On the doubling of variables technique in first order Hamilton-Jacobi equations

TL;DR

This paper revisits the doubling variables technique in first order Hamilton-Jacobi equations, demonstrating how tuning penalization affects proofs and regularity assumptions, with applications to Wasserstein spaces.

Contribution

It introduces a new approach to the doubling variables method by adjusting penalization, linking regularity to geometric properties, and extends it to Wasserstein space equations.

Findings

Tuning penalization alters proof strategies.

Regularity hypotheses can be replaced by geometric properties.

Application to Wasserstein space equations demonstrates versatility.

Abstract

In this paper, we revisit the technique of doubling variables in first order Hamilton-Jacobi equations, especially when the equations arise in optimal control. We show that by tuning the penalization between the two points, we can change drastically the proof, somehow shifting the regularity hypotheses into geometrical properties of the penalization. We present this idea in a finite dimensional setting and then exploit it on equations posed on Wasserstein spaces.