Approximate homogenization of convex nonlinear elliptic PDEs
Chris Finlay, Adam M. Oberman · Communications in Mathematical Sciences · 2018
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 H(Q) as an average of the operator against the optimal invariant measure, generalizing the linear case.Several nontrivial analytic formulas for H(Q) 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.