Homogenization of elliptic equations with principal part not in divergence form and hamiltonian with quadratic growth

In this paper, we consider the following problem: Here the coefficients aij and bi are smooth, periodic with respect to the second variable, and the matrix (aij)ij is uniformly elliptic. The Hamiltonian H is locally Lipschitz continuous with respect to uϵ and Duϵ, and has quadratic growth with respect to Duϵ. The Hamilton-Jacobi-Beliman equations of some stochastic control problems are of this type. Our aim is to pass to the limit in (0ϵ) as ϵ tends to zero. We assume the coefficients bi to be centered with respect to the invariant measure of the problem (see the main assumption (3.13)). Then we derive L∞, H and W, p0 > 2, estimates for the solutions of (0ϵ). We also prove the following corrector's result: This allows us to pass to the limit in (0ϵ) and to obtain This problem is of the same type as the initial one. When (0ϵ) is the Hamilton-Jacobi-Bellman equation of a stochastic control problem, then (00) is also a Hamilton-Jacobi-Bellman equation but one corresponding to a modified set of controls.