Asymptotic behaviour of renormalized solutions to elliptic equations with measure data and G-converging operators

Abstract

We study the limit as n goes to +∞ of the renormalized solutions u ⁿ to the nonlinear elliptic problems

$$ -\textrm{div}(a_n({x \nabla{u_n}}))=\mu,\ \textrm{in}\Omega, \quad {u_n} = 0\,\ \textrm{on}\partial\Omega,$$

where Ω is a bounded open set of ℝ^N, N≥ 2, and μ is a Radon measure with bounded variation in Ω. Under the assumption of G-convergence of the operators $\mathcal{A}_n(v)=-\mathrm{div}(a_n({x,\ {\nabla_v}}))$, defined for $v\in W^{1,p}_0(\Omega), p>1$, to the operator $\mathcal{A}_0(v)=-\mathrm{div}(a_0({x,\ {\nabla_v}}))$, we shall prove that the sequence (u _n ) admits a subsequence converging almost everywhere in Ω to a function u which is a renormalized solution to the problem

$$ -\textrm{div}(a_0(x,\ {\nabla_u})) = \mu,\ \textrm{in}\ \Omega,\quad u=0\,\ \textrm{on} \partial\Omega.$$