A Sobolev gradient flow for the area-normalised Dirichlet energy of $H^1$ maps

TL;DR

This paper introduces a Sobolev gradient flow for the area-normalized Dirichlet energy of $H^1$ maps, proving long-term existence and convergence to circles, thus establishing an isoperimetric inequality in this setting.

Contribution

It develops a Sobolev gradient flow for the normalized Dirichlet energy of $H^1$ maps and proves its solutions' global existence and convergence to circles.

Findings

Solutions with positive initial area exist for all time.

Solutions converge to (possibly multiply-covered) circles.

Establishes an isoperimetric inequality for $H^1(du)$ maps.

Abstract

In this article we study the $H^1(du)$-gradient flow for the energy $E[X] = Q[X]/A[X]$ where $Q[X]$ is the Dirichlet energy of $X$, $A[X]$ is the signedenclosed area of $X$, and $X:\mathbb{S}\rightarrow\mathbb{R}^2$ is a $H^1(du)$ map. We prove that solutions with initially positive signed enclosed area exist eternally, and converge as $t\rightarrow\infty$ to a (possibly multiply-covered) circle. In this way we recover a parametrised isoperimetric inequality for $H^1(du)$ maps.