Integral representation and $\Gamma$-convergence of variational integrals with $p\left(x\right)$-growth

Coscia, Alessandra; Mucci, Domenico

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 $\int f(x,Du)\mathrm{d}x$ under nonstandard growth conditions of $(p,q)$-type: namely, we assume that

$${\left|z\right|}^{p\left(x\right)}\le f(x,z)\le {L(1+|z|}^{p\left(x\right)}).$$

Under weak assumptions on the continuous function $p\left(x\right)$, we prove $\Gamma$-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 $p\left(x\right)$ to be discontinuous on nice sets.

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

[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 | Zbl

[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 | Numdam | MR | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

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

[13] A. Coscia and G. Mingione, Hölder continuity of the gradient of $p\left(x\right)$-harmonic mappings. C. R. Acad. Sci. Paris 328 (1999) 363-368. | MR | Zbl

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

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

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

[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 | Zbl

[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 | Numdam | Zbl

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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