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Optimal Hardy inequalities in cones

Part of: Partial differential equations Spectral theory and eigenvalue problems Elliptic equations and systems

Published online by Cambridge University Press:  04 November 2016

Baptiste Devyver ,
Yehuda Pinchover  and
Georgios Psaradakis
Baptiste Devyver
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2 , Canada (devyver@math.ubc.ca)
Yehuda Pinchover
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel (pincho@techunix.technion.ac.il; georgios@techunix.technion.ac.il)
Georgios Psaradakis
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel (pincho@techunix.technion.ac.il; georgios@techunix.technion.ac.il)
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Let Ω be an open connected cone in ℝn 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.

Keywords

Hardy inequalityminimal growthpositive solutions

MSC classification

Secondary: 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals 35B09: Positive solutions 35J20: Variational methods for second-order elliptic equations 35P05: General topics in linear spectral theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 1 , February 2017 , pp. 89 - 124
Copyright
Copyright © Royal Society of Edinburgh 2017