AI 中文总结
研究图形平移孤子方程超定边值问题，通过反射线性化等多种原理及方法，证明\(\Gamma\geq0\)时常数狄利克雷数据使有界区域为球，还证双连通区域球壳刚性定理，并给出反例说明相关假设的尖锐性。
AI 中文摘要
我们研究图形平移孤子方程的超定边值问题\[ -\text{div}\!\left(\frac{Du}{\sqrt{1 + |Du|^2}}\right)=\frac{1}{\sqrt{1 + |Du|^2}}\quad\text{在}\Omega\text{内}, \qquad \partial_\nu u=\Gamma H + C\quad\text{在}\partial\Omega\text{上}\]，其中\(\Gamma, C\)为常数，\(H\)是\(\mathbb{R}^n\)中规则有界区域\(\Omega\)边界\(\partial\Omega\)的平均曲率，且\(H_{\partial B_R} = -1/R\)。对于\(\Gamma\geq0\)，我们证明常数狄利克雷数据迫使有界区域为球。还证明了具有两个有序边界高度且内部\(a < u < b\)的双连通区域的球壳刚性定理。论证结合了反射下的线性化、赖歇尔的临界平面和环形延拓原理、曲率比较、塞林的角引理以及一个排除单高度问题中环形替代的径向常微分方程。最后给出明确反例表明符号、排序、连通性和嵌套假设的尖锐性。
英文摘要
We study overdetermined boundary problems for the graphical translating-soliton equation \[ -\operatorname{div}\!\left(\frac{Du}{\sqrt{1+|Du|^2}}\right) =\frac{1}{\sqrt{1+|Du|^2}} \quad\text{in }Ω, \qquad \partial_νu=ΓH+C \quad\text{on }\partialΩ, \] where $Γ, C$ are constants, $H$ is the mean curvature of the boundary $\partialΩ$ of the regular bounded domain in $R^n$ such that $H_{\partial B_R}=-1/R$. For $Γ\geq0$, we prove that constant Dirichlet data force a bounded domain to be a ball. We also prove a spherical-shell rigidity theorem for a doubly connected domain with two ordered boundary heights and $a<u<b$ in the interior. The argument combines linearization under reflection, Reichel's critical-plane and annular continuation principles, curvature comparison, Serrin's corner lemma, and a radial ODE that excludes the annular alternative in the one-height problem. Finally, we give explicit counterexamples showing the sharpness of the sign, ordering, connectedness, and nesting assumptions.
