AI 中文总结
本文在三维情形下对2≤k<n的齐次k-Hessian方程外部超定问题，结合参数q与尺度函数，通过不同积分机制建立球刚性结果，将构造从凸域扩展至严格星形域。
AI 中文摘要
本文在三维情形下研究齐次k-Hessian方程的外部超定问题：$$\sigma_k(D^2u)=0\quad\text{在}~\mathbb{R}^n\setminus\overline{\Omega}\text{内}$$。对于$2\le k<n$且光滑严格星形域$\Omega\Subset\mathbb{R}^n$，我们在所有三维情形下建立了球刚性结果。若$2\leq k<\frac{n}{2}$，在$\partial\Omega$的严格$(k-1)$凸性条件下，研究边界值为$-1$且无穷远处极限为0的解；若$k=\frac{n}{2}$，在$\partial\Omega$的严格$(k-1)$凸性条件下，研究边界值为0且无穷远处呈对数增长的解；若$k>\frac{n}{2}$，在$\partial\Omega$的严格$(k-1)$凸性条件下，研究边界值为1且无穷远处呈基本幂次增长的解。每种情形下，若$\partial\Omega$上$|Du|=c>0$（$c$为正常数），则强制$\Omega$为欧氏球并确定解的显式形式。三个论证由单一参数$q=(n-k)/k$和尺度函数$F_q(t)=\frac{t^{1-q}-1}{1-q}$（$F_1(t)=\log t$）组织，归一化后$U=F_q(v)$且$M_q[v]=vD^2v-qDv\otimes Dv$满足$\sigma_k(M_q[v])=0$。统一接触点计算给出尖锐界$|Dv|\le b$，边界粘性接触则得$H_k\ge qbH_{k-1}$。闭合论证的积分机制在临界指数处发生变化：$q>1$时使用Rellich–Pohozaev恒等式，$q=1$时使用尺度不变Wronskian–Newton流，$q<1$时使用重归一化Newton–Jacobi质量。当$q\leq 1$时，我们还通过Minkowski gauge获得全局严格k凸定义函数，这有助于将外部构造从凸域扩展到严格星形域。
英文摘要
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.
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