General monotone formula for homogeneous $k$-Hessian equation in the exterior domain and its applications

TL;DR

This paper introduces a new approach to solve overdetermined problems for the $k$-Hessian equation in exterior domains, establishing monotone formulas and geometric inequalities for $k$-admissible solutions, extending previous methods.

Contribution

It develops a novel perspective combining integral identities and geometric inequalities to address the $k$-Hessian exterior problem, overcoming limitations of traditional methods.

Findings

Proved ball characterization for the $k$-Hessian equation in exterior domains.

Established general monotone formulas for $k$-admissible solutions.

Derived geometric inequalities related to $k$-convex, star-shaped domains.

Abstract

In this paper, we deal with an overdetermined problem for the $k$-Hessian equation ($1\leq k<\frac n2$) in the exterior domain and prove the corresponding ball characterizations. Since that Weinberger type approach seems to fail to solve the problem, we give a new perspective to solve exterior overdetermined problem by combining two integral identities and geometric inequalities inspired by Brandolini-Nitsch-Salani's results \cite{BNS}. Meanwhile, we establish general monotone formulas to derive geometric inequalities related to $k$-admissible solution $u$ in $\mathbb R^n\setminus\Omega$, where $\Omega$ is smooth, $k$-convex and star-shaped domain, which constructed by Ma-Zhang\cite{MZ} and Xiao\cite{xiao}.