Regularity theory of degenerate elliptic equations in nondivergence form with applications to homogenization

We will present a regularity result for degenerate elliptic equations in nondivergence form.

In joint work with Charlie Smart, we extend the regularity theory of Caffarelli to equations with possibly unbounded ellipticity-- provided that the ellipticity satisfies an averaging condition. As an application we obtain a stochastic homogenization result for such equations (and a new estimate for the effective coefficients) as well as an invariance principle for random diffusions in random environments. The degenerate equations homogenize to uniformly elliptic equations, and we give an estimate of the ellipticity.