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Chiado’ Piat, Valeria; Dal Maso, Gianni; Defranceschi, Anneliese

G-convergence of monotone operators. (English) Zbl 0731.35033

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7, No. 3, 123-160 (1990).
Summary: A general notion of G-convergence for sequences of maximal monotone operators of the form \({\mathcal A}_ hu=-div(a_ h(x,Du))\) is introduced in terms of the asymptotic behavior, as \(h\to +\infty\), of the solutions \(u_ h\) to the equations \({\mathcal A}_ hu_ h=f_ h\) and of their momenta \(a_ h(x,Du_ h)\). The main results of the paper are the local character of the G-convergence and the G-compactness of some classes of nonlinear monotone operators.

Cited in 4 Reviews
Cited in 45 Documents

MSC:

35J60 Nonlinear elliptic equations
47H05 Monotone operators and generalizations
35G30 Boundary value problems for nonlinear higher-order PDEs

Keywords:

Besov-Triebel-Lizorkin spaces; G-convergence; maximal monotone operators
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\textit{V. Chiado' Piat} et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7, No. 3, 123--160 (1990; Zbl 0731.35033)
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References:

[1] Ambrosetti, A.; Sbordone, C., Γ-convergenza e G-convergenza per problemi non lineari di tipo ellittico, Boll. Un. Mat. Ital., 13, 352-362, (1976) · Zbl 0345.49004
[2] Attouch, H., Introduction à l’homogénisation d’inéquations variationnelles, Rend. Sem. Mat. Univ. Politec. Torino, Vol. 40, 1-23, (1982) · Zbl 0523.35004
[3] Attouch, H., Variational convergence for functions and operators, (1984), Pitman London · Zbl 0561.49012
[4] Barbu, V.; Precupanu, Th., Convexity and optimization in Banach spaces, (1978), Editura Academiei Bucuresti · Zbl 0379.49010
[5] Bensoussan, A.; Lions, J.-L.; Papanicolaou, G., Asymptotic analysis for periodic structures, (1978), North Holland Amsterdam · Zbl 0411.60078
[6] Boccardo, L.; Gallouet, Th., Homogenization with jumping nonlinearities, Ann. Mat. Pura Appl., Vol. 138, 211-221, (1984) · Zbl 0568.35038
[7] Boccardo, L.; Murat, F., Remarques sur l’homogénéisation de certains problèmes quasi linéaires, Portugal. Math., Vol. 41, 535-562, (1982) · Zbl 0524.35042
[8] Boccardo, L.; Murat, F., Homogénéisation de problèmes quasi linéaires, studio di problemi-limite Della analisi funzionale (bressanone, 1981), 13-51, (1982), Pitagora editrice Bologna
[9] Brezis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans LES espaces de Hilbert, (1973), North-Holland Amsterdam · Zbl 0252.47055
[10] Browder, F. E., Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proceedings of Symposia Pure Math., 18, (1976), American Mathematical Society · Zbl 0176.45301
[11] Castaing, C.; Valadier, M., Convex analysis and measurable multifunctions, Lec. Notes Math., 580, (1977), Springer-Verlag Berlin · Zbl 0346.46038
[12] De Giorgi, E.; Spagnolo, S., Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital., Vol. 8, 391-411, (1973) · Zbl 0274.35002
[13] Fusco, N.; Moscariello, G., On the homogenization of quasilinear divergence structure operators, Ann. Mat. Pura Appl., Vol. 146, 1-13, (1987) · Zbl 0636.35027
[14] Fusco, N.; Moscariello, G., Further results on the homogenization of quasilinear operators, Ricerche Mat., 35, 231-246, (1986) · Zbl 0658.35016
[15] Hiai, F.; Umegaki, H., Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal., 7, 149-182, (1977) · Zbl 0368.60006
[16] Kuratowski, K., Topology, (1968), Academic Press New York · Zbl 0158.40901
[17] Matzeu, M., Su un tipo di continuità dell’operatore subdifferenziale, Boll. Un. Mat. Ital., Vol. 14-B, 480-490, (1977) · Zbl 0376.28004
[18] Murat, F., H-convergence, Séminaire d’analyse fonctionnelle et numérique, (1977), Université d’Alger
[19] Murat, F., Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Vol. 5, 489-507, (1978) · Zbl 0399.46022
[20] Murat, F.; Tartar, L., Homogénéisation d’opérateurs monotones, (1981), Manuscript
[21] Pascali, D.; Sburlan, S., Nonlinear mappings of monotone type, (1978), Editura Academiei Bucuresti · Zbl 0392.47026
[22] Raitum, U. E., On the G-convergence of quasilinear elliptic operators with unbounded coefficients, Soviet Math. Dokl., 24, 472-476, (1981) · Zbl 0501.35033
[23] Sanchez-Palencia, E., Non homogeneous media and vibration theory, Lec. Notes Phys., 127, (1980), Springer-Verlag Berlin · Zbl 0432.70002
[24] Spagnolo, S., Sul limite delle soluzioni di problemi di Cauchy relativi all’equazione del calore, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Vol. 21, 657-699, (1967) · Zbl 0153.42103
[25] Spagnolo, S., Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Vol. 22, 577-597, (1968) · Zbl 0174.42101
[26] Spagnolo, S., Convergence in energy for elliptic operators, Proc. Third Symp. Numer. Solut. Partial Diff. Equat., 469-498, (1976), Academic Press San Diego, (College Park, 1975)
[27] Tartar, L., Cours peccot, (1977), Collège de France Paris
[28] Tartar, L., Quelques remarques sur l’homogénéisation, Proceedings of the Japan-France seminar 1976, Functional analysis and numerical analysis, 469-482, (1978), Japan Society for the Promotion of Science
[29] Tartar, L., Compensated compactness and applications to partial differential equations, nonlinear analysis & mechanics, heriot-watt symposium, vol. IV, Res. Notes Math., Vol. 39, 136-211, (1979), Pitman London
[30] Zhikov, V. V.; Kozlov, S. M.; Oleinik, O. A.; T’en Ngoan, Kha, Averaging and G-convergence of differential operators, Russian Math. Surveys, Vol. 34, 69-147, (1979) · Zbl 0445.35096
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