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Elliptic translators on manifolds with multidimensional singularities. (English. Russian original) Zbl 1274.58006
Differ. Equ. 49, No. 4, 494-509 (2013); translation from Differ. Uravn. 49, No. 4, 513-527 (2013).
This paper deals with translators \(1+ T:H^s(Y,E)\to H^s(Y,E)\), acting on vector bundles. In general, the kernel and the cokernel of \(I+T\) are infinite-dimensional spaces. To ensure the Fredholm property of the problem under consideration, one should “rig a translator”, i.e., add a certain number of boundary and coboundary conditions on some submanifold \(X\). The authors define a rigging operator (translational morphism) \[ D_X= \begin{pmatrix} I+ T & C_{YX}\\ B_{XY} & D_X\end{pmatrix}, \] \(B_{XY}\) being a boundary operator, \(C_{YX}\) a coboundary operator, \(D_X\) is a pseudodifferential operator on \(X\), and its ellipticity. They prove the Fredholm property of \(JD_X\) in Theorem 1 and establish an index formula. In Theorem 3 it is shown that, for all \(s\) except for some finite set of singular exponents, there exists an ellipticity rigging of the translator \(I+T\) of the form \(D_X\).

58J32 Boundary value problems on manifolds
58J05 Elliptic equations on manifolds, general theory
58J20 Index theory and related fixed-point theorems on manifolds
35S15 Boundary value problems for PDEs with pseudodifferential operators
Full Text: DOI
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