A fully nonlinear Sobolev trace inequality

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Yi Wang , Johns Hopkins University

 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 conditionu=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 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.