Exterior Serrin Rigidity for the Homogeneous{k}-Hessian Equation

In this paper, we study the exterior overdetermined problems for the homogeneous k-Hessian equations $$σ_k(D^2u)=0\quad\text{in}~\mathbb{R}^n\setminus\overlineΩ$$ in three dimensional regimes. For $2\le k<n$ and smooth strictly star-shaped domain $Ω\Subset\R^n$, we establish ball rigidity results in all three dimensional regimes. If $2\leq k<\frac{n}{2}$, the solution with boundary value $-1$ and limit zero at infinity is treated under strict $(k-1)$-convexity of $\partialΩ$. If $k=\frac{n}{2}$, the solution with boundary value $0$ and logarithmic growth at infinity is considered under strict $(k-1)$-convexity of $\partialΩ$; if $k>\frac{n}{2}$, the solution with boundary $1$ and fundamental power-growth at infinity is considered under strict $(k-1)$-convexity of $\partialΩ$. In each case, for a positive constant $c$, $|Du|=c>0$ on $\partialΩ$ 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 \[ F_q(t)=\frac{t^{1-q}-1}{1-q},\qquad F_1(t)=\log t. \] After normalization, $U=F_q(v)$ and $M_q[v]=vD^2v-qDv\otimes Dv$ satisfies $σ_k(M_q[v])=0$. A unified contact-point calculation gives the sharp bound $|Dv|\le b$;a boundary viscosity contact then yields $H_k\ge qbH_{k-1}$. 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 $q\leq 1,$ 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.