## Illinois Journal of Mathematics

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

#### Abstract

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

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

Dates
First available in Project Euclid: 20 October 2009

https://projecteuclid.org/euclid.ijm/1256060690

Digital Object Identifier
doi:10.1215/ijm/1256060690

Mathematical Reviews number (MathSciNet)
MR1712521

Zentralblatt MATH identifier
0934.35056

#### Citation

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. https://projecteuclid.org/euclid.ijm/1256060690