Homogenization of elliptic equations with principal part not in divergence form and hamiltonian with quadratic growth
Alain Bensoussan, Lucio Boccardo, François Murat · Communications on Pure and Applied Mathematics · 1986
Abstract In this paper, we consider the following problem: equation image 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 and W , p 0 > 2, estimates for the solutions of (0 ϵ ). We also prove the following corrector's result: equation image This allows us to pass to the limit in (0 ϵ ) and to obtain equation image 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 (0 0 ) is also a Hamilton‐Jacobi‐Bellman equation but one corresponding to a modified set of controls.