Uniqueness for inhomogeneous Dirichlet problem for ellipticparabolic equations

We show the L1 contraction and comparison principle for weak (and, more generally, renormalized) solutions of the elliptic–parabolic problem j(v)t − div(∇w + F (w)) = f(t, x), w = ϕ(v) in (0, T) × Ω ⊂ R+ × RN with inhomogeneous Dirichlet boundary datum g ∈ L2(0, T;W 1,2(Ω)) for w (the boundary datum is taken in the sense w − g ∈ L2(0, T;H10 (Ω))) and initial datum jo ∈ L1(Ω) for j(v). Here ϕ and j are non-decreasing, and we assume that F is just continuous. Our proof consists in doubling of variables in the interior of Ω as introduced by Carrillo in 1999, and in a careful treatment of the flux term near the boundary of Ω. For the latter argument, the result is restricted to the linear dependence on ∇w of the diffusion term. The proof allows for a wide class of domains Ω, including, for example, domains of finite perimeter with uniform exterior cone condition or even domains with cracks. We obtain the corresponding results for the associated stationary problem and discuss on generalization of our technique to the case of nonlinear diffusion operators. 1