Plateau flow or the heat flow for half-harmonic maps

Using the interpretation of the half-Laplacian on S 1 as the Dirichletto-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 ⊂ Ê 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 ⊂ Ê 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.