On (discounted) global Eikonal equations in metric spaces

TL;DR

This paper investigates a novel global slope operator-based model for Eikonal equations in metric spaces, establishing existence, uniqueness, and stability of solutions, and offering new approximation methods and integration formulas.

Contribution

It introduces a new global slope operator approach to Eikonal equations in metric spaces, independent of local properties, and develops methods for solution existence, uniqueness, and stability analysis.

Findings

Existence and uniqueness of solutions established

A viscosity solution perspective with Perron's method employed

Stable solutions with respect to data and discount factor

Abstract

Eikonal equations in metric spaces have strong connections with the local slope operator (or the De Giorgi slope). In this manuscript, we explore and delve into an analogous model based on the global slope operator, expressed as $\lambda u + G[u] = \ell$, where $\lambda \geq 0$. In strong contrast with the classical theory, the global slope operator relies neither on the local properties of the functions nor on the structure of the space, and therefore new insights are developed in order to analyze the above equation. Under mild assumptions on the metric space $X$ and the given data $\ell$, we primarily discuss: $(a)$ the existence and uniqueness of (pointwise) solutions; $(b)$ a viscosity perspective and the employment of Perron's method to consider the maximal solution; $(c)$ stability of the maximal solution with respect to both, the data $\ell$ and the discount factor $\lambda$. Our techniques provide a method to approximate solutions of Eikonal equations in metric spaces and a new integration formula based on the global slope of the given function.