Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group

Abstract

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.

Hölder behavior of viscosity solutions of some fully nonlinear equations in the Heisenberg group

“…Regularity of viscosity solutions is a subject that attracts the interests of many researchers. Thus we like to point out the following less recent and recent results about some properties of the solutions of nonlinear equations in the elliptic degenerate case: [25], [22], [23], [5], [1] and, concerning the evolutive framework, [20].…”

Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces

“…In general, one cannot expect the convexity preserving property still hold in general geodesic spaces even for very simple equations. In fact, the solution of a first order linear PDE fails to preserve horizontal convexity in the first Heisenberg group, as shown in [26]. It is however not clear whether preservation of horizontal convexity holds even for the Hamilton-Jacobi flow (HJ).…”

An application of the theorem on Sums to viscosity solutions of degenerate fully nonlinear equations

“…We point out this aspect since, on the other hand, there exists also a literature that deal with intrinsic regularity results, see e.g. [14]. In particular those results are stated using intrinsic notions associated with the geometry of the operator considered.…”

“…We point out this aspect since there exists also a wide literature in which the obtained results are introduced by using an intrinsic notion of regularity, see for example, [16]. In particular, those results are stated by exploiting the intrinsic notions of distance and differentiability associated with the geometry of the operator.…”