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

2025/05/02 by Gabriela Jaramillo, Jaramillo, Gabriela
Computer Science · Engineering · Mathematics · #35J10 #46N20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2505.01534

openalex publication_date 2025/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Pinning and dipole asymptotics of locally deformed striped phases
• A rigorous derivation of the asymptotic wavenumber of spiral wave solutions of the complex Ginzburg-Landau equation
• Rotating spirals in oscillatory media with nonlocal interactions and their normal form
• Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations

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