Nonfattening condition for the generalized evolution by mean curvature and applications
This paper studies the problem of uniqueness for a class of geometric evolution equations of hypersurfaces in Euclidean space, that includes motion by mean curvature. The idea is as follows: using the level-set approach one can define the notion of visosity solution for a class of PDE's that include mean curvature flow (\textit{Y.-G. Chen, Y. Giga} and \textit{S. Goto} [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 this way given an initial hypersurface \(\Gamma_0\) in \(\mathbb{R}^n\) one obtains existence of a global viscosity solution \(\Gamma_t\), \(t\geq 0\) of the mean curvature flow starting from \(\Gamma_0\). In general, this solution is not unique, and the main result of the paper gives uniqueness under a certain technical assumption. As a corollary they prove the following result. Recall that a function \(u_0:\mathbb{R}^n\to\mathbb{R}\) is called convex at infinity if there is a ball in \(\mathbb{R}^n\) so that for any convex set \(C\) in the complement of the ball, the restriction \(u_0| _C\) is convex. Then they prove that for any continuous such \(u_0\) (without any growth assumption at infinity) there exists a unique global solution of the mean curvature equation for graphs, starting at the graph of \(u_0\). The existence of such a solution follows from \textit{K. Ecker} and \textit{G. Huisken} [Invent. Math. 105, No.~3, 547--569 (1991; Zbl 0707.53008)].
scientific article; zbMATH DE number 5302264