Gaussian Curvature estimates for the convex level sets of solutions for some nonlinear elliptic partial differential equations

TL;DR

This paper establishes lower bounds for the Gaussian curvature of convex level sets in solutions to certain nonlinear elliptic PDEs, including minimal graphs and semilinear equations, based on boundary data.

Contribution

It provides new curvature estimates for convex level sets of solutions to specific nonlinear elliptic PDEs, linking boundary conditions to interior geometric properties.

Findings

Lower bounds for Gaussian curvature of convex level sets

Curvature estimates depend on boundary gradient norms

Results apply to minimal graphs and semilinear elliptic equations

Abstract

We give a lower bound for the Gaussian curvature of convex level sets of minimal graphs and the solutions to semilinear elliptic equations with the norm of boundary gradient and the Gaussian curvature of the boundary.