The Concavity of the Gaussian Curvature of the Convex Level Sets of $p$-Harmonic Functions with Respect to the Height

Abstract

For the $p$-harmonic function with strictly convex level sets, we find an auxiliary function which comes from the combination of the norm of gradient of the $p$-harmonic function and the Gaussian curvature of the level sets of $p$-harmonic function. We prove that this curvature function is concave with respect to the height of the $p$-harmonic function. This auxiliary function is an affine function of the height when the $p$-harmonic function is the $p$-Green function on ball.

Keywords

$p$-harmonic function')">$p$-harmonic function / Level set / Gaussian curvature / Support function / Maximum prinicple

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