The Mixed Problem for the Laplacian in Lipschitz Domains

Katharine A. Ott; Russell M. Brown

doi:10.1007/s11118-012-9317-6

The Mixed Problem for the Laplacian in Lipschitz Domains

Katharine A. Ott, Russell M. Brown

Mathematics

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

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 L^p. Thus, if we specify Dirichlet data on D in the Sobolev space W^1,p(D) and Neumann data on N in L^p (N), the mixed problem with data f_D and f_N has a unique solution and the non-tangential maximal function of the gradient lies in L^p(∂Ω). We also obtain results for p = 1 when the data comes from Hardy spaces.

Original language	English
Pages (from-to)	1333-1364
Number of pages	32
Journal	Potential Analysis
Volume	38
Issue number	4
DOIs	https://doi.org/10.1007/s11118-012-9317-6
State	Published - 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).

Keywords

Laplacian
Mixed boundary value problem
Non-smooth domain

ASJC Scopus subject areas

Analysis

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10.1007/s11118-012-9317-6

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title = "The Mixed Problem for the Laplacian in Lipschitz Domains",

abstract = "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.",

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AU - Ott, Katharine A.

AU - Brown, Russell M.

N1 - 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).

PY - 2013/5

Y1 - 2013/5

N2 - 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.

AB - 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.

KW - Laplacian

KW - Mixed boundary value problem

KW - Non-smooth domain

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The Mixed Problem for the Laplacian in Lipschitz Domains

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Keywords

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