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.

Convexity of level sets of minimal graph on space form with nonnegative curvature

“…For Poisson equations and a class of semilinear elliptic partial differential equations, ) concluded that the level sets of their solutions are all convex with respect to the gradient direction, the curvature estimate of the level sets has been got by ), and in the same paper they also described the geometrical properties of the level sets of the minimal graph. In the sequel, following the technique in [28], Wang([38]) got the precise relation between the curvature of the convex level sets and the height of minimal graph of general dimensions which generalized the previous results of Longinetti ([25]).…”

“…The curvature estimate of the level sets of the solution to partial differential equations then have no new progress until recently, Ma-Ou-Zhang( [27]) got the Gaussian curvature estimates of the convex level sets of harmonic functions which depend on the Gaussian curvature of the boundary and the norm of the gradient on the boundary in R n . Furthermore, in [28] the concavity of the Gaussian curvature of the convex level sets of p-harmonic functions with respect to the height was derived to describe the variation of the curvature along the height of the function. In [18], the lower bound of the principal curvature of the convex level sets of the solution to a kind of fully nonlinear elliptic equations was derived.…”

Curvature estimates for the level sets of spatial quasiconcave solutions to a class of parabolic equations

“…Both of their test functions involved the Gaussian curvature of the boundary and the norm of the gradient on the boundary. Furthermore, for the p-harmonic function with strictly convex level sets, Ma-Zhang [19] obtained that the curvature function introduced in it is concave with respect to the height of the p-harmornic function. For the principal curvature estimates in higher dimension, in terms of the principal curvature of the boundary and the norm of the gradient on the boundary, Chang-Ma-Yang [8] obtained the lower bound estimates of principal curvature for the strictly convex level sets of higher dimensional harmonic functions and solutions to a class of semilinear elliptic equations under certain structural conditions in [4].…”