Classical Neumann Problems for Hessian Equations and Geometric Applications

The classic Neumann problem for Laplace equation has many geometric applications. For example, Reilly used its solution to give a new proof of Minkowski inequality. Prof. Xinan Ma and me, have proved the existence of the Neumann problems for Hessian equations in a uniformly convex domain in Rn. Motivated from Reilly and Ma-Qiu's work, Chao Xia and me also find geometric applications about classical Neumann problems for Hessian equations. We will talk about how to prove the existence of classical Neumann problems under the uniformly convex domain. Then we use the solution of the classical Neumann problem to give a new proof of a family of Alexandrov-Fenchel inequalities arising from convex geometry.