Maximum principles for the polyharmonic equation on Lipschitz domains

J. Pipher; G. C. Verchota

doi:10.1007/BF02345828

Maximum principles for the polyharmonic equation on Lipschitz domains

J. Pipher, G. C. Verchota

Department of Mathematics

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

28 Scopus citations

Abstract

The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in ℝ³. In ℝn, n≥4, the Agmon-Miranda maximum principle and L^p-Dirichlet estimates for certain p>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2 m+1 the L^p Dirichlet problem for Δ^m fails to be solvable for p>2(n−1)/(n−3).

Original language	English (US)
Pages (from-to)	615-636
Number of pages	22
Journal	Potential Analysis: An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis
Volume	4
Issue number	6
DOIs	https://doi.org/10.1007/BF02345828
State	Published - Dec 1995

Keywords

35J
Agmon-Miranda
Nonsmooth
dilation invariant
implicit functional

ASJC Scopus subject areas

Analysis

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10.1007/BF02345828

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abstract = "The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in ℝ3. In ℝn, n≥4, the Agmon-Miranda maximum principle and Lp-Dirichlet estimates for certain p>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2 m+1 the Lp Dirichlet problem for Δm fails to be solvable for p>2(n−1)/(n−3).",

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author = "J. Pipher and Verchota, {G. C.}",

year = "1995",

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T1 - Maximum principles for the polyharmonic equation on Lipschitz domains

AU - Pipher, J.

AU - Verchota, G. C.

PY - 1995/12

Y1 - 1995/12

N2 - The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in ℝ3. In ℝn, n≥4, the Agmon-Miranda maximum principle and Lp-Dirichlet estimates for certain p>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2 m+1 the Lp Dirichlet problem for Δm fails to be solvable for p>2(n−1)/(n−3).

AB - The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in ℝ3. In ℝn, n≥4, the Agmon-Miranda maximum principle and Lp-Dirichlet estimates for certain p>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2 m+1 the Lp Dirichlet problem for Δm fails to be solvable for p>2(n−1)/(n−3).

KW - 35J

KW - Agmon-Miranda

KW - Nonsmooth

KW - dilation invariant

KW - implicit functional

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SP - 615

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JO - Potential Analysis: An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis

JF - Potential Analysis: An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis

IS - 6

ER -

Maximum principles for the polyharmonic equation on Lipschitz domains

Abstract

Keywords

ASJC Scopus subject areas

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