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

In earlier work, we posed a stability question for Serrin's overdetermined problem under Dirichlet perturbations and proved that the answer is negative in dimensions $n\ge3$. Here we resolve the question in the planar convex class and obtain a sharp quantitative theory without any a priori geometric nondegeneracy. Let $u_\Omega$ solve $$-\Delta u_\Omega=1\ \text{in }\Omega,\qquad \partial_\nu u_\Omega=-\frac{|\Omega|}{P(\Omega)}\ \text{on }\partial\Omega, \qquad \int_{\partial\Omega}u_\Omega\,d\sigma=0,$$ and set $O(\Omega):=\text{osc}_{\partial \Omega}u_\Omega$. We construct fixed-area annuli with $O(\Omega_k)\to0$ that remain far from every disk, showing that convexity is essential in dimension two. By contrast, if $\Omega_k\subset\mathbb R^2$ are convex, $|\Omega_k|=\pi$, and $O(\Omega_k)\to0$, then, up to translations, $\Omega_k$ converges in Hausdorff distance to the unit disk. Moreover, $$R_\Omega-r_\Omega+\inf_{z\in\mathbb R^2}d_H(\Omega,B_1(z)) \le C\,O(\Omega)$$ for all planar convex $\Omega$ with $|\Omega|=\pi$ and sufficiently small $O(\Omega)$, 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 $$A(\Omega):=\frac1{P(\Omega)}\int_{\partial\Omega}u_\Omega,d\sigma-\min_{\partial\Omega}u_\Omega.$$ In the planar convex class, $A(\Omega_k)\to0$ still forces convergence to a disk, and $$R_\Omega-r_\Omega+\inf_z d_H(\Omega,B_1(z)) \le C A(\Omega)^{2/3}$$ for $|\Omega|=\pi$ and sufficiently small $A(\Omega)$.