Differ. Equations, 47, 890-893, (2011) · Zbl 1236.58034 [5] A. Savin and B. Sternin, in Pseudo-Differential Operators, Generalized Functions and Asymptotics (Birkhäuser, Basel, 2013), pp. 1-26. · Zbl 1269.58009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.