Classical Neumann Problems for Hessian equations and Alexandrov-Fenchel's inequalities

TL;DR

This paper advances the study of Neumann boundary value problems for Hessian equations in convex domains and applies these results to provide new proofs of Alexandrov-Fenchel inequalities in convex geometry.

Contribution

It establishes the existence of classical Neumann problems for Hessian equations in uniformly convex domains and uses these solutions to derive geometric inequalities.

Findings

Existence of classical Neumann problems for Hessian equations in convex domains.

New proof of Alexandrov-Fenchel inequalities using solutions to Hessian Neumann problems.

Extension of previous work on Neumann problems for Hessian equations.

Abstract

Recently, the first named author together with Xinan Ma \cite{ma2015neumann}, have proved the existence of the Neumann problems for Hessian equations. In this paper, we proceed further to study classical Neumann problems for Hessian equations. We prove here the existence of classical Neumann problems under the uniformly convex domain in R^n. As an application, 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. This geometric application is motivated from Reilly \cite{Reilly1980}.