On the Well-posedness of the Dirichlet Problem in Certain Classes of Nontangentially Accessible Domains

Abstract

We prove that if Ω ⊂ ℝⁿ is a bounded NTA domain (in the sense of Jerison and Kenig) with an Ahlfors regular boundary, and which satisfies a uniform exterior ball condition, then the Dirichlet problem

$$ \Delta u = 0 in \Omega , u|_{\partial \Omega } = f \in L^p (\partial \Omega , d\sigma ), $$

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, has a unique solution for any p ∈ (1,∞). This solution satisfies natural nontangential maximal function estimates and can be represented as

$$ u(y) = - \int_{\partial \Omega } {\partial _{\nu (x)} G(x,y)f(x) d\sigma (x),} y \in \Omega . $$

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. Above, ν denotes the outward unit normal to Ω and G(·,·) stands for the Green function associated with Ω.