Besov regularity for operator equations on patchwise smooth manifolds

TL;DR

This paper investigates the regularity of solutions to boundary operator equations on patchwise smooth manifolds, introducing Besov-type spaces based on wavelet bases to analyze approximation rates and improve adaptive numerical methods.

Contribution

It introduces a new class of Besov-type spaces tailored for patchwise smooth manifolds and establishes their embedding properties, enhancing understanding of solution regularity for boundary integral equations.

Findings

Established embeddings of Sobolev spaces into Besov-type spaces

Derived regularity results for boundary integral equations

Analyzed wavelet approximation rates for adaptive schemes

Abstract

We study regularity properties of solutions to operator equations on patchwise smooth manifolds $\partial\Omega$ such as, e.g., boundaries of polyhedral domains $\Omega \subset \mathbb{R}^3$. Using suitable biorthogonal wavelet bases $\Psi$, we introduce a new class of Besov-type spaces $B_{\Psi,q}^\alpha(L_p(\partial \Omega))$ of functions $u\colon\partial\Omega\rightarrow\mathbb{C}$. Special attention is paid on the rate of convergence for best $n$-term wavelet approximation to functions in these scales since this determines the performance of adaptive numerical schemes. We show embeddings of (weighted) Sobolev spaces on $\partial\Omega$ into $B_{\Psi,\tau}^\alpha(L_\tau(\partial \Omega))$, $1/\tau=\alpha/2 + 1/2$, which lead us to regularity assertions for the equations under consideration. Finally, we apply our results to a boundary integral equation of the second kind which arises from the double layer ansatz for Dirichlet problems for Laplace's equation in $\Omega$.