A fully nonlinear Sobolev trace inequality

TL;DR

This paper establishes a sharp Sobolev trace inequality for k-Hessian functions by constructing a boundary functional that extends the variational framework to non-zero boundary data, advancing the understanding of nonlinear boundary value problems.

Contribution

It introduces a natural boundary functional for the k-Hessian energy, enabling the analysis of solutions with general boundary data and deriving a new sharp Sobolev trace inequality.

Findings

Constructed a boundary functional for k-Hessian energy.

Proved a sharp Sobolev trace inequality for k-admissible functions.

Extended variational methods to non-vanishing boundary conditions.

Abstract

The $k$-Hessian operator $\sigma_k$ is the $k$-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the $k$-Hessian equation $\sigma_k(D^2u)=f$ with Dirichlet boundary condition $u=0$ is variational; indeed, this problem can be studied by means of the $k$-Hessian energy $-\int u\sigma_k(D^2u)$. We construct a natural boundary functional which, when added to the $k$-Hessian energy, yields as its critical points solutions of $k$-Hessian equations with general non-vanishing boundary data. As a consequence, we prove a sharp Sobolev trace inequality for $k$-admissible functions $u$ which estimates the $k$-Hessian energy in terms of the boundary values of $u$.