\(C^{1,\alpha }\) regularity of solutions to parabolic Monge-Ampère equations (Q2903090)

The authors study the interior regularity of viscosity solutions of the parabolic Monge-Ampère equation NEWLINE\[NEWLINE u_t = b(x,t)(\det D^2u)^p, \leqno(1) NEWLINE\]NEWLINE where \(p>0\), and \(b\) is bounded and measurable with \(\lambda\leq b(x,t)\leq \Lambda\) for some fixed constants \(\lambda>0\) and \(\Lambda<\infty\). The function \(u\) is assumed to be convex in \(x\) and increasing in \(t\).NEWLINENEWLINEEquations of this type arise in connection with the flow of convex \(n\)-dimensional hypersurfaces \(\Sigma^n(t)\) embedded in \(\mathbf{R}^{n+1}\), evolving under the Gauss curvature flow NEWLINE\[NEWLINE \frac{\partial X}{\partial t} = K^p \mathbf{N}, \leqno(2) NEWLINE\]NEWLINE where each point \(X\) moves in the inward normal direction \(\mathbf{N}\) with speed \(K^p\), where \(K\) is the Gauss curvature. If \(\Sigma^n(t)\) is expressed locally as a graph \(x_{n+1}=u(x,t)\), \(x\in \Omega\subset\mathbf{R}^n\), then \(u\) satisfies an equation of the form (1), with \(b=(1+|\nabla u|^2)^q\) where \(q=-[(n+2)p-1]/2\).NEWLINENEWLINEThe behaviour of solutions of (2) depends on the value of \(p\). \textit{B. Andrews} [Pac. J. Math. 195, No. 1, 1--34 (2000; Zbl 1028.53072)] showed if \(0<p\leq 1/n\), then any convex hypersurface evolving by (2) instantly becomes uniformly convex and smooth. On the other hand, if \(p>1/n\), convex hypersurfaces evolving by (2) may have flat sides that persist for some positive time. Furthermore, these solutions do not instantly become smooth, and may fail to be even of class \(C^{1,1}\) while the flat sides persist.NEWLINENEWLINEGeneralizing a result of \textit{L. A. Caffarelli} [Ann. Math. (2) 131, No. 1, 129--134 (1990; Zbl 0704.35045)] for the elliptic case, the authors prove that if at a time \(t\) the convex set \(D\) where the graph of a solution \(u\) of (1) agrees with a supporting plane contains a line segment, then either the extreme points of \(D\) lie on \(\partial\Omega\), or else \(u(\cdot,t)\) coincides with the initial data on \(D\) (as for example occurs with solutions with flat sides).NEWLINENEWLINEA key result is a further extension of this idea for angles rather than line segments. Namely, if at a time \(t\) the solution \(u\) admits a tangent angle from below, then either the set where \(u\) coincides with the edge of the angle has all its extreme points on \(\partial\Omega\), or else the same tangent angle touches the initial data from below.NEWLINENEWLINEThe \(C^{1,\alpha}\) regularity is closely related to this phenomenon. Clearly, \(C^1\) regularity cannot hold if angles persist for some time. It turns out that the exponent \(p=1/(n-2)\) is critical. If \(p<1/(n-2)\), then at any time after the initial time solutions are \(C^{1,\alpha}\) in the interior of any section of \(u(\cdot,t)\) which is contained in \(\Omega\). For \(p=1/(n-2)\) solutions are \(C^1\) with a logarithmic modulus of continuity for the gradient.NEWLINENEWLINEIn the case of arbitrary powers \(p>0\), \(C^{1,\alpha}\) estimates hold at any point where \(u\) separates from the initial data. Finally, if the initial data is \(C^{1,\beta}\) in some direction \(e\), then the solution \(u\) is \(C^{1,\alpha}\) in the same direction \(e\) for all later times, for some \(\alpha=\alpha(\beta)>0\).