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On general boundary-value problems for elliptic equations. (English. Russian original) Zbl 0926.35033
Sb. Math. 189, No. 10, 1573-1586 (1998); translation from Mat. Sb. 189, No. 10, 145-160 (1998).
Let $$M$$ be a compact manifold with a smooth boundary $$X:=\partial M$$. A “general” boundary value problem in the sense of the title is given by an elliptic operator $$\widehat D: H^s(M,E)\to H^{s-m}(M,F)$$ of order $$m$$ ($$E$$, $$F$$ are vector bundles over $$M$$ and $$H^s$$ denote $$L^2$$-Sobolev spaces) and a boundary operator $$\widehat B$$ mapping the jets $$j^{m-1}_X u\in\bigoplus_j H^{s-1/2-j}(X,i^*E)$$ of $$u\in H^s(M,E)$$ into some Banach space $${\mathcal L}$$. Thus $$\widehat B$$ replaces the pseudodifferential operators in the “classical” boundary value problems. The authors arrive at generalized Shapiro-Lopatinskij conditions which ensure the Fredholm property of the pair $$(\widehat D,\widehat B)$$. They remark that each elliptic operator $$\widehat D$$ may be augmented by a boundary operator $$\widehat B$$ such that this holds for $$(\widehat D,\widehat B)$$ whereas in the classical setting this is not always possible. This is illustrated by two examples (Cauchy-Riemann and Euler operator).
Reviewer: N.Weck (Essen)

##### MSC:
 35J40 Boundary value problems for higher-order elliptic equations 58J32 Boundary value problems on manifolds
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