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May  2016, 15(3): 761-794. doi: 10.3934/cpaa.2016.15.761

## A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space

 1 Department of mathematics, Korea university, 1 anam-dong, sungbuk-gu, Seoul 136-701, South Korea 2 Department of Mathematics, Ajou University, 206 Worldcup-ro, Yeontong-gu, Suwon 443-749, South Korea

Received  March 2015 Revised  December 2015 Published  February 2015

In this article we consider parabolic systems and $L_p$ regularity of the solutions. With zero boundary condition the solutions experience bad regularity near the boundary. This article addresses a possible way of describing the regularity nature. Our space domain is a half space and we adapt an appropriate weight into our function spaces. In this weighted Sobolev space setting we develop a Fefferman-Stein theorem, a Hardy-Littlewood theorem and sharp function estimations. Using these, we prove uniqueness and existence results for second-order elliptic and parabolic partial differential systems in weighed Sobolev spaces.
Citation: Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761
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