A generalized strange term in Signorini's type problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 5, p. 773-805

The limit behavior of the solutions of Signorini’s type-like problems in periodically perforated domains with period ε is studied. The main feature of this limit behaviour is the existence of a critical size of the perforations that separates different emerging phenomena as ε0. In the critical case, it is shown that Signorini’s problem converges to a problem associated to a new operator which is the sum of a standard homogenized operator and an extra zero order term (“strange term”) coming from the geometry; its appearance is due to the special size of the holes. The limit problem captures the two sources of oscillations involved in this kind of free boundary-value problems, namely, those arising from the size of the holes and those due to the periodic inhomogeneity of the medium. The main ingredient of the method used in the proof is an explicit construction of suitable test functions which provide a good understanding of the interactions between the above mentioned sources of oscillations.

DOI : https://doi.org/10.1051/m2an:2003055
Classification:  35B27,  35A25,  42C30
Keywords: Signorini's problem, homogenization, Tartar's method, variational inequality
@article{M2AN_2003__37_5_773_0,
     author = {Conca, Carlos and Murat, Fran\c cois and Timofte, Claudia},
     title = {A generalized strange term in Signorini's type problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {5},
     year = {2003},
     pages = {773-805},
     doi = {10.1051/m2an:2003055},
     zbl = {1040.35008},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2003__37_5_773_0}
}
Conca, Carlos; Murat, François; Timofte, Claudia. A generalized strange term in Signorini's type problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 5, pp. 773-805. doi : 10.1051/m2an:2003055. http://www.numdam.org/item/M2AN_2003__37_5_773_0/

[1] N. Ansini and A. Braides, Asymptotic analysis of periodically perforated non-linear media. J. Math. Pures Appl. 81 (2002) 439-451. | Zbl 1036.35021

[2] H. Attouch, Variational Convergence for Functions and Operators. Pitman, London (1984). | MR 773850 | Zbl 0561.49012

[3] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998). | MR 1684713 | Zbl 0911.49010

[4] H. Brézis, Problèmes unilatéraux. J. Math. Pures Appl. 51 (1972) 1-168. | Zbl 0237.35001

[5] C. Calvo Jurado and J. Casado Díaz, The limit of Dirichlet systems for variable monotone operators in general perforated domains. J. Math. Pures Appl. 81 (2002) 471-493. | Zbl 1035.35009

[6] J. Casado Díaz, Existence of a sequence satisfying Cioranescu-Murat conditions in homogenization of Dirichlet problems in perforated domains. Rend. Mat. Appl. 16 (1996) 387-413. | Zbl 0870.35013

[7] J. Casado-Díaz, Homogenization of Dirichlet problems for monotone operators in varying domains. Proc. Roy. Soc. Edinburgh Sect. A 127 (1997) 457-478. | Zbl 0877.35013

[8] D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979) 590-607. | Zbl 0427.35073

[9] D. Cioranescu and F. Murat, Un terme étrange venu d'ailleurs, I and II, in Nonlinear Differential Equations and Their Applications, Collège de France Seminar, H. Brézis and J.L. Lions, Eds., Vol. II, pp. 98-138, Vol. III, pp. 154-178. Pitman, London, Research Notes in Mathematics 60 and 70 (1982 and 1983). English translation: A Strange Term Coming from Nowhere, in Topics in the Mathematical Modelling of Composite Materials, A. Cherkaev and R.V. Kohn, Eds. Birkhäuser, Boston, Progr. Nonlinear Differential Equations Appl. 31 (1997) 45-93. | Zbl 0496.35030

[10] C. Conca and P. Donato, Non homogeneous Neumann problems in domains with small holes. RAIRO Modél. Math. Anal. Numér. 22 (1988) 561-607. | Numdam | Zbl 0669.35028

[11] C. Conca and M. Vanninathan, On uniform H 2 -estimates in periodic homogenization. Proc. Roy. Soc. Edinburgh A 131 (2001) 499-517. | Zbl 1005.35014

[12] G. Dal Maso, On the integral representation of certain local functionals. Ricerche Mat. 32 (1983) 85-113. | Zbl 0543.49001

[13] G. Dal Maso, Γ-convergence and μ-capacities. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14 (1987) 423-464. | Numdam | Zbl 0657.49005

[14] G. Dal Maso, An Introduction to Γ- Convergence. Birkhäuser, Boston, Progr. Nonlinear Differential Equations Appl. 8 (1993). | Zbl 0816.49001

[15] G. Dal Maso and A. Garroni, New results on the asymptotic behavior of Dirichlet problems in perforated domains. Math. Models Methods Appl. Sci. 4 (1994) 373-407. | Zbl 0804.47050

[16] G. Dal Maso and F. Murat, Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators. Ann. Scuola. Norm. Sup. Pisa Cl. Sci. 24 (1997) 239-290. | Numdam | Zbl 0899.35007

[17] G. Dal Maso and F. Murat, Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains. Ann. Inst. H. Poincaré, Anal. non linéaire (to appear). | Numdam | Zbl 1110.35008

[18] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Atti. Accad. Naz. Lincei Rend. Cl. Sci. Mat. Fis. Natur. 58 (1975) 842-850. | Zbl 0339.49005

[19] G. Duvaut and J.L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972). | Zbl 0298.73001

[20] G. Fichera, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Mem. Accad. Naz. Lincei, Serie 8 7 (1964) 91-140. | Zbl 0146.21204

[21] A. Kovalevsky, An effect of double homogenization for Dirichlet problems in variable domains of general structure. C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1151-1156. | Zbl 0931.35054

[22] J.L. Lions and G. Stampacchia, Variational inequalities. Comm. Pure Appl. Math. 20 (1967) 493-519. | Zbl 0152.34601

[23] V.A. Marcenko and E.Ja. Hrouslov, Boundary Value Problems in Domains with Fine-Grained Boundary (in russian). Naukova Dumka, Kiev (1974). | Zbl 0289.35002

[24] F. Murat, H-convergence, Séminaire d'Analyse Fonctionnelle et Numérique de l'université d'Alger 1977-1978, multigraphied, 34 p. English translation: F. Murat and L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials, A. Cherkaev and R.V. Khon, Eds. Birkhäuser, Boston, Progr. Nonlinear Differential Equations Appl. 31 (1997) 21-43. | Zbl 0920.35019

[25] A. Signorini, Sopra alcune questioni di Elastostatica. Atti della Soc. Ital. per il Progresso della Scienze (1963). | JFM 59.1413.02

[26] I.V. Skrypnik, Nonlinear Elliptic Boundary Value Problems. Teubner-Verlag, Leipzig (1986). | MR 915342 | Zbl 0617.35001

[27] L. Tartar, Problèmes d'homogénéisation dans les équations aux dérivées partielles. Cours Peccot, Collège de France (1977) (Partially written in [24]).

[28] L. Tartar, Quelques remarques sur l'homogénéisation, in Functional Analysis and Numerical Analysis, Proceedings of the Japan-France Seminar 1976, H. Fujita, Ed. Japan Society for the Promotion of Science, Tokyo (1978) 469-482.