On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions (Q321588)

This paper focuses on the microscopic spacetime convexity principle for the second fundamental forms of the spatial and spacetime level sets of the solutions to fully nonlinear parabolic equations. The author generalizes the constant rank theorem in [\textit{C. Q. Chen}, \textit{X. N. Ma} and \textit{P. Salani}, ``On the spacetime quasiconcave solutions of the heat equation'', Preprint, \url{arXiv:1405.6373}] to fully nonlinear parabolic equations, establishing the corresponding microscopic spacetime convexity principles for spacetime level sets. The results hold for fully nonlinear parabolic equations under a general structural assumption, including \(p\)-Laplacian parabolic equations for \(p>1\) and certain mean curvature-type parabolic equations.NEWLINENEWLINEIn more detail, the author considers the spacetime quasiconcave solution to fully nonlinear parabolic equations as NEWLINE\[NEWLINE \frac{\partial u}{\partial t}=F(\nabla^2 u,\nabla u, u,x,t) \;\;\text{ in } \Omega\times(0,T], NEWLINE\]NEWLINE where \(\Omega\) is a domain in \(\mathbb{R}^n\) and \(F\) and \(u\) are assumed to satisfy certain conditions.NEWLINENEWLINEThe main result of the paper states that, under certain additional assumptions, if \(u\in C^{3,1}(\Omega\times (0,T])\) is a spacetime quasiconcave solution to the above fully nonlinear parabolic equation, then the second fundamental form of spacetime level sets has the same constant rank in \(\Omega\) for each fixed \(t\in (0,T]\). A similar constant rank theorem is also obtained for the second fundamental form of the spatial level sets. It is also shown that both results hold for the spacetime quasiconcave solutions to \(p\)-Laplacian parabolic equations (\(p>1\)), and also certain mean curvature-type parabolic equations.