Electron. J. Diff. Equ.,
Vol. 2013 (2013), No. 93, pp. 124.
A gradient estimate for solutions to parabolic equations
with discontinuous coefficients
Jishan Fan, Kyoungsun Kim, Sei Nagayasu, Gen Nakamura
Abstract:
LiVogelius and LiNirenberg 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 402751, Korea
email: kskim@inha.ac.kr


Sei Nagayasu
Department of Mathematical Science
Graduate School of Material Science
University of Hyogo
2167 Shosha, Himeji, Hyogo 6712280, Japan
email: sei@sci.uhyogo.ac.jp


Gen Nakamura
Department of Mathematics, Inha University
Incheon 402751, Korea
email: nakamuragenn@gmail.com

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