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 |
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Kyoungsun Kim Department of Mathematics, Inha University Incheon 402-751, Korea email: kskim@inha.ac.kr |
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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 |
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Gen Nakamura Department of Mathematics, Inha University Incheon 402-751, Korea email: nakamuragenn@gmail.com |
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