Plateau flow or the heat flow for half-harmonic maps

Struwe, Michael

doi:10.2140/apde.2024.17.1397

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Plateau flow or the heat flow for half-harmonic maps

Michael Struwe

Vol. 17 (2024), No. 4, 1397–1438

DOI: 10.2140/apde.2024.17.1397

Abstract

Using the interpretation of the half-Laplacian on $S^{1}$ as the Dirichlet-to-Neumann operator for the Laplace equation on the ball $B$ , we devise a classical approach to the heat flow for half-harmonic maps from $S^{1}$ to a closed target manifold $N \subset ℝ^{n}$ , recently studied by Wettstein, and for arbitrary finite-energy data we obtain a result fully analogous to the author’s 1985 results for the harmonic map heat flow of surfaces and in similar generality. When $N$ is a smoothly embedded, oriented closed curve $Γ \subset ℝ^{n}$ , the half-harmonic map heat flow may be viewed as an alternative gradient flow for a variant of the Plateau problem of disc-type minimal surfaces.

Keywords

half-harmonic maps, harmonic map heat flow, Plateau problem

Mathematical Subject Classification

Primary: 35K55, 35K65, 53E99

Secondary: 53A10

Milestones

Received: 3 February 2022

Revised: 14 July 2022

Accepted: 19 August 2022

Published: 17 May 2024

Authors

Michael Struwe
Departement Mathematik ETH Zürich Zürich Switzerland

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Vol. 17, No. 4, 2024