Exterior Serrin Rigidity for the Homogeneous{k}-Hessian Equation
Abstract
In this paper, we study the exterior overdetermined problems for the homogeneous k-Hessian equations σk(D2u)=0 in ℝn∖Ω̅ in three dimensional regimes. For 2≤ k<n and smooth strictly star-shaped domain ΩSubsetRn, we establish ball rigidity results in all three dimensional regimes. If 2≤ k<n/2, the solution with boundary value −1 and limit zero at infinity is treated under strict (k−1)-convexity of ∂Ω. If k=n/2, the solution with boundary value $0$ and logarithmic growth at infinity is considered under strict (k−1)-convexity of ∂Ω; if k>n/2, the solution with boundary $1$ and fundamental power-growth at infinity is considered under strict (k−1)-convexity of ∂Ω. In each case, for a positive constant c, |Du|=c>0 on ∂Ω forces Ω to be a Euclidean ball and determines the solution explicitly. The three arguments are organized by the single parameter q=(n−k)/k and the scale function Fq(t)=(t1−q−1)/(1−q), F1(t)=log t. After normalization, U=Fq(v) and Mq[v]=vD2v−qDv⊗ Dv satisfies σk(Mq[v])=0. A unified contact-point calculation gives the sharp bound |Dv|≤ b;a boundary viscosity contact then yields Hk≥ qbHk−1. The integral mechanism that closes the argument changes at the critical exponent: Rellich--Pohozaev identities are used for q>1, a scale-invariant Wronskian--Newton current for q=1, and a renormalized Newton--Jacobi mass for q<1. When q≤ 1, we also obtain a global strict k-convex defining function by Minkowski gauge, this help us extend the exterior construction from convex domains to strictly star-shaped domains.
Related
- A Liouville type rigidity result for entire solutions of the k-Hessian equation
- Rigidity for Convex Entire Solutions to the k-Hessian Equation
- Heintze--Karcher-Type Inequalities for the p-Laplacian and Serrin-Type Rigidity
- Rigidity results for Euler flows in two dimensional exterior domains
- The Stokes Operator on Exterior Domains in Homogeneous Weighted Function Spaces:From Weak Theory to ℋ∞-calculus to Fractional Domains
BibTeX
@misc{zhang2026exterior,
title = {{Exterior Serrin Rigidity for the Homogeneousk-Hessian Equation}},
author = {Zhihui Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03339},
eprint = {2609.03339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03339},
}
RIS
TY - UNPB
TI - Exterior Serrin Rigidity for the Homogeneous{k}-Hessian Equation
AU - Zhihui Zhang
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03339
UR - https://arxiv.org/abs/2609.03339
AB - In this paper, we study the exterior overdetermined problems for the homogeneous k-Hessian equations σ_k(D^2u)=0 in ℝ^n∖Ω̅ in three dimensional regimes. For 2≤ k<n and smooth strictly star-shaped domain ΩSubsetR^n, we establish ball rigidity results in all three dimensional regimes. If 2≤ k<n/2, the solution with boundary value −1 and limit zero at infinity is treated under strict (k−1)-convexity of ∂Ω. If k=n/2, the solution with boundary value $0$ and logarithmic growth at infinity is considered under strict (k−1)-convexity of ∂Ω; if k>n/2, the solution with boundary $1$ and fundamental power-growth at infinity is considered under strict (k−1)-convexity of ∂Ω. In each case, for a positive constant c, |Du|=c>0 on ∂Ω forces Ω to be a Euclidean ball and determines the solution explicitly. The three arguments are organized by the single parameter q=(n−k)/k and the scale function F_q(t)=(t^1−q−1)/(1−q), F_1(t)=log t. After normalization, U=F_q(v) and M_q[v]=vD^2v−qDv⊗ Dv satisfies σ_k(M_q[v])=0. A unified contact-point calculation gives the sharp bound |Dv|≤ b;a boundary viscosity contact then yields H_k≥ qbH_k−1. The integral mechanism that closes the argument changes at the critical exponent: Rellich--Pohozaev identities are used for q>1, a scale-invariant Wronskian--Newton current for q=1, and a renormalized Newton--Jacobi mass for q<1. When q≤ 1, we also obtain a global strict k-convex defining function by Minkowski gauge, this help us extend the exterior construction from convex domains to strictly star-shaped domains.
ER -