Serrin's Problem under Dirichlet Perturbations: Geometric Compactness and Sharp Planar Stability

In earlier work [21], we posed a stability question for Serrin’s overdetermined problem under Dirichlet perturbations and proved that the answer is negative in dimensions $n\ge 3$. Here we resolve the question in the planar convex class and obtain a sharp quantitative theory without any a priori geometric nondegeneracy. Let $u_{\mathrm{\Omega }}$ solve $$-\mathrm{\Delta }{u}_{\mathrm{\Omega }}=1\text{ in }\mathrm{\Omega },{\mathrm{\partial }}_{\nu }{u}_{\mathrm{\Omega }}=-\frac{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}{P(\mathrm{\Omega })}\text{ on }\mathrm{\partial }\mathrm{\Omega },{\int }_{\mathrm{\partial }\mathrm{\Omega }}{u}_{\mathrm{\Omega }}d\sigma =0,$$ and set $O(\mathrm{\Omega }):={\text{osc}}_{\mathrm{\partial }\mathrm{\Omega }}{u}_{\mathrm{\Omega }}$. We construct fixed-area annuli with $O({\mathrm{\Omega }}_k)\to 0$ that remain far from every disk, showing that convexity is essential in dimension two. By contrast, if ${\mathrm{\Omega }}_k\subset {\mathbb{R}}^2$ are convex, $\mathrm{\mid }{\mathrm{\Omega }}_k\mathrm{\mid }=\pi$, and $O({\mathrm{\Omega }}_k)\to 0$, then, up to translations, ${\mathrm{\Omega }}_k$ converges in Hausdorff distance to the unit disk. Moreover, $$R_{\mathrm{\Omega }}-r_{\mathrm{\Omega }}+\underset{z\in {\mathbb{R}}^2}{\inf }{d}_H(\mathrm{\Omega },B_1(z))\le CO(\mathrm{\Omega })$$ for all planar convex $\mathrm{\Omega }$ with $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }=\pi$ and sufficiently small $O(\mathrm{\Omega })$, and the linear order is optimal. The proof combines a new mechanism excluding long-thin degeneration, the rough-domain Serrin rigidity theorem of Figalli–Zhang, new tangential-gradient and linear boundary-growth estimates, a boundary $P$-function estimate, and the reverse-Serrin identity of Magnanini–Molinarolo–Poggesi. We also study the weaker deficit $$A(\mathrm{\Omega }):=\frac{1}{P(\mathrm{\Omega })}{\int }_{\mathrm{\partial }\mathrm{\Omega }}{u}_{\mathrm{\Omega }},d\sigma -\underset{\mathrm{\partial }\mathrm{\Omega }}{\min }{u}_{\mathrm{\Omega }}\mathrm{.}$$ In the planar convex class, $A({\mathrm{\Omega }}_k)\to 0$ still forces convergence to a disk, and $$R_{\mathrm{\Omega }}-r_{\mathrm{\Omega }}+\underset{z}{\inf }{d}_H(\mathrm{\Omega },B_1(z))\le CA(\mathrm{\Omega }{)}^{2\mathrm{/}3}$$ for $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }=\pi$ and sufficiently small $A(\mathrm{\Omega })$.