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

Hongjie Dong

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.

  • AMS Subject Headings

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

  • Copyright

COPYRIGHT: © Global Science Press

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@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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