Homogenization of weakly coupled systems of Hamilton--Jacobi equations with fast switching rates

We consider homogenization for weakly coupled systems of Hamilton--Jacobi equations with fast switching rates. The fast switching rate terms force the solutions converge to the same limit, which is a solution of the effective equation. We discover the appearance of the initial layers, which appear naturally when we consider the systems with different initial data and analyze them rigorously. In particular, we obtain matched asymptotic solutions of the systems and rate of convergence. We also investigate properties of the effective Hamiltonian of weakly coupled systems and show some examples which do not appear in the context of single equations.Comment: final version, to appear in Arch. Ration. Mech. Ana

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

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.Comment: 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 example

Stochastic Homogenization of Monotone Systems of Viscous Hamilton--Jacobi Equations with Convex Nonlinearities

We consider the homogenization of monotone systems of viscous Hamilton--Jacobi equations with convex nonlinearities set in the stationary, ergodic setting. The primary focus of this paper is on collapsing systems which, as the microscopic scale tends to zero, average to a deterministic scalar Hamilton--Jacobi equation. However, our methods also apply to systems which do not collapse and, as the microscopic scale tends to zero, average to a deterministic system of Hamilton--Jacobi equations

Stochastic Homogenization of Monotone Systems of Viscous Hamilton--Jacobi Equations with Convex Nonlinearities

We consider the homogenization of monotone systems of viscous Hamilton--Jacobi equations with convex nonlinearities set in the stationary, ergodic setting. The primary focus of this paper is on collapsing systems which, as the microscopic scale tends to zero, average to a deterministic scalar Hamilton--Jacobi equation. However, our methods also apply to systems which do not collapse and, as the microscopic scale tends to zero, average to a deterministic system of Hamilton--Jacobi equations

Stochastic Homogenization of Monotone Systems of Viscous Hamilton--Jacobi Equations with Convex Nonlinearities

We consider the homogenization of monotone systems of viscous Hamilton--Jacobi equations with convex nonlinearities set in the stationary, ergodic setting. The primary focus of this paper is on collapsing systems which, as the microscopic scale tends to zero, average to a deterministic scalar Hamilton--Jacobi equation. However, our methods also apply to systems which do not collapse and, as the microscopic scale tends to zero, average to a deterministic system of Hamilton--Jacobi equations