On the $\sigma_k$ Hessian equation and its compatible boundary integral

It is well known that by using the Brenier's map, one can give a simple proof of the classical isoperimetric inequality with optimal transport method. It is however an open question if the general Alexandrov-Fenchel inequality can be proved in a similar manner. This relies on the solvability of $\sigma_k$ Hessian equation with a suitable boundary condition. In this talk, I will discuss the solvability of $\sigma_k$ Hessian equation with various boundary conditions. If time permits, I will also talk about the conformal invariant properties for the $k$-Yamabe problem with boundary, which shed light on how this PDE problem of the $\sigma_k$ Hessian operator is interplaying with the geometry of the (convex) body. This is joint work with Jeffrey Case.Non UBCUnreviewedAuthor affiliation: Johns Hopkins UniversityFacult