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A lagrangian approach to weakly coupled Hamilton-Jacobi systems
We study a class of weakly coupled Hamilton–Jacobi systems with a specific
aim to perform a qualitative analysis in the spirit of weak KAM theory. Our main
achievement is the definition of a family of related action functionals containing the
Lagrangians obtained by duality from the Hamiltonians of the system. We use them to
characterize, by means of a suitable estimate, all the subsolutions of the system, and
to explicitly represent some subsolutions enjoying an additional maximality property. A
crucial step for our analysis is to put the problem in a suitable random frame. Only
some basic knowledge of measure theory is required, and the presentation is accessible
to readers without background in probability