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Plateau flow or the heat flow for half-harmonic maps

Michael Struwe

Vol. 17 (2024), No. 4, 1397–1438
Abstract

Using the interpretation of the half-Laplacian on S1 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 S1 to a closed target manifold N 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 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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