The evolution of <i>H</i>-surfaces with a Plateau boundary condition

Duzaar, Frank; Scheven, Christoph

doi:10.1016/j.anihpc.2013.10.003

The evolution of H-surfaces with a Plateau boundary condition

Duzaar, Frank ; Scheven, Christoph

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 109-157

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Résumé

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.

MR Zbl

DOI : 10.1016/j.anihpc.2013.10.003

@article{AIHPC_2015__32_1_109_0,
     author = {Duzaar, Frank and Scheven, Christoph},
     title = {The evolution of {\protect\emph{H}-surfaces} with a {Plateau} boundary condition},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {109--157},
     publisher = {Elsevier},
     volume = {32},
     number = {1},
     year = {2015},
     doi = {10.1016/j.anihpc.2013.10.003},
     mrnumber = {3303944},
     zbl = {1328.53083},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2013.10.003/}
}

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Duzaar, Frank; Scheven, Christoph. The evolution of H-surfaces with a Plateau boundary condition. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 109-157. doi: 10.1016/j.anihpc.2013.10.003

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