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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)

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