A Dirichlet's principle for the k-Hessian

Abstract

The k-Hessian operator σk is the k-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the k-Hessian equation σk(D2u)=f with Dirichlet boundary condition u=0 is variational; indeed, this problem can be studied by means of the k-Hessian energy −∫uσk(D2u). 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 establish a Dirichlet's principle for k-admissible functions with prescribed Dirichlet boundary data.