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

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  • 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.

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


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