Besov regularity for operator equations on patchwise smooth manifolds

2013/12/10 by Stephan Dahlke, Dahlke, Stephan, Markus Weimar +1
Computer Science · Mathematics · #30H25 #35B65 #42C40 #45E99 #46E35 #47B38 #65T60 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1312.2734

openalex publication_date 2013/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study regularity properties of solutions to operator equations on patchwise smooth manifolds ∂Ω such as, e.g., boundaries of polyhedral domains Ω⊂ ℝ³. 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\colon∂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.

Citations

• Almost diagonal matrices and Besov-type spaces based on wavelet\n expansions

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