Integral representation and Γ-convergence of variational integrals with P(X)-growth

ESAIM - Control, Optimisation and Calculus of Variations

Volumn 7, Issue , 2002, Pages 495-519

  Coscia, Alessandra ^a   Mucci, Domenico ^a  

^a UNIVERSITY OF PARMA   (Italy)

Author keywords

Integral representation; Nonstandard growth conditions; convergence

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; FUNCTIONS; INTEGRAL EQUATIONS; SET THEORY; THEOREM PROVING;

BOREL FUNCTION; INTEGRAL FUNCTIONALS; NONSTANDARD GROWTH CONDITIONS; VARIATIONAL INTEGRALS;

VARIATIONAL TECHNIQUES;

EID: 0036412721     PISSN: 12928119     EISSN: 12623377     Source Type: Journal    
DOI: 10.1051/cocv:2002065     Document Type: Article

References (31)

1. E. Acerbi, G. Bouchitté and I. Fonseca, Relaxation of convex functionals: The gap phenomenon. Ann. Inst. H. Poincaré (2003).
2. E. Acerbi and G. Mingione, Regularity results for a class of functionals with non standard growth. Arch. Rational Mech. Anal. 156 (2001) 121-140.
3. E. Acerbi and G. Mingione, Regularity results for a class of quasiconvex functionals with non standard growth. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 XXX (2001) 311-339.
4. R.A. Adams, Sobolev spaces. Academic Press, New York (1975).
5. Yu.A. Alkutov, The Harnack inequality and the Holder property of solutions of nonlinear elliptic equations with a nonstandard growth condition. Differential Equations 33 (1998) 1653-1663.
6. G. Bouchitté, I. Fonseca and J. Malý, The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent. Proc. Roy. Soc. Edinburgh Ser. A 128 (1988) 463-479.
7. G. Buttazzo, Semicontinuity, relaxation and integral representation in the calculus of variations. Longman, Harlow, Pitman Res. Notes in Math. 207 (1989).
8. G. Buttazzo and G. Dal Maso, A characterization of nonlinear functionals on Sobolev spaces which admit an integral representation with a Caratheodory integrand. J. Math. Pures Appl. 64 (1985) 337-361.
9. G. Buttazzo and G. Dal Maso, Integral representation and relaxation of local functionals. Nonlinear Anal. 9 (1985) 515-532.
10. A. Braides and A. Defranceschi, Homogenization of multiple integrals. Oxford University Press, Oxford, Oxford Lecture Ser. in Maths. and its Appl. 12 (1998).
11. L. Carbone and C. Sbordone, Some properties of Γ-limits of integral functionals. Ann. Mat. Pura Appl. (iv) 122 (1979) 1-60.
12. V. Chiadò Piat and A. Coscia, Holder continuity of minimizers of functionals with variable growth exponent. Manuscripta Math. 93 (1997) 283-299.
13. A. Coscia and G. Mingione, Hölder continuity of the gradient of p(x)-harmonic mappings. C. R. Acad. Sci. Paris 328 (1999) 363-368.
14. G. Dal Maso, An introduction to r-convergence. Birkäuser, Boston, Prog. Nonlinear Differential Equations Appl. 8 (1993).
15. G. Dal Maso and L. Modica, A general theory for variational functionals. Quaderno S.N.S. Pisa, Topics in Funct. Anal. (1982).
16. E. De Giorgi, Sulla convergenza di alcune successioni di integrali di tipo dell’area. Rend. Mat. Univ. Roma 8 (1975) 277-294.
17. E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 58 (1975) 842-850.
18. E. De Giorgi and G. Letta, Une notion générale de convergence faible pour des fonctions croissantes d’ensemble. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1977) 61-99.
19. I. Ekeland and R. Temam, Convex analysis and variational problems. North Holland, Amsterdam (1978).
20. X. Fan and D. Zhao, A class of De Giorgi type and Holder continuity. Nonlinear Anal. T.M.A. 36 (1999) 295-318.
21. N. Fusco, On the convergence of integral functionals depending on vector-valued functions. Ricerche Mat. 32 (1983) 321-339.
22. P. Marcellini, Regularity and existence of solutions of elliptic equations with p, q-growth conditions. J. Differential Equations 90 (1991) 1-30.
23. P. Marcellini, Regularity for some scalar variational problems under general growth conditions. J. Optim. Theory Appl. 90 (1996) 161-181.
24. C.B. Morrey, Quasi-convexity and semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 25-53.
25. K.R. Rajagopal and M. Ruzicka, Mathematical modelling of electrorheological fluids. Cont. Mech. Therm. 13 (2001) 59-78.
26. M. Ruzicka, Electrorheological fluids: Modeling and mathematical theory. Springer, Berlin, Lecture Notes in Math. 1748 (2000).
27. V.V. Zhikov, On the passage to the limit in nonlinear variational problems. Russian Acad. Sci. Sb. Math. 76 (1993) 427-459.
28. V.V. Zhikov, On Lavrentiev’s phenomenon. Russian J. Math. Phys. 3 (1995) 249-269.
29. V.V. Zhikov, On some variational problems. Russian J. Math. Phys. 5 (1997) 105-116.
30. V.V. Zhikov, Meyers type estimates for solving the non linear Stokes system. Differential Equations 33 (1997) 107-114.
31. V.V. Zhikov, S.M. Kozlov and O.A. Oleinik, Homogenization of differential operators and integral functionals. Springer, Berlin (1994).