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Homogenization of a nonlinear parabolic problem corresponding to a Leray-Lions monotone operator with right-hand side measure. (English) Zbl 1445.35026

In the manuscript under review, the authors discuss a homogenisation problem associated to a Dirichlet problem for the heat equation with nonlinear and time-dependent conductivities. More preciseley, on the space-time cylinder \(Q=(0,T)\times \Omega\) with \(\Omega\subseteq \mathbb{R}^d\) open and bounded, the authors consider the sequence of problems \[ \partial_t u_n -\operatorname{div}(a_n(t,x,\operatorname{grad}u_n))=\mu_n, \] where the sequence of unknowns \((u_n)_n\) is subject to homogeneous Dirichlet boundary conditions, and \(L_1(\Omega)\) initial conditions. The right-hand sides \((\mu_n)_n\) form a a sequence of Radon measures with bounded variation in \(Q\) suitably converging (’narrow topology’) to some limit right-hand side \(\mu_0\). The driving elliptic part of the parabolic expression \[ A_n(v)=-\operatorname{div}a_n(t,x,\operatorname{grad}v) \] is supposed to converge with respect to \(G\)-convergence to some limit \(A_0\colon L_p(0,T;W_{p,0}^{1}(\Omega))\to L_{p'}(0,T;W_{p'}^{-1}(\Omega))\), \(p>1\) (with spatial duality paired over \(L_2(\Omega)\)), represented by \(a_0\) in the way that \[ A_0(v)=-\operatorname{div}a_0(t,x,\operatorname{grad}v). \] The main result of the authors then shows that a suitable subsequence of renormalised solutions \((u_n)_n\) of the nonlinear parabolic problem indexed by \(n\) converges to a limit renormalised solution \(u_0\) of \[ \partial_t u_0 -\operatorname{div}(a_0(t,x,\operatorname{grad}u_0))=\mu_0. \]

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35R06 PDEs with measure
35K55 Nonlinear parabolic equations
35R05 PDEs with low regular coefficients and/or low regular data
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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