Approximate homogenization of fully nonlinear elliptic PDEs: estimates and numerical results for Pucci type equations

Abstract

We are interested in the shape of the homogenized operator $\overline F(Q)$ for PDEs which have the structure of a nonlinear Pucci operator. A typical operator is $H^{a_1,a_2}(Q,x) = a_1(x) \lambda_{\min}(Q) + a_2(x)\lambda_{\max}(Q)$. Linearization of the operator leads to a non-divergence form homogenization problem, which can be solved by averaging against the invariant measure. We estimate the error obtained by linearization based on semi-concavity estimates on the nonlinear operator. These estimates show that away from high curvature regions, the linearization can be accurate. Numerical results show that for many values of $Q$, the linearization is highly accurate, and that even near corners, the error can be small (a few percent) even for relatively wide ranges of the coefficients.