Fredholm properties of singular elliptic operators arising in the study of point defects
Gabriela Jaramillo
TL;DR
This work addresses the Fredholm theory for singular elliptic operators in planar settings by introducing doubly weighted Sobolev spaces $H^{s}_{\sigma,\gamma}(\mathbb{R}^2)$ and $M^{s,p}_{\sigma,\gamma}(\mathbb{R}^2)$ to control behavior both near the origin and at infinity. It employs an angular Fourier decomposition to reduce the problems to radial ODEs, enabling precise descriptions of kernels and cokernels for operators $\Delta - \mathrm{Id}$ and $\Delta - \tfrac{1}{r^2}$ (and corresponding shifted operators), with explicit bases built from modified Bessel functions and power-type functions. The main contributions are sharp Fredholm criteria in terms of $(\sigma,\gamma)$, complete characterizations of kernels and cokernels via angular modes, and Green’s-function–style inverse constructions that underpin Lyapunov–Schmidt reductions for point defects in pattern-forming systems. These results advance rigorous defect analysis in unbounded domains and connect to classical weighted-elliptic theory by McOwen and Lockhart while addressing singular behavior at the origin relevant to defects. The findings have implications for core/far-field matching in models like Ginzburg–Landau and Swift–Hohenberg, facilitating rigorous existence proofs for defect-bearing patterns.
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.