Level set mean curvature flow with Neumann boundary conditions (Q6171998)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) with smooth boundary. The author considers a family of hypersurfaces \(\{\Gamma_t\}_{t\ge 0}\) in \(\overline\Omega\) evolving under the mean curvature flow with Neumann boundary conditions, \[ V=-H \mbox{ on } \Gamma_t, \qquad \Gamma_t \perp \partial\Omega \tag{1} \] where \(V\) is the normal velocity of \(\Gamma_t\) and \(H\) is the mean curvature of \(\Gamma_t\). This problem has been studied in the absence of the boundary condition (i.e., \(\Omega=\mathbb{R}^N\)) using several different formulations. There is the level set approach of \textit{Y.-G. Chen} et al. [J. Differ. Geom. 33, No. 3, 749--786 (1991; Zbl 0696.35087)] and \textit{L. C. Evans} and \textit{J. Spruck} [J. Differ. Geom. 33, No. 3, 635--681 (1991; Zbl 0726.53029)] in which the hypersurfaces \(\Gamma_t\) are the level sets of a function \(u\) solving the PDE \[ \partial_tu=|\nabla u| \mathrm{div} \left( \frac{\nabla u}{|\nabla u|} \right) \] in the viscosity sense. There is also \textit{K. Brakke}'s formulation using the theory of varifolds [The motion of a surface by its mean curvature. Princeton, New Jersey: Princeton University Press. Tokyo: University of Tokyo Press (1978; Zbl. 0386.53047)]. There are several existence and uniqueness results for (1), including those of \textit{M.-H. Sato} [Adv. Math. Sci. Appl. 4, No. 1, 249--264 (1994; Zbl 0811.35069)] for the level set formulation, and \textit{N. Edelen} [J. Reine Angew. Math. 758, 95--137 (2020; Zbl 1433.53124)] and \textit{M. Mizuno} and \textit{Y. Tonegawa} [SIAM J. Math. Anal. 47, No. 3, 1906--1932 (2015; Zbl 1330.35196)] for the Brakke formulation. \textit{L. C. Evans} and \textit{J. Spruck} [J. Geom. Anal. 5, No. 1, 79--116 (1995; Zbl 0829.53040)] showed that in the no boundary case, almost all level sets of the unique viscosity solution are unit density varifolds moving by mean curvature flow in Brakke's sense. Here the author proves an analogue of this result for (1), for appropriately chosen initial data. The main new point is the characterisation of the boundary condition that the Brakke flow satisfies: the first variation of the Brakke flow is bounded on \(\partial\Omega\) and perpendicular to \(\partial\Omega\) almost everywhere. Moreover, the class of admissible test functions is strictly larger than in previous work in that they are not required to satisfy \(\nabla\phi \cdot \nu_{\partial\Omega}=0\).