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

Kyeong-Hun  Kim; Kijung  Lee

doi:10.3934/cpaa.2016.15.761

Article Contents

2016, Volume 15, Issue 3: 761-794. Doi: 10.3934/cpaa.2016.15.761

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

Kyeong-Hun Kim^1, and
Kijung Lee^2,

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

Received: February 28, 2015

Revised: November 30, 2015

Published: April 2016

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K51, 35J57; Secondary: 42B37.

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