Periodic Homogenization for Weakly Elliptic Hamilton-Jacobi-Bellman Equations with Critical Fractional Diffusion

Adina Ciomaga (1) , Daria Ghilli (2) , Erwin Topp (3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 Dipartimento di Matematica Pura e Applicata [Padova]
3 Departamento de Matematica y C.C.

In this paper we establish periodic homogenization for Hamilton-Jacobi-Bellman (HJB) equations, associated to nonlocal operators of integro-differential type. We consider the case when the fractional diffusion has the same order as the drift term, and is weakly elliptic. The outcome of the paper is twofold. One one hand, we provide Lipschitz regularity results for weakly elliptic non-local HJB, extending the results previously obtained in [8]. On the other hand, we establish a convergence result, based on half relaxed limits and a comparison principle for the effective problem. The latter strongly relies on the regularity and the ellipticity properties of the effective Hamiltonian, for which a fine Lipschitz estimate of the corrector plays a crucial role.

Mots clés: regularity of generalized solutions; viscosity solutions; nonlinear elliptic equations; partial integro-differential equations; homogenization AMS Classification: 35D10; 35D40; 35J60; 35R09

Domaines: Equations aux dérivées partielles [math.AP]

Adina Ciomaga, Daria Ghilli, Erwin Topp. Periodic Homogenization for Weakly Elliptic Hamilton-Jacobi-Bellman Equations with Critical Fractional Diffusion. 2020. ⟨hal-02485630⟩