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

TL;DR

This paper reviews recent advances over 10-15 years in the $L_p$ theory for elliptic and parabolic equations with discontinuous coefficients, covering various coefficient regularities and equation types.

Contribution

It summarizes key developments in $L_p$ estimates and methods for equations with discontinuous or irregular coefficients, highlighting progress in non-local and nonlinear cases.

Findings

Development of $L_p$ estimates for equations with measurable coefficients

Extension to non-local elliptic and parabolic equations

Advances in fully nonlinear elliptic and parabolic equations

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.