Convexity of the smallest principal curvature of the convex level sets of some quasi-linear elliptic equations with respect to the height

TL;DR

This paper proves the convexity of the smallest principal curvature of convex level sets of certain quasi-linear elliptic equations, including p-harmonic functions and minimal graphs, with respect to the function's height.

Contribution

It introduces a new test function combining gradient norm and curvature, demonstrating convexity of the curvature function relative to the height.

Findings

Curvature function is convex in the height for p-harmonic functions.

Test function is affine in the height for p-Green functions.

Similar convexity results are obtained for minimal graphs.

Abstract

For the $p$-harmonic function with strictly convex level sets, we find a test function which comes from the combination of the norm of gradient of the $p$-harmonic function and the smallest principal curvature of the level sets of $p$-harmonic function. We prove that this curvature function is convex with respect to the height of the $p$-harmonic function. This test function is an affine function of the height when the $p$-harmonic function is the $p$-Green function on the ball. For the minimal graph, we obtain a similar results.