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

TL;DR

This paper investigates the Fredholm properties of singular elliptic operators in weighted Sobolev spaces, providing conditions for injectivity, surjectivity, and isomorphism, with detailed kernel and range descriptions.

Contribution

It introduces a family of doubly weighted spaces tailored for analyzing elliptic operators with singular coefficients, advancing understanding of their Fredholm properties.

Findings

Conditions for injectivity, surjectivity, and isomorphism of operators.

Explicit descriptions of kernels and ranges.

Framework for analyzing elliptic operators with singularities.

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.