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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 $$D_X$$.

##### MSC:
 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
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##### References:
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