Resumen de Homogenization of a nonlinear parabolic problem corresponding to a Leray–Lions monotone operator with right-hand side measure
Mohammed Abdellaoui, Elhoussine Azroul
In this paper we deal with asymptotic behaviour of renormalized solutions un to the nonlinear parabolic problems whose model is ⎧⎩⎨⎪⎪(un)t−div(an(t,x,∇un))=μnun(t,x)=0un(0,x)=un0 in Q=(0,T)×Ω, on (0,T)×∂Ω, in Ω, where Ω is a bounded open set of RN, N≥1, T>0 and un0∈C∞0(Ω) that approaches u0 in L1(Ω). Moreover (μn)n∈N is a sequence of Radon measures with bounded variation in Q which converges to μ in the narrow topology of measures. The main result states that, under the assumption of G-convergence of the operators An(v)=−div(an(t,x,∇vn)), defined for vn∈Lp(0,T;W1,p0(Ω)) for p>1, to the operator A0(v)=−div(a0(t,x,∇v)) and up to subsequences, (un) converges a.e. in Q to the renormalized solution u of the problem ⎧⎩⎨⎪⎪ut−div(a0(t,x,∇u))=μu(t,x)=0u(0,x)=u0 in Q=(0,T)×Ω, on (0,T)×∂Ω, in Ω.
The proposed renormalized formulation differs from the usual one by the fact that truncated function Tk(un) (which depend on the solutions) are used in place of the solutions un. We prove existence of such a limit-solution and we discuss its main properties in connection with G-convergence, we finally show the relationship between the new approach and the previous ones and we extend this result using capacitary estimates and auxiliary test functions.
