On the $G$-convergence of Morrey operators

Formica, Maria Rosaria ; Sbordone, Carlo

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 14 (2003) no. 1, pp. 33-49.

Abstract (VO)

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

Riassunto

Seguendo Morrey [14], ad ogni matrice simmetrica $A(x)$ a coefficienti misurabili, tale che $$\frac{|\xi|^{2}}{K} \le (A(x) \xi,\xi) \le K |\xi|^{2} \quad \text{a.e.} \quad x \in \Omega, \, \forall \xi \in \mathbb{R}^{2},$$$\Omega \in \mathbb{R}^{2}$ e ad ogni $u \in W^{1,2}(\Omega)$ si può associare un'altra matrice simmetrica $\mathcal{A} = \mathcal{A}(A,u)$ con $det \, \mathcal{A} = 1$ e soddisfacente $$\frac{|\xi|^{2}}{K} \le (\mathcal{A}(x) \xi,\xi) \le K |\xi|^{2} \quad \text{a.e.} \quad x \in \Omega, \, \forall \xi \in \mathbb{R}^{2},$$ La principale proprietà di $\mathcal{A}$ è che $\mathcal{A} \nabla u = A \nabla u$, se $\nabla u \neq 0$. Si studiano le proprietà di $\mathcal{A}$ come funzione di $A$ e di $u$. In particolare, si dimostra che, se $A_{b} \rightarrow^{G} A$, $u_{b} \rightharpoonup u$, $\nabla u \neq 0$ and $div \, A_{b} \nabla u_{b} = 0$ then $\mathcal{A} (A_{b},u_{b}) \rightarrow^{G} \mathcal{A} (A, u)$.

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