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Homogenization of a transmission problem with Hamilton-Jacobi equations and a two-scale interface. Effective transmission conditions

Abstract : We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across a two-scale oscillatory interface. Typically, the amplitude of the oscillations is of the order of ε while the period is of the order of ε 2. As ε → 0, the interfaces tend to a straight line Γ. We study the asymptotic behavior of the value function as ε → 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.
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https://hal.archives-ouvertes.fr/hal-01556296
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Submitted on : Tuesday, July 4, 2017 - 10:23:20 PM
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Yves Achdou, Nicoletta Tchou. Homogenization of a transmission problem with Hamilton-Jacobi equations and a two-scale interface. Effective transmission conditions. Journal de Mathématiques Pures et Appliquées, Elsevier, 2019, 122, pp.164-197. ⟨10.1016/j.matpur.2018.04.005⟩. ⟨hal-01556296⟩

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