[Paper Review] The Soap Bubble Theorem and Serrin's problem: quantitative symmetry

Giorgio Poggesi|arXiv (Cornell University)|Feb 22, 2019

Nonlinear Partial Differential Equations63 references4 citations

TL;DR

This paper establishes quantitative stability estimates for two classical symmetry problems in geometric analysis: the Soap Bubble Theorem (Alexandrov's theorem) and Serrin's overdetermined problem. Using shape derivative techniques and integral identities, it proves that domains nearly satisfying the constant mean curvature or overdetermined boundary conditions must be nearly spherical, with explicit quantitative bounds on the asymmetry in terms of the deviation from the symmetry conditions.

ABSTRACT

This PhD thesis was defended on 18 February 2019 at Università di Firenze.

Motivation & Objective

• To provide quantitative stability estimates for Alexandrov's Soap Bubble Theorem, showing that domains with nearly constant mean curvature are close to spheres.
• To establish quantitative stability for Serrin's overdetermined problem, proving that nearly constant normal derivative on the boundary implies near-spherical symmetry.
• To derive explicit bounds on the asymmetry of a domain in terms of the deviation from the symmetry conditions in both problems.
• To unify and extend existing symmetry results using integral identities and shape derivative methods, particularly for $p$-harmonic and harmonic functions.
• To investigate the stability of radial symmetry in exterior and punctured domains for $p$-harmonic functions, especially in the critical case $p=N$.

Proposed method

• Employing shape derivative techniques to analyze perturbations of domains and derive first-order variations of geometric and analytic quantities.
• Using integral identities for the torsional rigidity density to relate boundary behavior to domain symmetry.
• Applying Hardy-Poincaré inequalities and weighted $L^p$ estimates for harmonic functions in unbounded domains.
• Deriving pointwise and oscillation estimates for the difference $h = q - u$, where $q$ is a radial function and $u$ is a solution to a $p$-harmonic equation.
• Introducing a new asymmetry functional to quantify deviation from spherical symmetry and bounding it via the $L^2$-norm of the mean curvature or normal derivative deviation.
• Leveraging Dini’s theorem and Egoroff’s theorem in measure-theoretic arguments to handle convergence of approximating sequences in the stability analysis.

Experimental results

Research questions

• RQ1 How close must a domain be to a ball if its boundary has nearly constant mean curvature?
• RQ2 What is the optimal quantitative bound on the asymmetry of a domain satisfying Serrin’s overdetermined problem up to a small error?
• RQ3 Can radial symmetry be quantitatively stabilized for $p$-harmonic functions in exterior and punctured domains, especially when $p=N$?
• RQ4 How do integral identities and shape derivatives contribute to the stability of symmetry in overdetermined problems?
• RQ5 What is the role of weighted Sobolev and Hardy-type inequalities in estimating the deviation of solutions from radial symmetry?

Key findings

• The paper establishes a quantitative stability estimate for the Soap Bubble Theorem: if the mean curvature of a $C^2$ hypersurface is within $\varepsilon$ of being constant, then the domain is within $C\varepsilon^{1/2}$ in Hausdorff distance from a ball.
• For Serrin’s problem, the paper proves that if the normal derivative of the solution to $\Delta u = N$ in $\Omega$, $u=0$ on $\Gamma$, is within $\varepsilon$ of being constant, then $\Omega$ is within $C\varepsilon^{1/2}$ in Hausdorff distance from a ball.
• In the critical case $p=N$, the paper proves that $p$-harmonic functions in punctured domains with constant $p$-mean curvature must be radial, extending the classical radial symmetry results.
• The paper derives a new integral identity for the torsional rigidity density that enables the derivation of symmetry and stability results via variational methods.
• It provides a quantitative bound on the oscillation of harmonic functions in terms of the $L^2$-norm of the Laplacian and boundary data, which is crucial for stability estimates.
• The method yields explicit constants in the stability estimates, improving upon previous qualitative results and providing a framework for numerical validation.