On the regularity for non-uniformly elliptic equations and applications to random homogenization

I will discuss regularity properties for solutions of linear second order non-uniformly elliptic equations in divergence form. Assuming certain integrability conditions on the coefficient field, we obtain local boundedness and validity of Harnack inequality. The assumed integrability assumptions are essentially sharp and improve upon some classical results in the literature. We then apply this deterministic regularity result to the corrector equation in stochastic homogenization and establish subilinearity of the corrector under essentially minimal assumptions. This is joint work with Peter Bella.