In this paper, we revisit the convexity preserving properties for the level set mean curvature flow equation by using the game-theoretic approximation established by Kohn and Serfaty (2006). Our new proofs are based on investigating game strategies or iterated applications of dynamic programming principle, without invoking deep partial differential equation theory. We also use this method to study convexity preserving for the Neumann boundary problem.

Abstract. In this paper we study viscosity solutions of semilinear parabolic equations in the Heisenberg group. We show uniqueness of viscosity solutions with exponential growth at space infinity. We also study Lipschitz and horizontal convexity preserving properties under appropriate assumptions. Counterexamples show that in general such properties that are well-known for semilinear and fully nonlinear parabolic equations in the Euclidean spaces do not hold in the Heisenberg group.

No abstract

This paper introduces in a natural way a notion of horizontal convex envelopes of continuous functions in the Heisenberg group. We provide a convexification process to find the envelope in a constructive manner. We also apply the convexification process to show h-convexity of viscosity solutions to a class of fully nonlinear elliptic equations in the Heisenberg group satisfying a certain symmetry condition. Our examples show that in general one cannot expect h-convexity of solutions without the symmetry condition.

We extend Bony's propagation of support argument [2] to C 1 solutions of the non-homogeneous sub-elliptic p−Laplacian associated to a system of smooth vector fields satisfying Hörmander's finite rank condition. As a consequence we prove a strong maximum principle and strong comparison principle that generalize results of Tolksdorf [7].