The Concavity of the Gaussian Curvature of the Convex Level Sets of $$p$$ 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.