Gaussian curvature estimates for the convex level sets of p‐harmonic functions

Abstract

We give a positive lower bound for the Gaussian curvature of the convex level sets of p-harmonic functions with the Gaussian curvature of the boundary and the norm of the gradient on the boundary. Combining the deformation process, this estimate gives a new approach to studying the convexity of the level sets of the p-harmonic function.

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

“…Under the same assumptions of Theorem 1.2, Ma-Ou-Zhang [19] proved the following statement: f or n ≥ 2 and 1 < p < +∞, the function |∇u| n+1−2 p K attains its minimum on the boundary.…”

“…The estimates above contain the norm of the gradient, i.e., |∇u|, but it is well-known that |∇u| attains its maximum and minimum on the boundary [18,19].…”

“…Jost-Ma-Ou [11] and Ma-Ye-Ye [20] proved that the Gaussian curvature and the principal curvature of the convex level sets of 3-dimensional harmonic function attain its minimum on the boundary. Then, Ma-Ou-Zhang [19] and Chang-Ma-Yang [6] got the Gaussian curvature and principal curvature estimates of the convex level sets of higher-dimensional harmonic function (in [6], they also treated a class of semilinear elliptic equations) in terms of the Gaussian curvature or principal curvature of the boundary and the norm of the gradient on the boundary. For more recent results on curvature estimates, please see the papers [7,10,[25][26][27] and the references therein.…”

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

“…Longinetti also studied the precise relation between the curvature of the convex level curves and the height of 2-dimensional minimal surface in [17]. Ma-Ou-Zhang [18] got the Gaussian curvature estimates of the convex level sets on higher dimensional p-harmonic function, and Wang-Zhang [23] got the similar curvature estimates of some quasilinear elliptic equations under certain structural conditions in [4]. Both of their test functions involved the Gaussian curvature of the boundary and the norm of the gradient on the boundary.…”

The Convexity Estimates for the Solutions of Two Elliptic Equations

“…The deformation process is well-known and can be found in Ma and Xu[16] or Ma et al[15].Downloaded by [Heriot-Watt University] at 05:20 07 October 2014 Suppose 0 ∈ . At the initial time, let the domain be the standard unit ball B 1 0 and…”