Fredholm properties of singular elliptic operators arising in the study of point defects

Abstract

Motivated by the dynamics of defects in planar pattern-forming systems, we study Fredholm properties of elliptic operators with singular coefficients in weighted Sobolev spaces. In particular, we consider a family of doubly weighted spaces that encode algebraic decay/growth of functions at infinity, and near the origin. Our results give conditions on the weights under which the operators are either injective, surjective, or isomorphisms. We also give a precise description of the kernel and range of these operators.