Geodesic
A gradient estimate for solutions to parabolic equations with discontinuous coefficients
Electronic journal of differential equations, Tome 2013 (2013)
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 emphmanifolds 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.
Classification : 35K10, 35B65
Keywords: parabolic equations, discontinuous coefficients, gradient estimate 
@article{EJDE_2013__2013__a6,
     author = {Fan,  Jishan and Kim,  Kyoungsun and Nagayasu,  Sei and Nakamura,  Gen},
     title = {A gradient estimate for solutions to parabolic equations with discontinuous coefficients},
     journal = {Electronic journal of differential equations},
     year = {2013},
     volume = {2013},
     zbl = {1288.35129},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2013__2013__a6/}
}
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%A Nakamura,  Gen
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%J Electronic journal of differential equations
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%F EJDE_2013__2013__a6
Fan,  Jishan; Kim,  Kyoungsun; Nagayasu,  Sei; Nakamura,  Gen. A gradient estimate for solutions to parabolic equations with discontinuous coefficients. Electronic journal of differential equations, Tome 2013 (2013). http://geodesic.mathdoc.fr/item/EJDE_2013__2013__a6/