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.

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 $\Gamma \subset {ℝ}^3$, 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 $u_o{|}_{\partial B}:\partial B\to \Gamma$ is a homeomorphism, we consider the geometric flow

$${\partial }_{t}u-\Delta u=-2(H\circ u)D_{1}u\times D_{2}u\text{in}B\times (0,\infty ),$$

$$u(·,0)=u_o\text{on}B,u(·,t){|}_{\partial B}:\partial B\to \Gamma \text{is}\text{weakly}\text{monotone}\text{for}\text{all}t>0.$$

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.

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