Besov regularity for operator equations on patchwise smooth manifolds

• Published in: Foundations of computational mathematics Vol. 15; no. 6; pp. 1533 - 1569
• Main Authors: Dahlke, Stephan, Weimar, Markus
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2015; Springer
• Subjects: Analysis; Applications of Mathematics; Computer Science; Economics; Linear and Multilinear Algebras; Manifolds (Mathematics); Math Applications in Computer Science; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Transformations (Mathematics); United States; Double layer; 30H25; Besov spaces; Manifolds; Wavelets; 45E99; 47B38; Adaptive methods; 46E35; Integral equations; 42C40; 35B65; Regularity; 65T60; Weighted Sobolev spaces; Nonlinear approximation
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-015-9273-9

We study regularity properties of solutions to operator equations on patchwise smooth manifolds  ∂ Ω , e.g., boundaries of polyhedral domains Ω ⊂ R 3 . Using suitable biorthogonal wavelet bases Ψ , we introduce a new class of Besov-type spaces B Ψ , q α ( L p ( ∂ Ω ) ) of functions 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 ∂ Ω into B Ψ , τ α ( L τ ( ∂ 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 .