The evolution of H-surfaces with a Plateau boundary condition

Frank Duzaar; Christoph Scheven

doi:10.1016/j.anihpc.2013.10.003

JournalsaihpcVol. 32, No. 1pp. 109–157

The evolution of H-surfaces with a Plateau boundary condition

Frank Duzaar
Department Mathematik, Universität Erlangen–Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany
Christoph Scheven
Department Mathematik, Universität Duisburg–Eseen, Thea-Leymann-Strasse 9, 45127 Essen, Germany

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Abstract

In this paper we consider the heat flow associated to the classical Plateau problem for surfaces of prescribed mean curvature. To be precise, for a given Jordan curve $Γ \subset R^{3}$ , a given prescribed mean curvature function $H : R^{3} \to R$ and an initial datum $u_{o} : B \to R^{3}$ satisfying the Plateau boundary condition, i.e. that $u_{o} ∣_{\partial B} : \partial B \to Γ$ is a homeomorphism, we consider the geometric flow

\partial_{t} u - Δ u = - 2 (H \circ u) D_{1} u \times D_{2} u in B \times (0, \infty),

u (\cdot, 0) = u_{o} on B, u (\cdot, t) ∣_{\partial B} : \partial B \to Γ is weakly monotone for all t > 0.

We show that an isoperimetric condition on H ensures the existence of a global weak solution. Moreover, we establish that these global solutions sub-converge as $t \to \infty$ to a conformal solution of the classical Plateau problem for surfaces of prescribed mean curvature.

Cite this article

Frank Duzaar, Christoph Scheven, The evolution of H-surfaces with a Plateau boundary condition. Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 1, pp. 109–157

DOI 10.1016/J.ANIHPC.2013.10.003