A Dirichlet's principle for the κ-Hessian

The kappa-Hessian operator Crk is the kappa-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the kappa-Hessian equation sigma(kappa)(D(2)u) = f with Dirichlet boundary condition u = 0 is variational; indeed, this problem can be studied by means of the kappa-Hessian energy - integral u sigma(k)(D(2)u). We construct a natural boundary functional which, when added to the kappa-Hessian energy, yields as its critical points solutions of kappa-Hessian equations with general non-vanishing boundary data. As a consequence, we establish a Dirichlet's principle for kappa-admissible functions with prescribed Dirichlet boundary data. (C) 2018 Elsevier Inc. All rights reserved.