Optimal Hardy inequalities in cones

Let Ω be an open connected cone in ℝ n with vertex at the origin. Assume that the Operator is subcritical in Ω, where δΩ is the distance function to the boundary of Ω and μ ⩽ 1/4. We show that under some smoothness assumption on Ω the improved Hardy-type inequality holds true, and the Hardy-weight λ...

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Published inProceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 147; no. 1; pp. 89 - 124
Main Authors Devyver, Baptiste, Pinchover, Yehuda, Psaradakis, Georgios
Format Journal Article
LanguageEnglish
Published Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.02.2017
Cambridge University Press
Subjects
Boundaries
Cones
Inequalities
Mathematical functions
Operators
Optimization
Origins
Smoothness
minimal growth
positive solutions
35J20
35P05
Hardy inequality
Secondary 35B09
Primary 35A23
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Summary:Let Ω be an open connected cone in ℝ n with vertex at the origin. Assume that the Operator is subcritical in Ω, where δΩ is the distance function to the boundary of Ω and μ ⩽ 1/4. We show that under some smoothness assumption on Ω the improved Hardy-type inequality holds true, and the Hardy-weight λ(μ)|x|–2 is optimal in a certain definite sense. The constant λ(μ) > 0 is given explicitly.
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ISSN:0308-2105
1473-7124
DOI:10.1017/S0308210516000056