On the $G$-convergence of Morrey operators

Formica, Maria Rosaria ; Sbordone, Carlo

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 14 (2003), p. 33-49 / Harvested from Biblioteca Digitale Italiana di Matematica

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 $\mathcal{A} (A_{b},u_{b}) \rightarrow^{G} \mathcal{A} (A, u)$.

Résumé

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 $\mathcal{A} (A_{b},u_{b}) \rightarrow^{G} \mathcal{A} (A, u)$.

Détail

Publié le : 2003-03-01

MR 2057273 | Zbl 1105.35030

Bibliographie

[1] Capone, C., Quasiharmonic fields and Beltrami operators. Comm. Math. Univ. Carolinae, 43, 2, 2002, 363-377. | MR 1922134 | Zbl 1069.35024

[2] De Giorgi, E., Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. U.M.I., 1, 1968, 135-137. | MR 227827 | Zbl 0155.17603

[3] De Arcangelis, - Donato, P., On the convergence of Laplace-Beltrami operators associated to quasiregular mappings. Studia mathematica, t. LXXXVI, 1987, 189-204. | MR 917047 | Zbl 0646.30024

[4] Formica, M.R., On the $\Gamma$-convergence of Laplace-Beltrami operators in the plane. Annales Academiae Scientiarum Fennicae Matematica, vol. 25, 2000, 423-438. | MR 1762427 | Zbl 0955.30016

[5] Formica, M.R., Degenerate Elliptic Operators with coefficients in EXP. Ph.D. thesis, Università di Napoli «Federico II», 2001. | Zbl 1053.47510

[6] Formica, M.R., Beltrami operators in the plane. Preprint n. 6, Aracne editrice, Roma2002. | MR 1979563 | Zbl 1098.47517

[7] Francfort, G.A. - Murat, F., Optimal bounds for conduction in two-dimensional, two-phase, anisotropic media. In: R.J. Knops - A.A. Lacey (eds.), «Nonclassical» continuum mechanics. London, Math. Soc. Lecture Notes Series, 122, Cambridge 1987, 197-212. | MR 926503 | Zbl 0668.73018

[8] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems. Ann. of Math. Studies, 105, Princeton Univ. Press, 1983. | MR 717034 | Zbl 0516.49003

[9] Greco, L. - Sbordone, C., Sharp upper bounds for the degree of regularity of the solutions to an elliptic equation. Comm. P.D.E., 27, 5-6, 2002, 945-952. | MR 1916553 | Zbl 1019.35021

[10] Iwaniec, T. - Sbordone, C., Quasiharmonic fields. Ann. Inst. H. Poincaré - AN 18, 5, 2001, 519-572. | MR 1849688 | Zbl 1068.30011

[11] John, O. - Maly, J. - Stará, J., Nowhere continuous solutions to elliptic systems. Comm. Math. Univ. Carolinae, 30, 1, 1989, 33-43. | MR 995699 | Zbl 0691.35024

[12] Koshelev, A.I., Regularity of solutions of quasilinear elliptic systems. Russian Math. Survey, 33, 1978, 1-52. | MR 510669 | Zbl 0413.35033

[13] Maly, J., Finding equation from solutions. Draft1999.

[14] Morrey, C.B., On the solution of quasilinear elliptic partial differential equations. Trans. Amer. Math. Soc., 43, 1938, 126-166. | JFM 64.0460.02

[15] Murat, F., Compacité par Compensation. Ann. Sc. Norm. Pisa, 5, 1978, 489-507. | MR 506997 | Zbl 0399.46022

[16] Piccinini, L. - Spagnolo, S., On the Hölder Continuity of Solutions of Second Order Elliptic Equations in two Variables. Ann. Sc. Norm. Pisa, 26, 1972, 391-402. | MR 361422 | Zbl 0237.35028

[17] Soucek, J., Singular solutions to linear elliptic systems. Comm. Math. Univ. Carolinae, 25, 1984, 273-281. | MR 768815 | Zbl 0564.35008

[18] Spagnolo, S., Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Sc. Norm Sup. Pisa, vol. 22, fasc. 4, 1968, 571-597. | MR 240443 | Zbl 0174.42101

[19] Spagnolo, S., Some convergence problems. Symposia Mathematica, XVIII, 1976, 391-397. | MR 509184 | Zbl 0332.46020

[20] Tartar, L., Homogéneization et compacité par compensation. Cours Peccot, Collège de France, 1977. | Zbl 0544.47042