an:06200328
Zbl 1274.58006
Savin, A. Yu.; Sternin, B. Yu.
Elliptic translators on manifolds with multidimensional singularities
EN
Differ. Equ. 49, No. 4, 494-509 (2013); translation from Differ. Uravn. 49, No. 4, 513-527 (2013).
0012-2661 1608-3083
2013
j
58J32 58J05 58J20 35S15
manifold with multidimensional singularities; elliptic boundary value problem; index formula; elliptic translator
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 \(D_X\).
Petar Popivanov (Sofia)