Recent progress in the $L_p$ theory for elliptic and parabolic equations with discontinuous coefficients

Abstract

In this paper, we review some results over the last 10-15 years on elliptic and parabolic equations with discontinuous coefficients. We begin with an approach given by N. V. Krylov to parabolic equations in the whole space with VMOx coefficients. We then discuss some subsequent development including elliptic and parabolic equations with coefficients which are allowed to be merely measurable in one or two space directions, weighted Lp estimates with Muckenhoupt (Ap) weights, non-local elliptic and parabolic equations, as well as fully nonlinear elliptic and parabolic equations.