Convergence of layer potentials and Riemann-Hilbert problem on extension domains

TL;DR

This paper establishes the convergence of layer potential operators and related boundary integral operators for harmonic problems on sequences of extension domains, generalizing classical concepts like Cauchy integrals.

Contribution

It introduces a framework for convergence analysis of boundary operators on extension domains using dyadic approximations and generalizes fundamental integral transforms.

Findings

Layer potential operators converge on extension domains.

Neumann-Poincaré operators and Calderón projectors converge.

Generalization of Cauchy integrals and Hilbert transforms for extension domains.

Abstract

We prove the convergence of layer potential operators for the harmonic transmission problem over a sequence of converging two-sided extension domains. Consequently, the Neumann-Poincar{\'e} operators, Calder{\'o}n projectors, and associated Neumann series converge in this setting. As a result, we generalize the notion of Cauchy integrals and, in a sense, of Hilbert transforms for a class of extension domains. Our approach relies on dyadic approximations of arbitrary open sets, considering convergence in terms of characteristic functions, Hausdorff distance, and compact sets.