Volume 36, Issue 2
Recent Progress in the $L_p$ Theory for Elliptic and Parabolic Equations with Discontinuous Coefficients

Anal. Theory Appl., 36 (2020), pp. 161-199.

Published online: 2020-06

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• 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.

• Keywords

Elliptic and parabolic equations and systems, nonlocal equations, fully nonlinear equations, VMO and partially VMO coefficients, weighted estimates, Muckenhoupt weights.

35R05, 35B45, 35B65, 42B37, 35K20, 35J15, 35R11, 35K10, 35K45, 35J60, 35K55, 60J75

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• TXT
@Article{ATA-36-161, author = {Dong , Hongjie}, title = {Recent Progress in the $L_p$ Theory for Elliptic and Parabolic Equations with Discontinuous Coefficients}, journal = {Analysis in Theory and Applications}, year = {2020}, volume = {36}, number = {2}, pages = {161--199}, 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.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-0021}, url = {http://global-sci.org/intro/article_detail/ata/17129.html} }
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 -

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.

Hongjie Dong. (2020). Recent Progress in the $L_p$ Theory for Elliptic and Parabolic Equations with Discontinuous Coefficients. Analysis in Theory and Applications. 36 (2). 161-199. doi:10.4208/ata.OA-0021
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