The Mixed Problem for the Laplacian in Lipschitz Domains

Katharine A. Ott, Russell M. Brown

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


We consider the mixed boundary value problem, or Zaremba's problem, for the Laplacian in a bounded Lipschitz domain Ω in Rn, n ≥ 2. We decompose the boundary ∂Ω= D ∪ N with D and N disjoint. The boundary between D and N is assumed to be a Lipschitz surface in ∂Ω. We find an exponent q0 > 1 so that for p between 1 and q0 we may solve the mixed problem for Lp. Thus, if we specify Dirichlet data on D in the Sobolev space W1,p(D) and Neumann data on N in Lp (N), the mixed problem with data fD and fN has a unique solution and the non-tangential maximal function of the gradient lies in Lp(∂Ω). We also obtain results for p = 1 when the data comes from Hardy spaces.

Original languageEnglish
Pages (from-to)1333-1364
Number of pages32
JournalPotential Analysis
Issue number4
StatePublished - May 2013

Bibliographical note

Funding Information:
Research supported, in part, by the National Science Foundation.

Funding Information:
This work was partially supported by a grant from the Simons Foundation (#195075 to Russell Brown).


  • Laplacian
  • Mixed boundary value problem
  • Non-smooth domain

ASJC Scopus subject areas

  • Analysis


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