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

In this paper, we consider the following problem: (Formula Presented.) Here the coefficients a_ij and b_i are smooth, periodic with respect to the second variable, and the matrix (a_ij)_ij is uniformly elliptic. The Hamiltonian H is locally Lipschitz continuous with respect to u^ϵ and Du^ϵ, and has quadratic growth with respect to Du^ϵ. The Hamilton‐Jacobi‐Beliman equations of some stochastic control problems are of this type. Our aim is to pass to the limit in (0_ϵ) as ϵ tends to zero. We assume the coefficients b_i to be centered with respect to the invariant measure of the problem (see the main assumption (3.13)). Then we derive L^∞, H 10 and W1,p 0, p₀ > 2, estimates for the solutions of (0_ϵ). We also prove the following corrector's result: (Formula Presented.) This allows us to pass to the limit in (0_ϵ) and to obtain (Formula Presented.) This problem is of the same type as the initial one. When (0_ϵ) 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. Copyright © 1986 Wiley Periodicals, Inc., A Wiley Company