Hölder continuity of ρ(x)-superharmonic functions

A. Lyaghfouri

doi:10.1016/j.na.2010.06.016

Hölder continuity of ρ(x)-superharmonic functions

A. Lyaghfouri

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

13 Citations (Scopus)

Abstract

In this paper we show that a solution of the equation -Δ _ρ(x)u = μ is Hölder continuous with exponent α if and only if the nonnegative Radon measure μ satisfies the growth condition μ(Br(x))≤ Crn-^{ρ(x)+α(ρ(x)-1)} for any ball Br(x)⊂ Ω, with r small enough. This extends an old result of Lewy and Stampacchia for the Laplace operator, and a recent result of Kilpelinen and Zhong for the p-Laplace operator with p constant.

Original language	English
Pages (from-to)	2433-2444
Number of pages	12
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	73
Issue number	8
DOIs	https://doi.org/10.1016/j.na.2010.06.016
Publication status	Published - Oct 15 2010
Externally published	Yes

Keywords

ρ(x)-Laplace operator
ρ(x)-superharmonic functions
Hölder continuity
Radon measure
Variable exponent Sobolev spaces

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.na.2010.06.016

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abstract = "In this paper we show that a solution of the equation -Δ ρ(x)u = μ is H{\"o}lder continuous with exponent α if and only if the nonnegative Radon measure μ satisfies the growth condition μ(Br(x))≤ Crn-ρ(x)+α(ρ(x)-1) for any ball Br(x)⊂ Ω, with r small enough. This extends an old result of Lewy and Stampacchia for the Laplace operator, and a recent result of Kilpelinen and Zhong for the p-Laplace operator with p constant.",

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PY - 2010/10/15

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N2 - In this paper we show that a solution of the equation -Δ ρ(x)u = μ is Hölder continuous with exponent α if and only if the nonnegative Radon measure μ satisfies the growth condition μ(Br(x))≤ Crn-ρ(x)+α(ρ(x)-1) for any ball Br(x)⊂ Ω, with r small enough. This extends an old result of Lewy and Stampacchia for the Laplace operator, and a recent result of Kilpelinen and Zhong for the p-Laplace operator with p constant.

AB - In this paper we show that a solution of the equation -Δ ρ(x)u = μ is Hölder continuous with exponent α if and only if the nonnegative Radon measure μ satisfies the growth condition μ(Br(x))≤ Crn-ρ(x)+α(ρ(x)-1) for any ball Br(x)⊂ Ω, with r small enough. This extends an old result of Lewy and Stampacchia for the Laplace operator, and a recent result of Kilpelinen and Zhong for the p-Laplace operator with p constant.

KW - ρ(x)-Laplace operator

KW - ρ(x)-superharmonic functions

KW - Hölder continuity

KW - Radon measure

KW - Variable exponent Sobolev spaces

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SP - 2433

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JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 8

ER -

Hölder continuity of ρ(x)-superharmonic functions

Abstract

Keywords

ASJC Scopus subject areas

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