The oblique derivative problem for the heat equation in lipschitz cylinders

We consider a class of initial-boundary value problems for the heatequation on (0, T) x Ω with Ω a bounded Lipschitz domain in Rⁿ. On thelateral boundary, (0, T) x Ω = ∑_T, we specify (α, ∇v) where ∇v denotesthe spatial gradient of the solution and α: ∑_T→(x: ∣x∣u = 1) is a continuousvector field satisfying (α, v) ≥ p > 0 with v the unit normal to Ω. On theinitial surface, (0) xΩ, we require that the solution vanish. The lateral datais taken from L^p∑_T(I.t) ¦ For p ϵ (2 - ϵ, ∞), we show existence and uniquenessof solutions to this problem with estimates for the parabolic maximal functionof the spatial gradient of the solution.