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.
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Finlay, C., Oberman, A.M. Approximate Homogenization of Fully Nonlinear Elliptic PDEs: Estimates and Numerical Results for Pucci Type Equations. J Sci Comput 77, 936–949 (2018). https://doi.org/10.1007/s10915-018-0730-x
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DOI: https://doi.org/10.1007/s10915-018-0730-x