Approximate Homogenization of Fully Nonlinear Elliptic PDEs: Estimates and Numerical Results for Pucci Type Equations

We are interested in the shape of the homogenized operator 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 ) λ min ( Q ) + a 2 ( x ) λ 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.