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### Operatori ellittici degeneri a coefficienti in EXP

Bollettino dell'Unione Matematica Italiana

### On the $G$-convergence of Morrey operators

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Following Morrey [14] we associate to any measurable symmetric $2×2$ matrix valued function $A\left(x\right)$ such that $\frac{{\left|\xi \right|}^{2}}{K}\le \left(A\left(x\right)\xi ,\xi \right)\le K{\left|\xi \right|}^{2}\text{a.e.}x\in \mathrm{\Omega },\mathrm{\forall }\xi \in {\mathbb{R}}^{2},$ $\mathrm{\Omega }\in {\mathbb{R}}^{2}$ and to any $u\in {W}^{1,2}\left(\mathrm{\Omega }\right)$ another symmetric $2×2$ matrix valued function $\mathcal{A}=\mathcal{A}\left(A,u\right)$ with $det\mathcal{A}=1$ and satisfying $\frac{{\left|\xi \right|}^{2}}{K}\le \left(\mathcal{A}\left(x\right)\xi ,\xi \right)\le K{\left|\xi \right|}^{2}\text{a.e.}x\in \mathrm{\Omega },\mathrm{\forall }\xi \in {\mathbb{R}}^{2},$ The crucial property of $\mathcal{A}$ is that $\mathcal{A}\nabla u=A\nabla u$, if $\nabla u\ne 0$. We study the properties of $\mathcal{A}$ as a function of $A$ and $u$. In particular, we show that, if $A{}_{b}\to {}^{G}A$, ${u}_{b}⇀u$, $\nabla u\ne 0$ and $div{A}_{b}\nabla {u}_{b}=0$ then $\mathcal{A}\left(A{}_{b},u{}_{b}\right)\to {}^{G}\mathcal{A}\left(A,u\right)$.