Besov regularity for operator equations on patchwise smooth manifolds

Abstract

We study regularity properties of solutions to operator equations on patchwise smooth manifolds $\partialΩ$ such as, e.g., boundaries of polyhedral domains $Ω\subset \mathbb{R}^3$. Using suitable biorthogonal wavelet bases $Ψ$, we introduce a new class of Besov-type spaces $B_{Ψ,q}^α(L_p(\partial Ω))$ of functions $u\colon\partialΩ\rightarrow\mathbb{C}$. 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 $\partialΩ$ into $B_{Ψ,τ}^α(L_τ(\partial Ω))$, $1/τ=α/2 + 1/2$, 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 $Ω$.