A Serrin-type over-determined problem for Hessian equations in the exterior domain

TL;DR

This paper investigates a Serrin-type over-determined boundary value problem for Hessian equations in exterior domains, establishing the existence and uniqueness of a convex solution under specific boundary and asymptotic conditions.

Contribution

It introduces a new existence and uniqueness result for a class of Hessian equations with over-determined boundary conditions in exterior domains.

Findings

Unique bounded domain exists for the problem

Strictly convex solution is unique

Solution satisfies prescribed asymptotic behavior

Abstract

In this paper, we consider the Hessian equations in some exterior domain with prescribed asymptotic behavior at infinity and Dirichlet-Neumann conditions on its interior boundary. We obtain that there exists a unique bounded domain such that the over-determined problem admits a unique strictly convex solution.