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Joint Princeton-Rutgers Seminar on Geometric PDE's

A fully nonlinear Sobolev trace inequality

Location:  Other - Princeton - Fine Hall 224
Date & time: Friday, 21 October 2016 at 3:30PM - 3:31PM

Yi Wang, Johns Hopkins University: The \(k\)-Hessian operator \(\sigma_k\) is the \(k\)-th elementarysymmetric 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 naturalboundary functional which, when added to the \(k\)-Hessian energy, yields asits critical points solutions of \(k\)-Hessian equations with generalnon-vanishing boundary data. As a consequence, we prove a sharp Sobolevtrace inequality for \(k\)-admissible functions \(u\) which estimates the\(k\)-Hessian energy in terms of the boundary values of \(u\).

This is jointwork with Jeffrey Case.

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