End-point non-regularity of the Dirichlet problem on Lipschitz domains -- An elementary proof

Abstract

We construct a plane Lipschitz domain for which the range of the Laplace operator from $H^{3/2}\cap H_0^1$ to $H^{-1/2}$ is not closed. We show that rapid oscillations of the boundary create a boundary layer that leads to unbounded $H^{3/2}$ norm of the solution of the Dirichlet problem. Our proof is elementary in the sense that it is directly based on homogeneity arguments and does not use deep results of harmonic analysis such as estimates of area integrals or harmonic measures of Dahlberg or Jerison and Kenig.

arXiv comment21 pages, 6 figures