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We consider a family of optimal control problems in the plane with dynamicsand running costs possibly discontinuous across an oscillatory interface$\Gamma_\epsilon$. The oscillations of the interface have small period andamplitude, both of the order of $\epsilon$, and the interfaces$\Gamma_\epsilon$ tend to a straight line $\Gamma$. We study the asymptoticbehavior as $\epsilon\to 0$. We prove that the value function tends to thesolution of Hamilton-Jacobi equations in the two half-planes limited by$\Gamma$, with an effective transmission condition on $\Gamma$ keeping track ofthe oscillations of $\Gamma_\epsilon$.

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