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 $\epsilon\to 0$. 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 $\Gamma_\epsilon$.