Aubry sets for weakly coupled systems of Hamilton--Jacobi equations
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.
Keywords
Cite
@article{arxiv.1211.1245,
title = {Aubry sets for weakly coupled systems of Hamilton--Jacobi equations},
author = {Andrea Davini and Maxime Zavidovique},
journal= {arXiv preprint arXiv:1211.1245},
year = {2016}
}
Comments
35 pages v.2 the introduction has been rewritten and shortened. Some proofs simplified. Corrections and references added. Corollary 5.3 added stating antisymmetry of the Ma\~n\'e matrix on points of the Aubry set. Section 6 contains a new examples