Besov regularity for operator equations on patchwise smooth manifolds

Abstract

We study regularity properties of solutions to operator equations on patchwise smooth manifolds $$\partial \Omega $$?Ω, e.g., boundaries of polyhedral domains $$\Omega \subset \mathbb {R}^3$$Ω?R3. Using suitable biorthogonal wavelet bases $$\Psi $$?, we introduce a new class of Besov-type spaces $$B_{\Psi ,q}^\alpha (L_p(\partial \Omega ))$$B?,q?(Lp(?Ω)) of functions $$u:\partial \Omega \rightarrow \mathbb {C}$$u:?Ω?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 $$\partial \Omega $$?Ω into $$B_{\Psi ,\tau }^\alpha (L_\tau (\partial \Omega )), 1/\tau =\alpha /2 + 1/2$$B?,??(L?(?Ω)),1/?=?/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 $$\Omega $$Ω.