Ball and Spherical-Shell Rigidity from Overdetermined Translating Solitons

We study overdetermined boundary problems for the graphical translating-soliton equation $$-\mathrm{div}\text{\hspace{0.17em}⁣}\left(\frac{Du}{\sqrt{1+\mathrm{\mid }Du{\mathrm{\mid }}^2}}\right)=\frac{1}{\sqrt{1+\mathrm{\mid }Du{\mathrm{\mid }}^2}}\text{in }\mathrm{\Omega },{\mathrm{\partial }}_{\nu }u=\mathrm{\Gamma }H+C\text{on }\mathrm{\partial }\mathrm{\Omega },$$ where $\mathrm{\Gamma },C$ are constants, $H$ is the mean curvature of the boundary $\mathrm{\partial }\mathrm{\Omega }$ of the regular bounded domain in $R^n$ such that $H_{\mathrm{\partial }{B}_R}=-1\mathrm{/}R$. For $\mathrm{\Gamma }\ge 0$, 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.