Power- and Log-concavity of viscosity solutions to some elliptic Dirichlet problems

Michael Kühn

doi:10.3934/cpaa.2018131

Article Contents

2018, Volume 17, Issue 6: 2773-2788. Doi: 10.3934/cpaa.2018131

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Power- and Log-concavity of viscosity solutions to some elliptic Dirichlet problems

Michael Kühn

Mathematisches Institut der Universität zu Köln, Weyertal 86-90, Cologne, 50931, Germany

Received: September 30, 2017

Revised: January 31, 2018

Published: June 2018

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Abstract

In this article we consider a special type of degenerate elliptic partial differential equations of second order in convex domains that satisfy the interior sphere condition. We show that any positive viscosity solution $u$ of

$-|\nabla u|^α Δ^N_p u = 1$

has the property that $u^\frac{α+1}{α+2}$ is a concave function. Secondly we consider positive solutions of the eigenvalue problem

$-|\nabla u|^α Δ^N_p u = λ |u|^α u, $

in which case $\log u$ turns out to be concave. The methods provided include a weak comparison principle and a Hopf-type Lemma.

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Mathematics Subject Classification: 49K20; Secondary: 35D40, 35J60, 35J70.

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