Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains

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Abstract

For D, a bounded Lipschitz domain in Rn, n ≥ 2, the classical layer potentials for Laplace's equation are shown to be invertible operators on L2(∂D) and various subspaces of L2(∂D). For 1 < p ≤ 2 and data in Lp(∂D) with first derivatives in Lp(∂D) it is shown that there exists a unique harmonic function, u, that solves the Dirichlet problem for the given data and such that the nontangential maximal function of ▽u is in Lp(∂D). When n = 2 the question of the invertibility of the layer potentials on every Lp(∂D), 1 < p < ∞, is answered.

Original languageEnglish (US)
Pages (from-to)572-611
Number of pages40
JournalJournal of Functional Analysis
Volume59
Issue number3
DOIs
StatePublished - Dec 1984
Externally publishedYes

ASJC Scopus subject areas

  • Analysis

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