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A fully nonlinear Sobolev trace inequality

Banff International Research Station for Mathematical Innovation and Discovery

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 $\sigma_k(D^2 u)=f$ with Dirichlet boundary condition $u=0$ is variational; indeed, this problem can be studied by means of the $k$-Hessian energy $\int -u \sigma_k(D^2 u)$. 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 $u$. This is joint work with Jeffrey Case.

Mathematics; Partial differential equations; Calculus of variations and optimal control; optimization; Global analysis

video/mp4

Author affiliation: Johns Hopkins University

2017-02-01

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60302

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/