A fully nonlinear Sobolev trace inequality
A fully nonlinear Sobolev trace inequality

Yi Wang , Johns Hopkins University
Fine Hall 224
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 nonvanishing 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.