A microscopic convexity principle for spacetime convex solutions of fully nonlinear parabolic equations
The authors investigate the following fully nonlinear parabolic equation: \[ \frac{\, \partial u \,}{\, \partial t \,} = F(D^2 u, Du, u, x, t), \qquad (x,t) \in \Omega \times (0,T], \] where \(F\) is smooth enough and elliptic, and \(\Omega\) is a domain in~\(\mathbb R^n\). Attention is devoted to solutions \(u\) that are convex in the space-time variable \((x,t)\), i.e., solutions whose Hessian matrix \(\widehat{D^2 u}\) with respect to \((x,t)\) is positive semidefinite. Constant-rank theorems for \(\widehat{D^2 u}\) are proved, in the spirit of \textit{L. A. Caffarelli} and \textit{A. Friedman} [Duke Math. J. 52, 431--456 (1985; Zbl 0599.35065)], \textit{I. M. Singer} et al. [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 319--333 (1985; Zbl 0603.35070)] and \textit{N. J. Korevaar} and \textit{J. L. Lewis} [Arch. Ration. Mech. Anal. 97, 19--32 (1987: Zbl 0624.35031)]. To be more specific, in Theorem~1.3 the authors prove that if \[ \text{\(F(A^{-1},p,u,x,t)\) is locally convex in \((A,u,x,t)\) for each fixed \(p \in \mathbb R^n\),} \tag{1} \] then the rank of \(\widehat{D^2 u}\) is constant in~\(x\) for each fixed \(t \in (0,T]\), and therefore it is a function \(l(t)\). Such a function is also shown to be monotone non-decreasing in~\(t\). The subsequent Theorem~1.4 deals with solutions that are not only convex in \((x,t)\), but also strictly convex in the space variable~\(x\), i.e., solutions whose Hessian matrix \(D^2 u\) with respect to the space variable~\(x\) is positive definite. In this case, assumption (1) is weakened: indeed, the theorem requires that the function \((A,u,x,t) \mapsto F(A^{-1},p,u,x,t)\) has convex level subsets for every fixed \(p \in \mathbb R^n\). Now, the conclusion is different than before: if the matrix \(\widehat{D^2 u}\) (which has order \(n + 1\)) attains the rank \(n\) at some point \((x_0,t_0) \in \Omega \times (0,T]\), then it has rank~\(n\) throughout \(\Omega \times (0,T]\). The proof is based on results by \textit{B. Bian} and \textit{P. Guan} [Invent. Math. 177, No. 2, 307--335 (2009; Zbl 1168.35010)]. Essentially, the claim follows by applying the strong maximum principle for parabolic equations to suitable functions defined in terms of the symmetric functions \(\sigma_k\) of the eigenvalues \(\lambda_i\) of the matrix \(\widehat{D^2 u}\). Consequences for parabolic convexity and mean curvature flow are derived.
scientific article; zbMATH DE number 6149087
scientific article