# Calendar

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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