A proof of a Parabolic Maximum Principle

Define the following parabolic operator in divergence form Lu ≡ ∂u ∂t − ∂ ∂x j a ij (x, t) ∂u ∂x i for (x, t) ∈ Ω T = Ω × (0, T ], where Ω is a bounded simply connected subset of R m for m ≥ 2 and T > 0 is a fixed but arbitrary positive number. We also assume that the a ij are bounded and measurable in ¯ Ω T and satisfy the uniform parabolicity condition: 1/ν |z| 2 ≤ m i,j=1 a ij (x, t)z i z j ≤ ν |z| 2 (1) for some ν > 0, almost everywhere in Ω T and all z ∈ R m. Let Γ = (∂Ω × [0, T ])∩(Ω × {t = 0}). We consider the following problem Lu = ∇ · F − f, (x, t) ∈ Ω T u = φ(x, t), on Γ (2) Our goal here is to furnish a proof of the following maximum principle whose statement can be found in [1] Theorem 1 (Maximum Principle). Let u be a smooth solution of (2) in ¯