Homogenization of convex functionals which are weakly coercive and not equi-bounded from above

Briane, Marc; Casado-Díaz, Juan

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 4, pp. 547-571.

This paper deals with the homogenization of nonlinear convex energies defined in $W_0^{1,1}\left(\Omega \right)$, for a regular bounded open set Ω of ${ℝ}^N$, the densities of which are not equi-bounded from above, and which satisfy the following weak coercivity condition: There exists $q>N-1$ if $N>2$, and $q⩾1$ if $N=2$, such that any sequence of bounded energy is compact in $W_0^{1,q}\left(\Omega \right)$. Under this assumption the Γ-convergence of the functionals for the strong topology of $L^{\infty }\left(\Omega \right)$ is proved to agree with the Γ-convergence for the strong topology of $L^1\left(\Omega \right)$. This leads to an integral representation of the Γ-limit in $C_0^1\left(\Omega \right)$ thanks to a local convex density. An example based on a thin cylinder with very low and very large energy densities, which concentrates to a line shows that the loss of the weak coercivity condition can induce nonlocal effects.

Classification : 35B27, 35B50, 35J60
Mots clés : Homogenization, Convex functionals, Nonlinear elliptic equations, Weak coercivity, Maximum principle

[1] E. Acerbi, V. Chiadò Piat, G. Dal Maso, D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal. 18 no. 5 (1992), 481-496 | MR | Zbl

[2] Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 no. 6 (1992), 1482-1518 | MR | Zbl

[3] J.P. Aubin, Un théorème de compacité, C. R. Math. Acad. Sci. Paris 256 (1963), 5042-5044 | MR | Zbl

[4] M. Bellieud, G. Bouchitté, Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects, Ann. Sc. Norm. Super. Pisa Cl. Sci. 26 no. 4 (1998), 407-436 | EuDML | Numdam | MR | Zbl

[5] A. Beurling, J. Deny, Espaces de Dirichlet, Acta Matematica 99 (1958), 203-224 | MR | Zbl

[6] A. Braides, Γ-Convergence for Beginners, Oxford University Press, Oxford (2002) | MR

[7] A. Braides, M. Briane, Homogenization of non-linear variational problems with thin low-conducting layers, Appl. Math. Optim. 55 no. 1 (2007), 1-29 | MR | Zbl

[8] A. Braides, M. Briane, J. Casado-Díaz, Homogenization of non-uniformly bounded periodic diffusion energies in dimension two, Nonlinearity 22 (2009), 1459-1480 | MR | Zbl

[9] A. Braides, V. Chiadò Piat, A. Piatnitski, A variational approach to double-porosity problems, Asymptot. Anal. 39 no. 3–4 (2004), 281-308 | MR | Zbl

[10] A. Braides, A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press, Oxford (1998) | MR | Zbl

[11] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York (2011) | MR | Zbl

[12] M. Briane, Nonlocal effects in two-dimensional conductivity, Arch. Ration. Mech. Anal. 182 no. 2 (2006), 255-267 | MR | Zbl

[13] M. Briane, J. Casado-Díaz, Two-dimensional div-curl results. Application to the lack of nonlocal effects in homogenization, Comm. Partial Differential Equations 32 (2007), 935-969 | MR | Zbl

[14] M. Briane, J. Casado-Díaz, Asymptotic behavior of equicoercive diffusion energies in two dimension, Calc. Var. Partial Differential Equations 29 no. 4 (2007), 455-479 | MR | Zbl

[15] G. Buttazzo, G. Dal Maso, Γ-limits of integral functionals, J. Anal. Math. 37 (1980), 145-185 | MR | Zbl

[16] M. Camar-Eddine, P. Seppecher, Closure of the set of diffusion functionals with respect to the Mosco-convergence, Math. Models Methods Appl. Sci. 12 no. 8 (2002), 1153-1176 | MR | Zbl

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

[18] G. Dal Maso, An Introduction to Γ-Convergence, Birkhäuser, Boston (1993) | MR

[19] E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dellʼarea, Rend. Mat. Roma 8 (1975), 277-294 | MR | Zbl

[20] E. De Giorgi, Γ-convergenza e G-convergenza, Boll. Unione Mat. Ital. A 14 (1977), 213-220 | MR

[21] E. De Giorgi, T. Franzoni, Su un tipo di convergenza variazionale, Rend. Acc. Naz. Lincei Roma 58 no. 6 (1975), 842-850 | MR | Zbl

[22] V.N. Fenchenko, E.Ya. Khruslov, Asymptotic of solution of differential equations with strongly oscillating matrix of coefficients which does not satisfy the condition of uniform boundedness, Dokl. AN Ukr. SSR 4 (1981) | Zbl

[23] B. Franchi, R. Serapioni, F. Serra Cassano, Irregular solutions of linear degenerate elliptic equations, Potential Anal. 9 (1998), 201-216 | MR | Zbl

[24] E.Ya. Khruslov, Homogenized models of composite media, G. Dal Maso, G.F. DellʼAntonio (ed.), Composite Media and Homogenization Theory, Progr. Nonlinear Differential Equations Appl., Birkhäuser (1991), 159-182 | MR | Zbl

[25] E.Ya. Khruslov, V.A. Marchenko, Homogenization of Partial Differential Equations, Prog. Math. Phys. vol. 46, Birkhäuser, Boston (2006) | MR | Zbl

[26] K. Kuratowski, C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 397-403 | MR | Zbl

[27] J.J. Manfredi, Weakly monotone functions, J. Geom. Anal. 4 no. 3 (1994), 393-402 | MR | Zbl

[28] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123 no. 2 (1994), 368-421 | MR | Zbl

[29] F. Murat, H-convergence, Séminaire dʼAnalyse Fonctionnelle et Numérique, Université dʼAlger (1977)(1978) F. Murat, L. Tartar, H-convergence, L. Cherkaev, R.V. Kohn (ed.), Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl. vol. 31, Birkhäuser, Boston (1998), 21-43

[30] S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Sc. Norm. Super. Pisa Cl. Sci. 22 no. 3 (1968), 571-597 | EuDML | Numdam | MR | Zbl

[31] L. Tartar, The General Theory of Homogenization: A Personalized Introduction, Lect. Notes Unione Mat. Ital., Springer-Verlag, Berlin, Heidelberg (2009) | MR | Zbl

[32] V.V. Zhikov, Connectedness and averaging. Examples of fractal conductivity, Mat. Sb. 187 no. 8 (1996), 3-40, Sb. Math. 187 no. 8 (1996), 1109-1147 | MR | Zbl

[33] V.V. Zhikov, On an extension and an application of the two-scale convergence method, Mat. Sb. 191 no. 7 (2000), 31-72, Sb. Math. 191 no. 7–8 (2000), 973-1014 | MR | Zbl