# On the $G$-convergence of Morrey operators

• Volume: 14, Issue: 1, page 33-49
• ISSN: 1120-6330

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## Abstract

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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 $\mathcal{A}\left(A{}_{b},u{}_{b}\right)\to {}^{G}\mathcal{A}\left(A,u\right)$.

## How to cite

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Formica, Maria Rosaria, and Sbordone, Carlo. "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 14.1 (2003): 33-49. <http://eudml.org/doc/252340>.

@article{Formica2003,
abstract = {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 $\mathcal\{A\} (A_\{b\},u_\{b\}) \rightarrow^\{G\} \mathcal\{A\} (A, u)$.},
author = {Formica, Maria Rosaria, Sbordone, Carlo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Elliptic equations; G-convergence; Morrey matrices; -convergence},
language = {eng},
month = {3},
number = {1},
pages = {33-49},
publisher = {Accademia Nazionale dei Lincei},
title = {On the $G$-convergence of Morrey operators},
url = {http://eudml.org/doc/252340},
volume = {14},
year = {2003},
}

TY - JOUR
AU - Formica, Maria Rosaria
AU - Sbordone, Carlo
TI - On the $G$-convergence of Morrey operators
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2003/3//
PB - Accademia Nazionale dei Lincei
VL - 14
IS - 1
SP - 33
EP - 49
AB - 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 $\mathcal{A} (A_{b},u_{b}) \rightarrow^{G} \mathcal{A} (A, u)$.
LA - eng
KW - Elliptic equations; G-convergence; Morrey matrices; -convergence
UR - http://eudml.org/doc/252340
ER -

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