December  2011, 31(4): 1017-1021. doi: 10.3934/dcds.2011.31.1017

Ennio De Giorgi and $\mathbf\Gamma$-convergence

1. 

SISSA, via Bonomea 265, 34136 Trieste, Italy

Received  March 2009 Revised  December 2010 Published  September 2011

$\Gamma$-convergence was introduced by Ennio De Giorgi in a series of papers published between 1975 and 1983. In the same years he developed many applications of this tool to a great variety of asymptotic problems in the calculus of variations and in the theory of partial differential equations.
Citation: Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017
References:
[1]

G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property, Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday, Acta Appl. Math., 65 (2001), 9-33. doi: 10.1023/A:1010602715526.

[2]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. (4), 144 (1986), 347-389.

[3]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math., 368 (1986), 28-42.

[4]

E. De Giorgi, Sulla convergenza di alcune successioni d'integrali del tipo dell'area, Collection of articles dedicated to Mauro Picone on the occasion of his ninetieth birthday, Rend. Mat. (6), 8 (1975), 277-294.

[5]

E. De Giorgi, $\Gamma$-convergenza e $G$-convergenza, Boll. Un. Mat. Ital. A (5), 14 (1977), 213-220.

[6]

E. De Giorgi, Convergence problems for functionals and operators, in "Proc. Int. Meeting on Recent Methods in Nonlinear Analysis" (Rome, 1978) (eds. E. De Giorgi, E. Magenes and U. Mosco), Pitagora, Bologna, (1979), 131-188.

[7]

E. De Giorgi, $\Gamma$-limiti di ostacoli, in "Recent Methods in Nonlinear Analysis and Applications," Proc. SAFA IV (Napoli, 1980) (eds. A. Canfora, S. Rionero, C. Sbordone and G. Trombetti), Liguori, Napoli, (1981), 51-84.

[8]

E. De Giorgi, Generalized limits in calculus of variations, in "Topics in Functional Analysis" (1980-81), Quad. Sc. Norm. Sup. Pisa, Pisa, (1981), 117-148.

[9]

E. De Giorgi, Operatori elementari di limite ed applicazioni al Calcolo delle Variazioni, Atti del convegno "Studio di problemi limite della Analisi Funzionale" (Bressanone, 1981), Pitagora, Bologna, (1982), 101-116.

[10]

E. De Giorgi, $G$-operators and $\Gamma$-convergence, in "Proc. Internat. Congr. Math., Vol. 1,2" (Warsaw, 1983), PWN, Warsaw, (1984), 1175-1191.

[11]

E. De Giorgi and G. Buttazzo, Limiti generalizzati e loro applicazioni alle equazioni differenziali, (Italian) [Generalized limits and their applications to differential equations], Matematiche (Catania), 36 (1981), 53-64.

[12]

E. De Giorgi and G. Dal Maso, $\Gamma$-convergence and the calculus of variations, in "Mathematical Theory of Optimization" (Genova, 1981) (ed. T. Zolezzi), Lecture Notes in Math., 979, Springer, Berlin, (1983), 121-143.

[13]

E. De Giorgi, G. Dal Maso and P. Longo, $\Gamma$-limiti di ostacoli, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 68 (1980), 481-487.

[14]

E. De Giorgi, G. Dal Maso and L. Modica, Convergenza debole di misure su spazi di funzioni semicontinue, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. (8), 79 (1985), 98-106.

[15]

E. De Giorgi, G. Dal Maso and L. Modica, Weak convergence of measures on spaces of semicontinuous functions, Proc. of the Int. Workshop on Integral Functionals in the Calculus of Variations (Trieste, 1985), Rend. Circ. Mat. Palermo (2), 15 (1987), 59-100.

[16]

E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58 (1975), 842-850.

[17]

E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Rend. Sem. Mat. Brescia, 3 (1979), 63-101.

[18]

E. De Giorgi and T. Franzoni, Una presentazione sintetica dei limiti generalizzati, (Italian) [Generalized limits: A synthesis), Portugal. Math., 41 (1982), 405-436.

[19]

E. De Giorgi and L. Modica, $\Gamma$-convergenza e superfici minime, preprint Scuola Normale Superiore di Pisa, 1979.

[20]

E. De Giorgi and S. Spagnolo, Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4), 8 (1973), 391-411.

[21]

O. Savin, Symmetry of entire solutions for a class of semilinear elliptic equations, in "International Congress of Mathematicians. Vol. III," 257-265, Eur. Math. Soc., Zürich, 2006.

[22]

S. Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore, Ann. Scuola Norm. Sup. Pisa (3), 21 (1967), 657-699.

[23]

S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 571-597; errata, ibid. (3), 22 (1968), 673.

show all references

References:
[1]

G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property, Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday, Acta Appl. Math., 65 (2001), 9-33. doi: 10.1023/A:1010602715526.

[2]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. (4), 144 (1986), 347-389.

[3]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math., 368 (1986), 28-42.

[4]

E. De Giorgi, Sulla convergenza di alcune successioni d'integrali del tipo dell'area, Collection of articles dedicated to Mauro Picone on the occasion of his ninetieth birthday, Rend. Mat. (6), 8 (1975), 277-294.

[5]

E. De Giorgi, $\Gamma$-convergenza e $G$-convergenza, Boll. Un. Mat. Ital. A (5), 14 (1977), 213-220.

[6]

E. De Giorgi, Convergence problems for functionals and operators, in "Proc. Int. Meeting on Recent Methods in Nonlinear Analysis" (Rome, 1978) (eds. E. De Giorgi, E. Magenes and U. Mosco), Pitagora, Bologna, (1979), 131-188.

[7]

E. De Giorgi, $\Gamma$-limiti di ostacoli, in "Recent Methods in Nonlinear Analysis and Applications," Proc. SAFA IV (Napoli, 1980) (eds. A. Canfora, S. Rionero, C. Sbordone and G. Trombetti), Liguori, Napoli, (1981), 51-84.

[8]

E. De Giorgi, Generalized limits in calculus of variations, in "Topics in Functional Analysis" (1980-81), Quad. Sc. Norm. Sup. Pisa, Pisa, (1981), 117-148.

[9]

E. De Giorgi, Operatori elementari di limite ed applicazioni al Calcolo delle Variazioni, Atti del convegno "Studio di problemi limite della Analisi Funzionale" (Bressanone, 1981), Pitagora, Bologna, (1982), 101-116.

[10]

E. De Giorgi, $G$-operators and $\Gamma$-convergence, in "Proc. Internat. Congr. Math., Vol. 1,2" (Warsaw, 1983), PWN, Warsaw, (1984), 1175-1191.

[11]

E. De Giorgi and G. Buttazzo, Limiti generalizzati e loro applicazioni alle equazioni differenziali, (Italian) [Generalized limits and their applications to differential equations], Matematiche (Catania), 36 (1981), 53-64.

[12]

E. De Giorgi and G. Dal Maso, $\Gamma$-convergence and the calculus of variations, in "Mathematical Theory of Optimization" (Genova, 1981) (ed. T. Zolezzi), Lecture Notes in Math., 979, Springer, Berlin, (1983), 121-143.

[13]

E. De Giorgi, G. Dal Maso and P. Longo, $\Gamma$-limiti di ostacoli, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 68 (1980), 481-487.

[14]

E. De Giorgi, G. Dal Maso and L. Modica, Convergenza debole di misure su spazi di funzioni semicontinue, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. (8), 79 (1985), 98-106.

[15]

E. De Giorgi, G. Dal Maso and L. Modica, Weak convergence of measures on spaces of semicontinuous functions, Proc. of the Int. Workshop on Integral Functionals in the Calculus of Variations (Trieste, 1985), Rend. Circ. Mat. Palermo (2), 15 (1987), 59-100.

[16]

E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58 (1975), 842-850.

[17]

E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Rend. Sem. Mat. Brescia, 3 (1979), 63-101.

[18]

E. De Giorgi and T. Franzoni, Una presentazione sintetica dei limiti generalizzati, (Italian) [Generalized limits: A synthesis), Portugal. Math., 41 (1982), 405-436.

[19]

E. De Giorgi and L. Modica, $\Gamma$-convergenza e superfici minime, preprint Scuola Normale Superiore di Pisa, 1979.

[20]

E. De Giorgi and S. Spagnolo, Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4), 8 (1973), 391-411.

[21]

O. Savin, Symmetry of entire solutions for a class of semilinear elliptic equations, in "International Congress of Mathematicians. Vol. III," 257-265, Eur. Math. Soc., Zürich, 2006.

[22]

S. Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore, Ann. Scuola Norm. Sup. Pisa (3), 21 (1967), 657-699.

[23]

S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 571-597; errata, ibid. (3), 22 (1968), 673.

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