Uniqueness for inhomogeneous Dirichlet problem for elliptic-parabolic equations

International audienceWe show the L^1 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 = \phi(v) in (0, T)×\Omega ⊂ IR+×IR^N with inhomogeneous Dirichlet boundary datum g ∈ L^2(0,T;W^{1,2}(\Omega)) for w (which is taken in the sense w − g ∈ L^2(0, T;H^1_0 (\Omega))) and initial datum j_o ∈ L^1(\Omega) for j(v). Here \phi, j are nondecreasing, and we assume F just continuous. Our proof consists in doubling of variables in the interior of \Omega as introduced by J.Carrillo [9] (Arch. Rational Mech. Anal., vol.147, 1999), and in a careful treatment of the flux term near the boundary of \Omega. For this last argument, the result is restricted to the linear dependence on ∇w of the diffusion term. The proof allows for a wide class of domains \Omega, including e.g. weakly Lipschitz domains with Lipschitz 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