Aubry Sets for Weakly Coupled Systems of Hamilton--Jacobi Equations

Abstract

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.

Herglotz’ Generalized Variational Principle and Contact Type Hamilton-Jacobi Equations

“…-Along this line, the quatitative semiconcavity and convexity estimate of the associated fundamental solutions have been obtained in [7] recently, which is useful for the intrinsic study of the global propagation of singularities of the viscosity solutions of (5) and (6) ( [8,4,5,6]); -When the Lagrangian has the form L(x, v) − λ u, by solving the associated Carathéodory equation (9) directly, one gets the representation formula for the associated viscosity solutions immediately ( [19,40,43,13]). The representation formula bridges the PDE aspects of the problem with the dynamical ones; -Consider a family of Lagrangians in the form {L(x, v) + ∑ k i=1 a i j u i }, a problem of Herglotz' variational principle in the vector form is closely connected to certain stochastic model of weakly coupled Hamilton-Jacobi equations (see, for instance, [20,23,38]).…”

Weak KAM theory for a weakly coupled system of Hamilton–Jacobi equations

“…In [6], motivated by earlier publications [2,7,11], the authors extended the notion of viscosity solution to these problems and proved that their value functions are viscosity solutions of a weakly coupled system of Hamilton-Jacobi equations. Recently, several authors have investigated random switching problems, their weakly coupled Hamilton-Jacobi equations [19], the corresponding extensions of the weak KAM and Aubry-Mather theories [10,17], the long-time behavior of solutions [3,4,18,20,22], and homogenization questions [21]. In these references, as in the present paper, the state of the systems has different modes.…”

“…10 We say that f : A → R is semiconcave with a linear modulus of continuity if there exists C > 0 such that…”

Uniqueness structure of weakly coupled systems of ergodic problems of Hamilton–Jacobi equations

“…In [19], the authors studied homogenization for weakly coupled systems and the rate of convergence to matched solutions. On the other hand, [1] generalized the notion of Aubry sets for the case of systems and proved comparison principle with respect to their boundary data on Aubry sets. In [17], the authors characterized the subsolutions of the systems and showed explicit representation for subsolutions enjoying maximal property.…”