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

Loredana Lanzani, Zhongwei Shen

Research output: Contribution to journalArticle

34 Citations (Scopus)

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 - 2004
Externally publishedYes

Fingerprint

Robin Boundary Conditions
Lipschitz Domains
Laplace equation
Laplace's equation
Boundary conditions
Single Layer Potential
Pointwise Estimates
Maximal Function
Compatibility Conditions
Existence and Uniqueness Results
Bounded Domain
Non-negative
Estimate

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Applied Mathematics

Cite this

On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains. / Lanzani, Loredana; Shen, Zhongwei.

In: Communications in Partial Differential Equations, Vol. 29, No. 1-2, 2004, p. 91-109.

Research output: Contribution to journalArticle

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