Hamilton-Jacobi equations in metric spaces

TL;DR

This paper provides an overview of Hamilton-Jacobi equations in metric spaces, focusing on well-posedness and recent developments in metric viscosity solutions, especially for the eikonal equation.

Contribution

It introduces and reviews recent notions of metric viscosity solutions for Hamilton-Jacobi equations, expanding the theory beyond Euclidean spaces.

Findings

Several new notions of metric viscosity solutions are discussed.

The well-posedness of Hamilton-Jacobi equations in metric spaces is established.

Connections to classical viscosity solutions in Euclidean spaces are clarified.

Abstract

These are lecture notes for our minicourse at OIST Summer Graduate School "Analysis and Partial Differential Equations" on June 12-17, 2023. We give an overview and collect a few important results concerning the well-posedness of Hamilton-Jacobi equations in metric spaces, especially several recently proposed notions of metric viscosity solutions to the eikonal equation. Basic knowledge about metric spaces and a review of viscosity solution theory in the Euclidean spaces are also presented.