Elliptic boundary problems on noncompact manifolds

Thomas Krainer, PennState-Altoona

Abstract: We discuss Fredholm criteria and regularity results for elliptic boundary value problems on a particular class of noncompact manifolds. A simple example for our setup is Euclidean space with a noncompact obstacle removed, and the stationary Schroedinger operator with a (complex) potential and boundary conditions on the boundary of the obstacle.

More generally, the operators under consideration may be regarded as cusp operators on manifolds with corners after suitable compactification of the noncompact ends, and boundary conditions are imposed on some of the boundary hypersurfaces.

Cusp operators (with cusp degeneracy on all boundary hypersurfaces) were introduced by Richard Melrose and Victor Nistor in 1996 in the context of the index problem on manifolds with boundary (unpublished), and in the case of higher codimensions by Robert Lauter and Sergiu Moroianu (2002).