Free-boundary regularity on the focusing problem for the Gauss curvature flow with flat sides (Q5947237)

In recent years several authors have studied the evolution of compact weakly convex surfaces in \({\mathbb{R}}^3\) moving under the Gauss curvature flow. \textit{B.~Andrews} [Invent. Math. 138, 151--161 (1999; Zbl 0936.35080)] showed that if the initial surface \(\Sigma_0\subset{\mathbb{R}}^3\) is of class \(C^{1,1}\), then the flow has a \(C^{1,1}\) viscosity solution; moreover, the evolving surfaces \(\Sigma_t\) converge to a point and become round as the final time is approached. Of particular interest here is the case that the initial surface \(\Sigma_0\) has a flat portion, which we may take to be \(\Sigma_0\cap \{z=0\}\) with \(\Sigma_0\subset \{z\geq 0\}\). Near the flat portion the evolving surfaces can be represented as the graph of a solution \(u\) of the parabolic Monge-Ampère type equation \[ u_t = {{\det D^2u}\over{(1+| Du| ^2)^{3/2}}}. \] \textit{R. S.~Hamilton} [Discourses Math. Appl. 3, 69--78 (1994; Zbl 0882.52002)] showed that in this case the interface \(\Gamma_t=\partial(\Sigma_t\cap \{z=0\})\) behaves like a free boundary propagating with finite speed. \textit{D.~Chopp, L. C.~Evans} and \textit{H.~Ishii} [Indiana Univ. Math. J. 48, 311--334 (1999; Zbl 0995.53050)] showed that if \(\Sigma_0\) is of class \(C^{3,1}\) (which implies that \(u_0\) vanishes in a cubic fashion near \(\Gamma_0\)), then the interface \(\Gamma_t\) does not move at all for a short time interval. The regularity of \(\Gamma_t\) for short times was studied by \textit{P.~Daskalopoulos} and \textit{R. S.~Hamilton} [J. Reine Angew. Math. 510, 187--227 (1999; Zbl 0931.53031)] in the case that \(u_0\) vanishes quadratically near \(\Gamma_0\). Under certain regularity assumptions they showed that if \(\Gamma_0\) is smooth and strictly convex, then this is also true of \(\Gamma_t\) for a short time interval \(0\leq t\leq \tau\), and furthermore, the strictly convex part of \(\Sigma_t\) is smooth up to \(\Gamma_t\) for \(0\leq t\leq \tau\). Here the authors consider, in the rotationally symmetric case, the regularity of the interfaces \(\Gamma_t\) up to the focusing time, that is, the time when the flat side shrinks to a point. Expressing the strictly convex part of \(\Sigma_t\) near \(\Gamma_t\) as the graph of a function \(z=f(r,t)\), \(r=\sqrt{x^2+y^2}\), they show that if at time \(t=0\), \(g=\sqrt{f}\) vanishes linearly at the flat side, then \(g\) is smooth up to the interface for \(t>0\) and remains smooth up to the focusing time. They also show that at the focusing time \(g\) is of class \(C^{1,\beta}\) for \(\beta<1/4\) and is no better than \(C^{1,2/5}\). The authors also find the exact self-similar profile of \(g=\sqrt{f}\) at its focusing time for rotationally symmetric solutions of for the parabolic Monge-Ampère equation \[ f_t=\det D^2f. \]