Differential games and Hamilton-Jacobi equations in the Heisenberg group

TL;DR

This paper studies Hamilton-Jacobi equations on the Heisenberg group, proving Lipschitz regularity of solutions and introducing game theory to analyze the sub-Riemannian structure, leading to new regularity results and solution representations.

Contribution

It introduces a novel approach combining Hamilton-Jacobi equations and game theory within the Heisenberg group, establishing Lipschitz regularity and viscosity solution representations.

Findings

Proved Lipschitz continuity of solutions with respect to Korányi distances.

Established Lipschitz regularity for value functions in a zero-sum game on the Heisenberg group.

Provided a representation formula for viscosity solutions of Hamilton-Jacobi equations.

Abstract

The purpose of this work is twofold. First we study the solutions of a Hamilton-Jacobi equation of the form $u_t(t,x)+\mathcal{H}(t,x,\nabla_H u(t,x))=0$, where $\nabla_H u$ represents the horizontal gradient of a function $u$ defined on the Heisenberg group ${I\!\!H}$. Motivated by the recent paper by Liu, Manfredi and Zhou (\cite{LiMaZh2016}), we prove a Lipschitz continuity preserving property for $u$ with respect to the Kor\'anyi homogeneous distances $d_G$ in ${I\!\!H}$. Secondly, we are keenly interested in introducing the game theory in ${I\!\!H}$, taking into account its Sub-Riemannian structure: inspired by ideas in the paper of Evans and Souganidis (see \cite{EvSo1984}), in the paper of and Balogh, Calogero and Pini \cite{BaCaPi2014}, we prove $d_G$-Lipschitz regularity results for the lower and the upper value functions of a zero game with horizontal curves as its trajectories, and we study the Hamilton-Jacobi-Isaacs equations associated to such zero game. As a consequence, we also provide a representation of the viscosity solution of the initial Hamilton-Jacobi equation.