The authors study a Cauchy-Dirichlet problem associated to the heat flow for surfaces of prescribed mean curvature on the unit ball \(B\subset\mathbb{R}^2\), namely
\[ \partial_t u -\Delta u = - 2(H\circ u)D_1u \times D_2u \quad\text{in}\quad B\times (0,\infty), \qquad u=u_0 \quad\text{on}\quad \partial_{\mathrm{par}}(B\times(0,\infty)), (*) \]
with solutions \(u:B\times(0,\infty) \rightarrow \mathbb{R}^3\), given Cauchy-Dirichlet datum \(u_0\), and a bounded continuous function \(H:\mathbb{R}^3 \rightarrow \mathbb{R}\). The parabolic boundary is \(\partial_{\mathrm{par}}(B\times(0,\infty))=\partial B\times (0,\infty) \cup \overline{B}\times \{0\}\).
The stationary case of \((*)\) has been studied intensively, with various existence results obtained by Heinz, Hildebrandt, Gulliver and Spruck, Steffen, and Wente.
The parabolic problem \((*)\) has received much less attention. O. Rey [Math. Ann. 291, No. 1, 123–146 (1991; Zbl 0761.58052)] showed that a Hildebrandt-type condition \(\|u_0\|\leq R\), \(\|H\|_{L^\infty} \leq R^{-1}\), ensures the existence of a classical solution. This result has recently been extended to higher-dimensional \(H\)-flows by C. Leone et al. [J. Math. Pures Appl. (9) 97, No. 3, 282–294 (2012; Zbl 1241.35053)], for bounded Lipschitz continuous \(H\).
In general, one cannot expect the existence of a global solution on \(B\times(0,\infty)\) without some smallness condition on \(H\), due to energy concentration phenomena, i.e., “bubbles” forming in finite time.
Here, the authors prove the existence of a global weak solution by imposing an isoperimetric condition on \(H\) analogous to that used by Steffen. As a special case of their main theorem, the problem \((*)\) has a weak solution with values in \(B_R\subset \mathbb{R}^3\) for data \(u_0\in W^{1,2}(B,B_R)\) provided \(H\) is bounded and continuous and satisfies \[ \int_{\{\xi\in B_R:|H(\xi)|\geq \frac{3}{2R}\}} |H|^3\, d\xi < \frac{9\pi}{2}, \qquad |H(a)|\leq \frac{1}{R} \quad\text{for}\quad a\in \partial B_R. \]
The second main result concerns the long-time behaviour of the weak solution constructed in the first theorem. Under the same assumptions the authors show that there exists a sequence \(t_j\rightarrow\infty\) such that \(u(\cdot,t_j)\rightharpoonup u_\infty\) weakly in \(W^{1,2}(B,\mathbb{R}^3)\) to a solution \(u_\infty\in W^{1,2}(B,\mathbb{R}^3)\) of the stationary problem corresponding to \((*)\). Furthermore, if \(u_0|_{\partial B}\) is continuous, then \(u_\infty\in C^0(\overline{B},\mathbb{R}^3)\cap C_{\mathrm{loc}}^{1,\alpha}(B,\mathbb{R}^3)\), and if \(H\) is Hölder continuous, then \(u_\infty\in C_{\mathrm{loc}}^{2,\alpha}(B,\mathbb{R}^3)\) and \(u_\infty\) is a classical solution.
The construction of weak solutions uses a time discretization procedure that relies on ideas introduced by N. Kikuchi in [NATO ASI Ser., Ser. C, Math. Phys. Sci. 332, 195–199 (1991; Zbl 0850.76043)], which has been used by J.- Haga et al. [Comput. Vis. Sci. 7, No. 1, 53–59 (2004; Zbl 1120.53304)] to reprove results concerning the existence of weak solutions to the harmonic map heat flow previously obtained by Ginzburg-Landau approximations.