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.