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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 $Γ_ε$. The oscillations of the interface have small period and amplitude, both of the order of $ε$, and the interfaces $Γ_ε$ tend to a straight line $Γ$. We study the asymptotic behavior as $ε\to 0$. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by $Γ$, with an effective transmission condition on $Γ$ keeping track of the oscillations of $Γ_ε$.