Aubry sets for weakly coupled systems of Hamilton--Jacobi equations

TL;DR

This paper extends the concept of Aubry sets to weakly coupled Hamilton--Jacobi systems on the torus, characterizing regions where critical subsolutions fail to be strict and establishing uniqueness and rigidity properties.

Contribution

It introduces a new Aubry set notion for coupled systems, characterizes it, and proves existence of smooth, strict subsolutions outside this set, advancing understanding of critical solutions.

Findings

Aubry set for coupled systems is characterized as obstruction region.

Existence of smooth, strict critical subsolutions outside the Aubry set.

The Aubry set acts as a uniqueness set for the critical system.

Abstract

We introduce a notion of Aubry set for weakly coupled systems of Hamilton--Jacobi equations on the torus and characterize it as the region where the obstruction to the existence of globally strict critical subsolutions concentrates. As in the case of a single equation, we prove the existence of critical subsolutions which are strict and smooth outside the Aubry set. This allows us to derive in a simple way a comparison result among critical sub and supersolutions with respect to their boundary data on the Aubry set, showing in particular that the latter is a uniqueness set for the critical system. We also highlight some rigidity phenomena taking place on the Aubry set.