A geometric characterization of a sharp Hardy inequality

Roger T. Lewis; Junfang Li; Yanyan Li

doi:10.1016/j.jfa.2012.01.015

A geometric characterization of a sharp Hardy inequality

Roger T. Lewis, Junfang Li, Yanyan Li

School of Arts and Sciences, Mathematics

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

35 Scopus citations

Abstract

In this paper, we prove that the distance function of an open connected set in R ⁿ⁺¹ with a C ² boundary is superharmonic in the distribution sense if and only if the boundary is weakly mean convex. We then prove that Hardy inequalities with a sharp constant hold on weakly mean convex C ² domains. Moreover, we show that the weakly mean convexity condition cannot be weakened. We also prove various improved Hardy inequalities on mean convex domains along the line of Brezis and Marcus (1997) [7].

Original language	English (US)
Pages (from-to)	3159-3185
Number of pages	27
Journal	Journal of Functional Analysis
Volume	262
Issue number	7
DOIs	https://doi.org/10.1016/j.jfa.2012.01.015
State	Published - Apr 1 2012

All Science Journal Classification (ASJC) codes

Analysis

Keywords

Distance function
Hardy inequality
Superharmonic

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10.1016/j.jfa.2012.01.015

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title = "A geometric characterization of a sharp Hardy inequality",

abstract = "In this paper, we prove that the distance function of an open connected set in R n+1 with a C 2 boundary is superharmonic in the distribution sense if and only if the boundary is weakly mean convex. We then prove that Hardy inequalities with a sharp constant hold on weakly mean convex C 2 domains. Moreover, we show that the weakly mean convexity condition cannot be weakened. We also prove various improved Hardy inequalities on mean convex domains along the line of Brezis and Marcus (1997) [7].",

keywords = "Distance function, Hardy inequality, Superharmonic",

author = "Lewis, {Roger T.} and Junfang Li and Yanyan Li",

note = "Funding Information: ✩ Research of the second author was supported in part by NSF DMS-1007223. Research of the third author was supported in part by NSF DMS-0701545. * Corresponding author. E-mail addresses: [email protected] (R.T. Lewis), [email protected] (J.F. Li), [email protected] (Y.Y. Li).",

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doi = "10.1016/j.jfa.2012.01.015",

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T1 - A geometric characterization of a sharp Hardy inequality

AU - Lewis, Roger T.

AU - Li, Junfang

AU - Li, Yanyan

N1 - Funding Information: ✩ Research of the second author was supported in part by NSF DMS-1007223. Research of the third author was supported in part by NSF DMS-0701545. * Corresponding author. E-mail addresses: [email protected] (R.T. Lewis), [email protected] (J.F. Li), [email protected] (Y.Y. Li).

PY - 2012/4/1

Y1 - 2012/4/1

N2 - In this paper, we prove that the distance function of an open connected set in R n+1 with a C 2 boundary is superharmonic in the distribution sense if and only if the boundary is weakly mean convex. We then prove that Hardy inequalities with a sharp constant hold on weakly mean convex C 2 domains. Moreover, we show that the weakly mean convexity condition cannot be weakened. We also prove various improved Hardy inequalities on mean convex domains along the line of Brezis and Marcus (1997) [7].

AB - In this paper, we prove that the distance function of an open connected set in R n+1 with a C 2 boundary is superharmonic in the distribution sense if and only if the boundary is weakly mean convex. We then prove that Hardy inequalities with a sharp constant hold on weakly mean convex C 2 domains. Moreover, we show that the weakly mean convexity condition cannot be weakened. We also prove various improved Hardy inequalities on mean convex domains along the line of Brezis and Marcus (1997) [7].

KW - Distance function

KW - Hardy inequality

KW - Superharmonic

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EP - 3185

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 7

ER -

A geometric characterization of a sharp Hardy inequality

Abstract

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