Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across an oscillatory interface Gamma(epsilon). The oscillations of the interface have small period and amplitude, both of the order of epsilon, and the interfaces Gamma(epsilon) tend to a straight line Gamma. We study the asymptotic behavior as epsilon -> 0. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by Gamma, with an effective transmission condition on Gamma keeping track of the oscillations of Gamma(epsilon). (C) 2016 Elsevier Masson SAS. All rights reserved.