The inhomogeneous Dirichlet problem in Lipschitz domains
The inhomogeneous Dirichlet problem \[ \Delta u= f\qquad\text{on}\quad \Omega,\qquad u= 0\qquad\text{on}\quad \partial\Omega, \] with data in Sobolev spaces of domains \(\Omega\) in \(\mathbb{R}^n\) with Lipschitz boundary, is studied. There are certain special difficulties in this kind of domains, e.g. the customary approach of reducing the inhomogeneous problem to a homogeneous one, in the case of smooth domains, can fail in the case of Lipschitz domains for certain exceptional function spaces. In this work, the authors present a complete description of all Sobolev spaces for which estimates hold, and the best possible estimates in Besov spaces. Some counterexamples in borderline cases are also constructed.
scientific article; zbMATH DE number 764042