2026/07/24 by Liang Cheng, Li Ma
#math.AP
We study overdetermined boundary problems for the graphical translating-soliton equation -div ((Du)/(√(1+|Du|2))) =(1)/(√(1+|Du|2)) \quadin Ω, ∂νu=ΓH+C \quadon ∂Ω, where Γ, C are constants, H is the mean curvature of the boundary ∂Ω of the regular bounded domain in Rn such that H∂ BR=-1/R. For Γ≥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.