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Homogenization of elliptic equations with principal part not in divergence form and Hamiltonian with quadratic growth. (English) Zbl 0602.35030
We consider the following problem: \[ (0_{\epsilon})\quad - a_{ij}(x,x/\epsilon)\partial^ 2u^{\epsilon}/\partial x_ i \partial x_ j-(1/\epsilon)b_ i(x,x/\epsilon)\partial u^{\epsilon}/\partial x_ i=H(x,x/\epsilon),u^{\epsilon},Du^{\epsilon}) \] in \({\mathcal O}\), \(u_{\epsilon}=0\) on \(\Gamma\).
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 and Du, and has quadratic growth in Du. The Hamilton- Jacobi-Bellman equations of some stochastic control problems are of this type.
Our aim is to pass to the limit in \((0_{\epsilon})\) as \(\epsilon\) 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 \(L^{\infty}\), \(H^ 1_ 0\) and \(W^{1,p_ 0}(p_ 0>2)\) estimates for the solutions of \((0_{\epsilon})\). We also prove the following corrector’s result: \[ Du^{\epsilon}-Du+D\chi (x,x/\epsilon)Du\to 0\quad in\quad (L^ 2({\mathcal O}))^ N\quad strongly. \] This allows one to pass to the limit in \((0_{\epsilon})\) and to obtain: \[ (0_ 0)\quad -q_{ij}(x) \partial^ 2u/\partial x_ i \partial x_ j-r_ i(x) \partial u/\partial x_ i={\mathcal H}(x,u,Du)\quad in\quad {\mathcal O},\quad u=0\quad on\quad \Gamma. \] This problem is of the same type as the initial one. When \((0_{\epsilon})\) is the Hamilton-Jacobi-Bellman equation of a stochastic control problem, then \((0_ 0)\) is also a Hamilton-Jacobi-Bellman equation but corresponding to modified set of controls.

MSC:
35J60 Nonlinear elliptic equations
49L99 Hamilton-Jacobi theories
35B40 Asymptotic behavior of solutions to PDEs
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