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

Martin Costabel
2017 arXiv   pre-print
We construct a bounded C^1 domain Ω in R^n for which the H^3/2 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 Δ u=f in Ω and either u=0 on ∂Ω or ∂_n u=0 on ∂Ω is contained in H^3/2(Ω) but not in H^3/2+ε(Ω) for any ϵ>0. An analogous result holds for L^p Sobolev spaces with p∈(1,∞).
arXiv:1711.07179v1 fatcat:6eclaya35vdghctxcwulma3lxq