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 VMO x 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 L p estimates with Muckenhoupt (A p ) weights, non-local elliptic and parabolic equations, as well as fully nonlinear elliptic and parabolic equations.

$ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations

“…When full regularity is not available, partial regularity results can be found, see [1]. For recent progress and a survey of the latest regularity results for equations with rough coefficients, see [10].…”