Weak solutions to the heat flow for surfaces of prescribed mean curvature

Abstract:

In this paper we establish the existence of global weak solutions to the heat flow for surfaces of prescribed mean curvature, i.e. the existence for the Cauchy-Dirichlet problem to parabolic systems of the type \begin{equation*} \left \{ \begin {array}{c} \partial _t u-\Delta u =-2 (H\circ u)D_1u\times D_2u\quad \mbox {in $B\times (0,\infty )$,}\\[3pt] u=u_o\quad \mbox {on $\partial _\textrm {par} \big (B\times (0,\infty )\big )$}, \end{array} \right . \end{equation*} where $H\colon \mathbb {R}^3\to R$ is a bounded continuous function satisfying an isoperimetric condition, $B$ is the unit ball in $\mathbb {R}^2$ and $u\colon B\times (0,\infty )\to \mathbb {R}^3$. As one of the possible applications we show that the problem has a solution with values in $B_R\subset \mathbb {R}^3$, whenever $u_o(B)\subseteq B_R$ and furthermore there holds \begin{equation*} \int _{\{ \xi \in B_R: |H(\xi )|\ge \frac {3}{2R}\}}|H|^3 d\xi <\frac {9\pi }{2}, \qquad |H(a)|\le \tfrac {1}{R}\quad \mbox {for $a\in \partial B_R$.} \end{equation*}