Electron. J. Diff. Equ., Vol. 2013 (2013), No. 93, pp. 1-24.

A gradient estimate for solutions to parabolic equations with discontinuous coefficients

Jishan Fan, Kyoungsun Kim, Sei Nagayasu, Gen Nakamura

Abstract:
Li-Vogelius and Li-Nirenberg gave a gradient estimate for solutions of strongly elliptic equations and systems of divergence forms with piecewise smooth coefficients, respectively. The discontinuities of the coefficients are assumed to be given by manifolds of codimension 1, which we called them \emph{manifolds of discontinuities}. Their gradient estimate is independent of the distances between manifolds of discontinuities. In this paper, we gave a parabolic version of their results. That is, we gave a gradient estimate for parabolic equations of divergence forms with piecewise smooth coefficients. The coefficients are assumed to be independent of time and their discontinuities are likewise the previous elliptic equations. As an application of this estimate, we also gave a pointwise gradient estimate for the fundamental solution of a parabolic operator with piecewise smooth coefficients. Both gradient estimates are independent of the distances between manifolds of discontinuities.

Submitted November 11, 2012. Published April 11, 2013.
Math Subject Classifications: 35K10, 35B65.
Key Words: Parabolic equations; discontinuous coefficients; gradient estimate.

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Jishan Fan
Department of Applied Mathematics
Nanjing Forestry University
Nanjing 210037, China
email: fanjishan@njfu.edu.cn
Kyoungsun Kim
Department of Mathematics, Inha University
Incheon 402-751, Korea
email: kskim@inha.ac.kr
Sei Nagayasu
Department of Mathematical Science
Graduate School of Material Science
University of Hyogo
2167 Shosha, Himeji, Hyogo 671-2280, Japan
email: sei@sci.u-hyogo.ac.jp
Gen Nakamura
Department of Mathematics, Inha University
Incheon 402-751, Korea
email: nakamuragenn@gmail.com

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