Optimal solvability for the Dirichlet and Neumann problem in dimension two

2000/12/27 by Atanas Stefanov, Stefanov, Atanas, Gregory Verchota +2
Computer Science · Mathematics · #35J25 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.CA #msc:35J25

paper · pdf · doi:10.48550/arxiv.math/0012254

arxiv created 2000/12/27 · openalex publication_date 2000/12/27 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We show existence and uniqueness for the solutions of the regularity and the Neumann problems for harmonic functions on Lipschitz domains with data in the Hardy spaces H^p, p>2/3, where This in turn implies that solutions to the Dirichlet problem with data in the Holder class C^1/2(∂ D) are themselves in C^1/2(D). Both of these results are sharp. In fact, we prove a more general statement regarding the H^p solvability for divergence form elliptic equations with bounded measurable coefficients. We also prove similar solvability result for the regularity and Dirichlet problem for the biharmonic equation on Lipschitz domains.

Citations

• The neumann problem for elliptic equations with non-smooth coefficients

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