Calculus of Variations and Geometric Measure Theory

A. Davini - M. Zavidovique

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

created by davini on 12 Jul 2023
modified on 13 Jul 2023


Published Paper

Inserted: 12 jul 2023
Last Updated: 13 jul 2023

Journal: SIAM J. Math. Anal.
Volume: 46
Number: 5
Pages: 3361–3389
Year: 2014

ArXiv: 1211.1245 PDF


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.