Abstract. In this paper we present some new ideas to derive a priori second order estiamtes for a wide class of fully nonlinear parabolic equations. Our methods, which produce new existence results for the initial-boundary value problems in R n , are powerful enough to work in general Riemannian manifolds.Mathematical Subject Classification (2010): 35K10, 35K55, 58J35, 35B45.

Recent work has shown that mouse and human fibroblasts can be reprogrammed to cardiomyocyte-like cells with a combination of transcription factors. Current research has focused on improving the efficiency and mechanisms for fibroblast reprogramming. Previously, it has been reported that hypoxia enhances fibroblast cell reprogramming to pluripotent stem cells. In this study, we observed that 6 h of hypoxic conditions (2% oxygen) on newborn mouse dermal fibroblasts can improve the efficiency of reprogramming to cardiomyocyte-like cells. Expression of cardiac-related genes and proteins increased at 4 weeks after transfer of three transcription factors (Gata4/Mef2c/Tbx5 [GMT]). However, beating cardiomyocyte cells were not detected. The epigenetic mechanism of hypoxia-induced fibroblast reprogramming to cardiomyocyte cells requires further study.

We prove a constant rank theorem for the second fundamental form of the spatial convex level surfaces of solutions to equations ut = F (∇ 2 u, ∇u, u, t) under a structural condition, and give a geometric lower bound of the principal curvature of the spatial level surfaces.

In this paper, for the solutions of two elliptic equations we find the auxiliary curvature functions which attain respective minimum on the boundary. These results are the generalization of the classical ones in Makar-Limanov [17] for the torsion equation and Acker et al. [1] for the first eigenfunction of the Laplacian in convex domains of dimension 2. Then we get the new proof of the specific convexity of the solutions of the above two elliptic equations. As a consequence, for the elliptic equation v v = − 1 + v 2 in a smooth, bounded and strictly convex domain in n with homogeneous Dirichlet boundary value condition, we also get a sharply lower bound estimate of the Gaussian curvature for the solution surface by the curvature of the boundary of the domain.