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A free boundary problem associated with the isoperimetric inequality

Ar. Abanov, C. Beneteau, D. Khavinson, R. Teodorescu

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.

A free boundary problem associated with the isoperimetric inequality

Abstract

This paper proves a 30 year old conjecture that disks and annuli are the only domains where analytic content - the uniform distance from 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.

Paper Structure

This paper contains 10 sections, 8 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Four physical problems
  3. Serrin's Problem
  4. The shape of an electrified droplet
  5. Incompressible flows in 2D and generalized Rankine vortices
  6. Classical 2D incompressible vortex flows
  7. Chiral fields in conformal theories with several boundary components
  8. Reduction to the doubly-connected case
  9. Solution for the doubly-connected case
  10. Concluding Remarks

Key Result

Theorem 1.2

Let $\Omega$ and $\Gamma$ be as above. The following are equivalent: (i) $\lambda=\frac{2Area(\Omega)}{P(\Gamma)};$ (ii) There is $\varphi$ analytic in $\overline{\Omega}$ such that $\bar{z}(s)-i\lambda\frac{d\bar{z}}{ds}=\varphi(z(s))$ on $\Gamma,$ where $s$ is the arc-length parameter; (iii) The f holds for all bounded analytic functions $f$ in $\Omega,$ where $dA$ denotes area measure in $\math

Figures (1)

  • Figure 1: The domain $\Omega$ and its boundary components, shown with their orientations relative to $\Omega$ (clockwise for the interior contours, counterclockwise for the exterior one).

Theorems & Definitions (18)

  • Definition 1.1
  • Conjecture 1
  • Theorem 1.2: K87GuKh
  • Theorem 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 8 more