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

TL;DR

This paper investigates viscosity solutions of semilinear parabolic equations in the Heisenberg group, focusing on uniqueness, Lipschitz and convexity preservation, and highlighting differences from Euclidean cases.

Contribution

It establishes uniqueness of solutions with exponential growth and explores Lipschitz and convexity preservation properties specific to the Heisenberg group, including counterexamples.

Findings

Uniqueness of viscosity solutions with exponential growth.

Lipschitz and convexity preservation properties do not always hold in the Heisenberg group.

Counterexamples demonstrate differences from Euclidean space results.

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.