On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains

Martin Costabel

Mathematische Nachrichten, 2019, vol. 292, issue 10, 2165-2173

Abstract: We construct a bounded C1 domain Ω in Rn for which the H3/2 regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists f in C∞(Ω¯) such that the solution of Δu=f in Ω and either u=0 on ∂Ω or ∂nu=0 on ∂Ω is contained in H3/2(Ω) but not in H3/2+ε(Ω) for any ε>0. An analogous result holds for Lp Sobolev spaces with p∈(1,∞).

Date: 2019
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