Illinois Journal of Mathematics

The $L^{p}$ regularity problem for the heat equation in non-cylindrical domains

Steven Hofmann and John L. Lewis

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We consider the Dirichlet problem for the heat equation in domains with a minimally smooth, time-varying boundary. Our boundary data is taken to belong to a parabolic Sobolev space having a tangential (spatial) gradient, and $1/2$ of a time derivative, in $L^{p}$, $1 \lt p \lt 2 + \epsilon$. We obtain sharp $L^{p}$ estimates for the parabolic non-tangential maximal function of the gradient of our solutions.

Article information

Illinois J. Math., Volume 43, Issue 4 (1999), 752-769.

First available in Project Euclid: 20 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K05: Heat equation
Secondary: 31B10: Integral representations, integral operators, integral equations methods 35B65: Smoothness and regularity of solutions


Hofmann, Steven; Lewis, John L. The $L^{p}$ regularity problem for the heat equation in non-cylindrical domains. Illinois J. Math. 43 (1999), no. 4, 752--769. doi:10.1215/ijm/1256060690.

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