[Paper Review] Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration

Giulio Ciraolo, Rolando Magnanini+1 more|arXiv (Cornell University)|Jul 4, 2013

Advanced Mathematical Modeling in Engineering11 references4 citations

TL;DR

This paper establishes a quantitative stability result for radial symmetry in elliptic PDEs: if a positive solution to the torsion problem is nearly constant on a surface parallel to the boundary, then the domain must be nearly radially symmetric, with the deviation from spherical shape linearly controlled by the oscillation of the solution on that surface. The proof extends the method of moving planes using Harnack's inequality and refined boundary estimates to achieve sharp quantitative control in the elliptic setting.

ABSTRACT

Positive solutions of homogeneous Dirichlet boundary value problems or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of their level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. In fact, we show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls $B_{r_e}$ and $B_{r_i}$, with the difference $r_e-r_i$ (linearly) controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and enhances arguments developed in a paper by Aftalion, Busca and Reichel.

Motivation & Objective

• To establish a quantitative stability result for radial symmetry in elliptic boundary value problems.
• To investigate how close a domain must be to a ball if a solution is nearly constant on a surface parallel to the boundary.
• To extend the method of moving planes to a quantitative, stability framework for the torsion problem.
• To provide explicit linear bounds on domain deviation from radial symmetry in terms of solution oscillation on the parallel surface.
• To refine prior logarithmic stability estimates by achieving a power-law (linear) dependence via enhanced Harnack and boundary estimates.

Proposed method

• Adapts the method of moving planes to a quantitative setting using Harnack's inequality for harmonic functions in subdomains.
• Applies refined versions of Hopf's lemma and Serrin's corner lemma to control solution gradients and second derivatives.
• Uses the Minkowski sum construction Ω = G + B_R to define a domain with a level surface ∂G parallel to ∂Ω.
• Employs a weighted function w^m = u(x^m) - u(x) to measure asymmetry during the moving plane process.
• Derives a Harnack-type inequality with exponential dependence on domain size and curvature to bound solution variation.
• Introduces a barrier function based on the normal derivative and C^2 norm to control distance to the boundary in terms of solution oscillation.

Experimental results

Research questions

• RQ1 Under what conditions does a solution being nearly constant on a surface parallel to the boundary imply the domain is nearly radially symmetric?
• RQ2 Can the qualitative radial symmetry result of Serrin and others be strengthened to a quantitative stability estimate with explicit error bounds?
• RQ3 How does the deviation of the domain from spherical shape depend on the oscillation of the solution on the parallel surface?
• RQ4 Can the logarithmic stability estimate from Aftalion, Busca, and Reichel be improved to a linear or power-law dependence?
• RQ5 What role do curvature and regularity of the boundary play in the stability of radial configurations?

Key findings

• If the seminorm [u]_∂G is small, then the domain Ω is contained between two concentric balls B_{r_e} and B_{r_i} with r_e - r_i ≤ C [u]_∂G.
• The constant C depends only on N, the C^{2,α} regularity of ∂G, diam(G), and the radius R of the Minkowski sum.
• The stability estimate is linear in the oscillation of the solution, improving upon the logarithmic dependence in prior work.
• The proof relies on a refined application of Harnack's inequality and a new barrier estimate involving the C^2 norm of u.
• The method is robust and extends to semilinear elliptic equations with locally Lipschitz nonlinearities.
• The result holds uniformly across all directions, allowing the construction of a center of symmetry within O([u]_∂G) of the true center.