2026/07/13 by Qinfeng Li, Weihong Xie, Hang Yang
#math.AP
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≥3. Here we resolve the question in the planar convex class and obtain a sharp quantitative theory without any a priori geometric nondegeneracy. Let uΩ solve -ΔuΩ=1 in Ω, ∂νuΩ=-(|Ω|)/(P(Ω)) on ∂Ω, ∫∂ΩuΩ dσ=0, and set O(Ω):=osc∂ ΩuΩ. We construct fixed-area annuli with O(Ωk)→0 that remain far from every disk, showing that convexity is essential in dimension two. By contrast, if Ωk⊂\mathbb R2 are convex, |Ωk|=π, and O(Ωk)→0, then, up to translations, Ωk converges in Hausdorff distance to the unit disk. Moreover, RΩ-rΩ+infz∈\mathbb R2dH(Ω,B1(z)) ≤ C O(Ω) for all planar convex Ω with |Ω|=π and sufficiently small O(Ω), 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(Ω):=\frac1P(Ω)∫∂ΩuΩ,dσ-min∂ΩuΩ. In the planar convex class, A(Ωk)→0 still forces convergence to a disk, and RΩ-rΩ+infz dH(Ω,B1(z)) ≤ C A(Ω)2/3 for |Ω|=π and sufficiently small A(Ω).