TY - JOUR
T1 - Recent Progress in the $L_p$ Theory for Elliptic and Parabolic Equations with Discontinuous Coefficients
AU - Dong , Hongjie
JO - Analysis in Theory and Applications
VL - 2
SP - 161
EP - 199
PY - 2020
DA - 2020/06
SN - 36
DO - http://doi.org/10.4208/ata.OA-0021
UR - https://global-sci.org/intro/article_detail/ata/17129.html
KW - Elliptic and parabolic equations and systems, nonlocal equations, fully nonlinear equations, VMO and partially VMO coefficients, weighted estimates, Muckenhoupt weights.
AB - <p style="text-align: justify;">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 $\rm{VMO}_x$ 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 $L_p$&nbsp;estimates with Muckenhoupt ($A_p$)<sub> </sub>weights, non-local elliptic and parabolic equations, as well as fully nonlinear elliptic and parabolic equations.</p>