[Paper Review] Symmetry and linear stability in Serrin's overdetermined problem via the stability of the parallel surface problem

Giulio Ciraolo, Rolando Magnanini+1 more|arXiv (Cornell University)|Jan 29, 2015

Nonlinear Partial Differential Equations28 references3 citations

TL;DR

This paper establishes optimal linear stability for Serrin's overdetermined problem by linking it to the stability of the parallel surface problem. Using a novel strategy based on Lipschitz regularity estimates on parallel surfaces, the authors prove that if the normal derivative is constant on the boundary, the domain must be a ball, and provide a sharp linear stability estimate in $C^{2, au}$ domains.

ABSTRACT

We consider the solution of the problem $$ -Δu=f(u) \ \mbox{ and } \ u>0 \ \ \mbox{ in } \ Ω, \ \ u=0 \ \mbox{ on } \ Γ, $$ where $Ω$ is a bounded domain in $\mathbb{R}^N$ with boundary $Γ$ of class $C^{2,τ}$, $00$, where $r_e$ and $r_i$ are the radii of a spherical annulus containing $Γ$, $Γ^δ$ is a surface parallel to $Γ$ at distance $δ$ and sufficiently close to $Γ$, and $[u]_{Γ^δ}$ is the Lipschitz semi-norm of $u$ on $Γ^δ$; secondly, if in addition $u_ν$ is constant on $Γ$, show that $$ [u]_{Γ^δ}=o(C_δ)\ \mbox{ as } \ δ o 0^+. $$ In this paper, we prove that this strategy is successful. As a by-product of this method, for $C^{2,τ}$-regular domains, we also obtain a linear stability estimate for Serrin's symmetry result. Our result is optimal and greatly improves the similar logarithmic-type estimate of [ABR] and the Hölder estimate of [CMV] that was restricted to convex domains.

Motivation & Objective

• To resolve a conjecture in [CMS2] that Serrin’s symmetry theorem can be derived via stability of the parallel surface problem.
• To establish a linear stability estimate for Serrin’s overdetermined problem in $C^{2, au}$-regular domains.
• To improve upon prior logarithmic and H"older-type stability estimates by achieving optimal linear bounds.
• To connect the stability of the parallel surface problem to the symmetry and stability of Serrin’s problem through geometric and analytic techniques.

Proposed method

• Derive a Lipschitz regularity estimate for the solution $u$ on a parallel surface $\Gamma^\delta$ at distance $\delta$ from $\Gamma$, using Taylor expansion and curvature bounds.
• Establish the key inequality (5.1) relating the Lipschitz seminorm $[u]_{\Gamma^\delta}$ to the seminorms of $u_\nu$ and $u_{\nu\nu}$ on $\Gamma$ and $\Gamma^t$.
• Use the $C^{2,\tau}$ regularity of $u$ to control the decay of $[u_{\nu\nu}]_{\Gamma^t}$ as $t \to 0^+$, ensuring integrability.
• Apply Theorem 4.2 on parallel surface stability to derive a quantitative annular containment $B_{r_i} \subset \Omega \subset B_{r_e}$ with $r_e - r_i$ bounded by an integral of $[u_{\nu\nu}]_{\Gamma^t}$.
• Pass to the limit as $\delta \to 0^+$ to obtain the linear stability estimate $r_e - r_i \leq 2C [u_\nu]_\Gamma$.
• Leverage the fact that $[u_{\nu\nu}]_{\Gamma^t} = O(t^{\tau-1})$ to ensure the integral vanishes in the limit, yielding symmetry when $u_\nu$ is constant.

Experimental results

Research questions

• RQ1 Can Serrin’s symmetry theorem be derived via a stability argument based on the parallel surface problem?
• RQ2 What is the optimal quantitative stability estimate for Serrin’s overdetermined problem in $C^{2,\tau}$ domains?
• RQ3 How does the Lipschitz regularity of $u$ on parallel surfaces $\Gamma^\delta$ control the geometry of $\Omega$?
• RQ4 Can the stability estimate be improved beyond logarithmic or H"older-type bounds to achieve linear stability?
• RQ5 What role does the $C^{2,\tau}$ regularity of $u$ play in the decay of $[u_{\nu\nu}]_{\Gamma^t}$ as $t \to 0^+$?

Key findings

• The authors confirm the conjecture in [CMS2] that Serrin’s symmetry result can be derived via stability of the parallel surface problem.
• A sharp linear stability estimate is established: $r_e - r_i \leq 2C [u_\nu]_\Gamma$, where $C$ is a universal constant depending only on $N$ and $\tau$.
• The stability estimate is optimal and improves upon the logarithmic estimate in [ABR] and the H"older estimate in [CMV], which required convexity.
• The method yields symmetry as a consequence of stability: if $u_\nu$ is constant on $\Gamma$, then $\Omega$ is a ball.
• The proof relies on a new estimate (5.1) that bounds the Lipschitz seminorm of $u$ on $\Gamma^\delta$ in terms of the normal derivatives on $\Gamma$ and $\Gamma^t$.
• The decay rate $[u_{\nu\nu}]_{\Gamma^t} = O(t^{\tau-1})$ ensures the integral term vanishes as $\delta \to 0^+$, enabling the linear bound.