Integral representation and Γ-convergence of variational integrals with p(x)-growth
ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 495-519.

We study the integral representation properties of limits of sequences of integral functionals like f(x,Du)dx under nonstandard growth conditions of (p,q)-type: namely, we assume that

|z| p(x) f(x,z)L(1+|z| p(x) ).
Under weak assumptions on the continuous function p(x), we prove Γ-convergence to integral functionals of the same type. We also analyse the case of integrands f(x,u,Du) depending explicitly on u; finally we weaken the assumption allowing p(x) to be discontinuous on nice sets.

DOI : https://doi.org/10.1051/cocv:2002065
Classification : 49J45,  49M20,  46E35
Mots clés : integral representation, Γ-convergence, nonstandard growth conditions
@article{COCV_2002__7__495_0,
     author = {Coscia, Alessandra and Mucci, Domenico},
     title = {Integral representation and $\sf \Gamma $-convergence of variational integrals with ${p(x)}$-growth},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {495--519},
     publisher = {EDP-Sciences},
     volume = {7},
     year = {2002},
     doi = {10.1051/cocv:2002065},
     zbl = {1036.49022},
     mrnumber = {1925039},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2002065/}
}
TY  - JOUR
AU  - Coscia, Alessandra
AU  - Mucci, Domenico
TI  - Integral representation and $\sf \Gamma $-convergence of variational integrals with ${p(x)}$-growth
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2002
DA  - 2002///
SP  - 495
EP  - 519
VL  - 7
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2002065/
UR  - https://zbmath.org/?q=an%3A1036.49022
UR  - https://www.ams.org/mathscinet-getitem?mr=1925039
UR  - https://doi.org/10.1051/cocv:2002065
DO  - 10.1051/cocv:2002065
LA  - en
ID  - COCV_2002__7__495_0
ER  - 
Coscia, Alessandra; Mucci, Domenico. Integral representation and $\sf \Gamma $-convergence of variational integrals with ${p(x)}$-growth. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 495-519. doi : 10.1051/cocv:2002065. http://www.numdam.org/articles/10.1051/cocv:2002065/

[1] E. Acerbi, G. Bouchitté and I. Fonseca, Relaxation of convex functionals: The gap phenomenon. Ann. Inst. H. Poincaré (2003). | Numdam | Zbl 1025.49012

[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. | MR 1814973 | Zbl 0984.49020

[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. | EuDML 84444 | Numdam | MR 1895714 | Zbl 1027.49031

[4] R.A. Adams, Sobolev spaces. Academic Press, New York (1975). | MR 450957 | Zbl 0314.46030

[5] Yu.A. Alkutov, The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth condition. Differential Equations 33 (1998) 1653-1663. | MR 1669915 | Zbl 0949.35048

[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. | MR 1632814 | Zbl 0907.49008

[7] G. Buttazzo, Semicontinuity, relaxation and integral representation in the calculus of variations. Longman, Harlow, Pitman Res. Notes in Math. 207 (1989). | MR 1020296 | Zbl 0669.49005

[8] G. Buttazzo and G. Dal Maso, A characterization of nonlinear functionals on Sobolev spaces which admit an integral representation with a Carathéodory integrand. J. Math. Pures Appl. 64 (1985) 337-361. | MR 839727 | Zbl 0582.46038

[9] G. Buttazzo and G. Dal Maso, Integral representation and relaxation of local functionals. Nonlinear Anal. 9 (1985) 515-532. | MR 794824 | Zbl 0527.49008

[10] A. Braides and A. Defranceschi, Homogenization of multiple integrals. Oxford University Press, Oxford, Oxford Lecture Ser. in Maths. and its Appl. 12 (1998). | MR 1684713 | Zbl 0911.49010

[11] L. Carbone and C. Sbordone, Some properties of Γ-limits of integral functionals. Ann. Mat. Pura Appl. (iv) 122 (1979) 1-60. | MR 565062 | Zbl 0474.49016

[12] V. Chiadò Piat and A. Coscia, Hölder continuity of minimizers of functionals with variable growth exponent. Manuscripta Math. 93 (1997) 283-299. | EuDML 156300 | MR 1457729 | Zbl 0878.49010

[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. | MR 1675954 | Zbl 0920.49020

[14] G. Dal Maso, An introduction to Γ-convergence. Birkäuser, Boston, Prog. Nonlinear Differential Equations Appl. 8 (1993). | MR 1201152 | Zbl 0816.49001

[15] G. Dal Maso and L. Modica, A general theory for variational functionals. Quaderno S.N.S. Pisa, Topics in Funct. Anal. (1982). | MR 671757 | Zbl 0493.49005

[16] E. De Giorgi, Sulla convergenza di alcune successioni di integrali di tipo dell'area. Rend. Mat. Univ. Roma 8 (1975) 277-294. | Zbl 0316.35036

[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. | MR 448194 | Zbl 0339.49005

[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. | EuDML 83746 | Numdam | Zbl 0405.28008

[19] I. Ekeland and R. Temam, Convex analysis and variational problems. North Holland, Amsterdam (1978). | MR 569206 | Zbl 0322.90046

[20] X. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity. Nonlinear Anal. T.M.A. 36 (1999) 295-318. | MR 1688232 | Zbl 0927.46022

[21] N. Fusco, On the convergence of integral functionals depending on vector-valued functions. Ricerche Mat. 32 (1983) 321-339. | MR 766684 | Zbl 0563.49007

[22] P. Marcellini, Regularity and existence of solutions of elliptic equations with p,q-growth conditions. J. Differential Equations 90 (1991) 1-30. | MR 1094446 | Zbl 0724.35043

[23] P. Marcellini, Regularity for some scalar variational problems under general growth conditions. J. Optim. Theory Appl. 90 (1996) 161-181. | MR 1397651 | Zbl 0901.49030

[24] C.B. Morrey, Quasi-convexity and semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 25-53. | MR 54865 | Zbl 0046.10803

[25] K.R. Rajagopal and M. Růžička, Mathematical modelling of electrorheological fluids. Cont. Mech. Therm. 13 (2001) 59-78. | Zbl 0971.76100

[26] M. Růžička, Electrorheological fluids: Modeling and mathematical theory. Springer, Berlin, Lecture Notes in Math. 1748 (2000). | MR 1810360 | Zbl 0962.76001

[27] V.V. Zhikov, On the passage to the limit in nonlinear variational problems. Russian Acad. Sci. Sb. Math. 76 (1993) 427-459. | MR 1187249 | Zbl 0791.35036

[28] V.V. Zhikov, On Lavrentiev's phenomenon. Russian J. Math. Phys. 3 (1995) 249-269. | Zbl 0910.49020

[29] V.V. Zhikov, On some variational problems. Russian J. Math. Phys. 5 (1997) 105-116. | MR 1486765 | Zbl 0917.49006

[30] V.V. Zhikov, Meyers type estimates for solving the non linear Stokes system. Differential Equations 33 (1997) 107-114. | MR 1607245 | Zbl 0911.35089

[31] V.V. Zhikov, S.M. Kozlov and O.A. Oleinik, Homogenization of differential operators and integral functionals. Springer, Berlin (1994). | MR 1329546 | Zbl 0838.35001

Cité par Sources :