The Oblique Derivative Problem for the Heat Equation in Lipschitz Cylinders
R. M. Brown · Proceedings of the American Mathematical Society · 1989
We consider a class of initial-boundary value problems for the heat equation on $(0.T) \times \Omega$ with $\Omega$ a bounded Lipschitz domain in ${{\mathbf {R}}^n}$. On the lateral boundary, $(0,T) \times \partial \Omega = {\Sigma _T}$, we specify $\left \langle {\alpha , abla u} \right \rangle$ where $ abla u$ denotes the spatial gradient of the solution and $\alpha :{\Sigma _T} \to \{ x:|x| = 1\}$ is a continuous vector field satisfying $\left \langle {\alpha , u } \right \rangle \geq \mu > 0$ with $ u$ the unit normal to $\partial \Omega$. On the initial surface, $\{ 0\} \times \Omega$, we require that the solution vanish. The lateral data is taken from ${L^p}({\Sigma _T})$. For $p \in (2 - ,\infty )$, we show existence and uniqueness of solutions to this problem with estimates for the parabolic maximal function of the spatial gradient of the solution.