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The inhomogeneous Dirichlet problem in Lipschitz domains. (English) Zbl 0832.35034
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.
Reviewer: H.Ding (Beijing)

MSC:
35J25 Boundary value problems for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
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