Approximate homogenization of convex nonlinear elliptic PDEs

Chris Finlay; Adam M. Oberman

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arXiv · 1710.10309

Approximate homogenization of convex nonlinear elliptic PDEs

Chris Finlay·Adam M. Oberman

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Abstract

We approximate the homogenization of fully nonlinear, convex, uniformly elliptic Partial Differential Equations in the periodic setting, using a variational formula for the optimal invariant measure, which may be derived via Legendre-Fenchel duality. The variational formula expresses $\bar H$ as an average of the operator against the optimal invariant measure, generalizing the linear case. Several nontrivial analytic formulas for $\bar H$ are obtained. These formulas are compared to numerical simulations, using both PDE and variational methods. We also perform a numerical study of convergence rates for homogenization in the periodic and random setting and compare these to theoretical results.

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Chris Finlay, Adam M. Oberman. 2017-10-27. Approximate homogenization of convex nonlinear elliptic PDEs. https://arxiv.org/abs/1710.10309

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