[Paper Review] A free boundary problem associated with the isoperimetric inequality

Ar. Abanov, Catherine Bénéteau+2 more|arXiv (Cornell University)|Jan 15, 2016

Mathematical Dynamics and Fractals30 references3 citations

TL;DR

This paper proves a 30-year-old conjecture that disks and annuli are the only domains where analytic content achieves its theoretical lower bound of $2\text{Area}(\Omega)/P(\Gamma)$, using conformal mapping, Schwarzian derivatives, and overdetermined boundary value problems. The result confirms that extremal domains for this free boundary problem are precisely annuli and disks, linking analytic content to classical isoperimetric and fluid dynamics problems.

ABSTRACT

This paper proves a 30 year old conjecture that disks and annuli are the only domains where analytic content - the uniform distance from $\bar{z}$ to analytic functions - achieves its lower bound. This problem is closely related to several well-known free boundary problems, in particular, Serrin's problem about laminar flow of incompressible viscous fluid for multiply-connected domains, and Garabedian's problem on the shape of electrified droplets. Some further ramifications and open questions, including extensions to higher dimensions, are also discussed.

Motivation & Objective

• To resolve a longstanding conjecture on the extremal domains for analytic content, which is tied to the isoperimetric inequality.
• To establish that only disks and annuli achieve the lower bound $2\text{Area}(\Omega)/P(\Gamma)$ for analytic content.
• To connect the analytic content problem to classical free boundary problems, including Serrin’s problem and Garabedian’s droplet problem.
• To extend the characterization of extremal domains beyond simply-connected cases to multiply-connected domains, particularly doubly-connected ones.
• To explore the implications and open questions in higher dimensions and for more general boundary conditions.

Proposed method

• Reduction of the free boundary problem to domains of connectivity at most two, leveraging symmetry and conformal invariance.
• Use of the Schwarzian derivative to analyze conformal maps from annuli to extremal domains, showing they must be linear or of the form $a/z + b$.
• Application of the quadrature identity (1.2) to link analytic content to harmonic and elliptic PDEs.
• Analysis of the overdetermined boundary value problem (1.3) with constant normal derivative and constant boundary values on each component.
• Proof via Riccati equations and homogeneity of the Schwarzian derivative to classify all possible conformal maps preserving extremality.
• Use of Möbius invariance and transformation laws for the Schwarzian derivative to constrain the form of the conformal map.

Experimental results

Research questions

• RQ1 Which domains $\Omega$ achieve the minimal analytic content $\lambda(\Omega) = 2\text{Area}(\Omega)/P(\Gamma)$?
• RQ2 Can the conjecture that only disks and annuli achieve this lower bound be rigorously proven for multiply-connected domains?
• RQ3 How are the conditions in Theorem 1.2—such as the boundary condition involving $\bar{z}$ and the arc-length derivative—related to physical free boundary problems?
• RQ4 What is the role of the Schwarzian derivative in characterizing extremal conformal maps from annuli to extremal domains?
• RQ5 Can the results be extended to higher dimensions, particularly for $\mathbb{R}^n$, where the lower bound may not be sharp?

Key findings

• The conjecture is fully resolved: disks and annuli are the only domains for which $\lambda(\Omega) = 2\text{Area}(\Omega)/P(\Gamma)$.
• For doubly-connected domains, the conformal map from an annulus to the extremal domain must be either linear or of the form $a/z + b$, implying the image is itself an annulus.
• The solution to the overdetermined problem (1.3) with constant normal derivative and constant boundary values on each component exists only for annuli and disks.
• The quadrature identity (1.2) holds for all bounded analytic functions in $\Omega$ if and only if $\Omega$ is a disk or annulus.
• The Schwarzian derivative of the conformal map satisfies a homogeneous functional equation, forcing it to be $c/z^2$, which leads to the classification of the map's form.
• The result confirms that the extremal domains are precisely those for which the analytic content achieves the isoperimetric lower bound, linking complex analysis, PDEs, and geometric function theory.