Monge solution for discontinuous Hamilton-Jacobi equations in Carnot groups

Fares Essebei (IMATI Pavia)

Abstract.
In the seminar, I first introduce the setting of sub-Riemanninan geometry. Then, I present the notion of Monge solution for stationary Hamilton-Jacobi equations associated with discontinuous Hamiltonians, in the framework of Carnot groups.
The sub-Riemannian Hamilton–Jacobi equation is the form
\begin{equation}
(1) \qquad\qquad H(x,Xu)=0,
\end{equation}
where $\Omega$ is a subdomain of a Carnot group $\mathbb{G}$ of rank $m$, $Xu$ is the horizontal gradient associated to $\mathbb{G}$ and the Hamiltonian $H:\Omega\times \mathbb{R}^m \longrightarrow \mathbb{R}$ satisfies the following structural assumptions (H):

[H_1] $H: \Omega \times \mathbb{R}^m \to \mathbb{R}$ is Borel measurable.

[H_2] The set
\[
Z(x):=\{p\in\mathbb{R}^m\,:\,H(x,p) \le 0\}
\]
is closed, convex and $\partial Z(x)=\{p\in\mathbb{R}^m\,:\,H(x,p)=0\}$ for any $x\in\Omega$.

[H_3] There exists $\alpha >1$ such that
$$\hat{B}_{\frac{1}{\alpha}}(0) \subset Z(x) \subset \hat{B}_{\alpha}(0)$$
for any $x\in\Omega$, where $\hat{B}_{\alpha}(0)$ is Euclidean ball of radius $\alpha$ centered at the origin in $\mathbb{R}^m$.

I will show the equivalence between Monge and viscosity solutions in the continuous setting. Then, I prove the existence and the uniqueness of the Dirichlet problem associated with (1), together with a comparison principle and a stability result.

This is a joint work with Gianmarco Giovannardi and Simone Verzellesi.