On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

Loredana Lanzani, Zhongwei Shen

Research output: Contribution to journalArticle

34 Scopus citations

Abstract

Let Ω be a bounded Lipschitz domain in Rn, n ≥ 3 with connected boundary. We study the Robin boundary condition ∂u/∂N + bu = f ∈ Lp(∂Ω) on ∂Ω for Laplace's equation δu = 0 in Ω, where b is a non-negative function on ∂Ω. For 1 < p < 2 + ε, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ∥(∇u)*∥p ≤ C∥f∥p, as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.

Original languageEnglish (US)
Pages (from-to)91-109
Number of pages19
JournalCommunications in Partial Differential Equations
Volume29
Issue number1-2
StatePublished - Jan 1 2004
Externally publishedYes

Keywords

  • Laplace's equation
  • Lipschitz domains
  • Robin boundary condition

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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