Equivalence between entropy and renormalized solutions for parabolic equations

If Ω is an open bounded set in RN, N≥2, with a connected Lipschitz boundary ∂Ω, a(x,ξ) is an operator of Leray-Lions type, β and γ are non decreasing continuous real functions, β(0)=γ(0)=0, then for every (f,g)∈L1(]0,T[×RN)×L1(]0,T[×∂Ω),(u0,v0)∈L1(RN)×L1(∂Ω), we prove that the entropy solution coincides with the renormalized solution to the following problem: {u′−div[a(.,∇u)]+β(u)=fon  ]0,T[×(RN∖∂Ω),(τu)′+[∂u∂νa]+γ(τu)=g  and  [u]=0on  ]0,T[×∂Ω,(u(0,.),τu(0,.))=(u0,v0)a.e. on  RN×∂Ω, where [u] and [∂u∂νa] are respectively the jump across ∂Ω of u and the normal derivative ∂u∂νa related to the operator a.