Abstract
We consider a class of initial-boundary value problems for the heat equation on ( 0. T ) × Ω (0.T) \times \Omega with Ω \Omega a bounded Lipschitz domain in R n {{\mathbf {R}}^n} . On the lateral boundary, ( 0 , T ) × ∂ Ω = Σ T (0,T) \times \partial \Omega = {\Sigma _T} , we specify ⟨ α , ∇ u ⟩ \left \langle {\alpha ,\nabla u} \right \rangle where ∇ u \nabla u denotes the spatial gradient of the solution and α : Σ T → { x : | x | = 1 } \alpha :{\Sigma _T} \to \{ x:|x| = 1\} is a continuous vector field satisfying ⟨ α , ν ⟩ ≥ μ > 0 \left \langle {\alpha ,\nu } \right \rangle \geq \mu > 0 with ν \nu the unit normal to ∂ Ω \partial \Omega . On the initial surface, { 0 } × Ω \{ 0\} \times \Omega , we require that the solution vanish. The lateral data is taken from L p ( Σ T ) {L^p}({\Sigma _T}) . For p ∈ ( 2 − , ∞ ) 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.
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