Calculus of Variations and Geometric Measure Theory

A. Malusa - L. Orsina

Asymptotic behaviour of renormalized solutions to elliptic equations with measure data.

created by malusa on 07 Mar 2008


Published Paper

Inserted: 7 mar 2008

Journal: Calc. Var. Partial Differential Equations
Volume: 27
Pages: 179-202
Year: 2006


We study the limit as $n$ goes to $+\infty$ of the renormalized solutions $\u_n$ to the nonlinear elliptic problems \begin{equation} -\textrm{div}(an(x,D un))=\mu,\ \textrm{in\ }\Omega\,, \qquad un=0\,\ \textrm{on\ } \partial\Omega\,, \end{equation} where $\Omega$ is a bounded open set of $R^N$, $N\geq 2$, and $\mu$ is a Radon measure with bounded variation in $\Omega$. Under the assumption of G--convergence of the operators $\mathcal{A}_n(v)=-\textrm{div}(a_n(x,D v))$, defined for $v\in W^{1,p}_0$, $p>1$, to the operator $\mathcal{A}_0(v)=-\textrm{div}(a_0(x,D v))$, we shall prove that the sequence $(u_n)$ admits a subsequence converging almost everywhere in $\Omega$ to a function $u$ which is a renormalized solution to the problem \begin{equation} -\textrm{div}(a0(x,D v))=\mu,\ \textrm{in\ }\Omega\,,\qquad u=0\,\ \textrm{on\ } \partial\Omega\,. \end{equation}