On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems

Sili, Ali

Comptes Rendus. Mathématique, Tome 360 (2022) no. G1, pp. 1-23.

Dans ce travail, nous analysons le spectre d’un opérateur dégénéré $A_{\epsilon }$ correspondant à l’équation de la chaleur stationnaire dans un milieu composite $\epsilon$-périodique ayant deux composantes avec des coefficients de conductivité à fort contraste. Nous montrons que bien que ${𝒜}_{\epsilon }$ soit un opérateur auto-adjoint à résolvante compacte, sa limite $A_0$ lorsque la période $\epsilon$ tend vers $0$ est un opérateur non auto-adjoint dont le spectre est borné par des constantes positives ne dépendant que de la première valeur propre du Laplacien uni-dimensionnel dans $H_0^1(0,L)$ et de la première valeur propre du Laplacien bi-dimensionnel avec conditions au bord mixtes sur la cellule de référence $C$. Nous montrons en outre que le problème homogénéisé et le problème limite obtenu après réduction de dimension $3d-1d$ intervenant localement sont identiques, à une condition aux limites près, la condition de Neumann homogène sur le bord de $C$ dans le problème $3d-1d$ devant être remplacée dans le problème homogénéisé par une condition de périodicité.

In this paper we perform the analysis of the spectrum of a degenerate operator $A_{\epsilon }$ corresponding to the stationary heat equation in a $\epsilon$-periodic composite medium having two components with high contrast diffusivity. We prove that although $A_{\epsilon }$ is a self-adjoint operator with compact resolvent, its limit $A_0$ when the size $\epsilon$ of the medium tends to zero is a non self-adjoint operator whose spectrum is bounded by positive constants depending on the first eigenvalue of the one-dimensional Laplacian in $H_0^1(0,L)$ and the first eigenvalue of the bi-dimensional Laplacian with mixed boundary conditions on the representative cell $C$. Furthermore, we show that the homogenized problem and the one-dimensional limit problem obtained by the reduction of dimension $3d-1d$ occurring locally are identical except for one boundary condition which is a homogeneous Neumann condition on the boundary of $C$ in the $3d-1d$ problem and a periodicity condition in the case of homogenization.

Classification : 35B25, 35B27, 35B40, 35B45, 35J25, 35J57, 35J70, 35P20

[1] Allaire, Grégoire Homogenization and Two-Scale Convergence, SIAM J. Math. Anal., Volume 23 (1992) no. 6, pp. 1482-1518 | DOI | MR | Zbl

[2] Allaire, Grégoire; Capdeboscq, Yves Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Eng., Volume 187 (2000) no. 1-2, pp. 91-117 | DOI | MR | Zbl

[3] Arbogast, Todd; Douglas, Jim; Hornung, Ulrich Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal., Volume 21 (1990) no. 4, pp. 823-836 | DOI | MR | Zbl

[4] Braides, Andrea; Chiadò Piat, Valeria; Piatnitski, Andrey A variational approach to double-porosity problems, Asymptotic Anal., Volume 39 (2004) no. 3-4, pp. 281-300 | MR | Zbl

[5] Brézis, Haïm Analyse Fonctionnelle, Théorie et applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, 1983

[6] Caillerie, Denis; Dinari, B. A perturbation problem with two small parameters in the framework of the heat conduction of a fiber reinforced body, Partial differential equations (Banach Center Publications), Volume 19, Polish Scientific Publishers, 1984, pp. 59-78 | Zbl

[7] Casado-Díaz, Juan Two-scale convergence for nonlinear Dirichlet problems, Proc. R. Soc. Edinb., Sect. A, Math., Volume 130 (2000) no. 2, pp. 249-276 | DOI

[8] Charef, Hamid; Sili, Ali The effective equilibrium law for a highly heterogeneous elastic periodic medium, Proc. R. Soc. Edinb., Sect. A, Math., Volume 143 (2013) no. 3, pp. 507-561 | DOI | MR | Zbl

[9] Cioranescu, Doina; Saint Jean Paulin, Jeannine Homogenization of reticulated structures, Applied Mathematical Sciences, 136, Springer, 1999 | DOI

[10] Gaudiello, Antonio; Sili, Ali Homogenization of highly oscillating boundaries with strongly contrasting diffusivity, SIAM J. Math. Anal., Volume 47 (2015) no. 3, pp. 1671-1692 | DOI | MR | Zbl

[11] Kesavan, Srinivasan Homogenization of elliptic eigenvalue problems. I, II, Appl. Math. Optim., Volume 5 (1979), p. 153-167, 197–216 | DOI | MR | Zbl

[12] Kesavan, Srinivasan; Sabu, Nicholas Two-dimensional approximation of eigenvalue problems in shell theory: Flexural shells, Chin. Ann. Math., Ser. B, Volume 21 (2000) no. 1, pp. 1-16 | DOI | MR | Zbl

[13] Kreĭn, Mark; Rutman, M. Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl. Ser., Volume 10 (1962), pp. 1-128

[14] Le Dret, Hervé Problèmes variationnels dans les multi-domaines: modélisation des jonctions et applications, Recherches en Mathématiques Appliquées, 19, Masson, 1991

[15] Leugering, Gunter; Nazarov, Sergeĭ A.; Taskinen, Jari The band-gap structures of the spectrum in a periodic medium of Masonry type, Netw. Heterog. Media, Volume 15 (2020) no. 4, pp. 555-580 | DOI | MR | Zbl

[16] Mel’nik, T. A.; Nazarov, Sergeĭ A. Asymptotics of the Neumann spectral problem solution in a domain of “thick comb” type, J. Math. Sci., New York, Volume 85 (1997) no. 6, pp. 2326-2346 | DOI

[17] Murat, François; Sili, Ali A remark about the periodic homogenization of certain composite fibered media, Netw. Heterog. Media, Volume 15 (2020) no. 1, pp. 125-142 | DOI | MR | Zbl

[18] Nguetseng, Gabriel A General Convergence Result for a Functional Related to the Theory of Homogenization, SIAM J. Math. Anal., Volume 20 (1989) no. 3, pp. 608-623 | DOI | MR | Zbl

[19] Panasenko, Grigory Multi-scale modelling for structures and composites, Springer, 2005

[20] Paroni, Roberto; Sili, Ali Nonlocal effects by homogenization or 3D-1D dimension reduction in elastic materials reinforced by stiff fibers, J. Differ. Equations, Volume 260 (2016) no. 3, pp. 2026-2059 | DOI | Zbl

[21] Sili, Ali Homogenization of a nonlinear monotone problem in an anisotropic medium, Math. Models Methods Appl. Sci., Volume 14 (2004) no. 3, pp. 329-353 | DOI | MR | Zbl

[22] Sili, Ali A diffusion equation through a highly heterogeneous medium, Appl. Anal., Volume 89 (2010) no. 6, pp. 893-904 | DOI | MR | Zbl

[23] Vanninathan, Muthusamy Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci., Math. Sci., Volume 90 (1981) no. 3, pp. 239-271 | DOI | MR | Zbl

[24] Zhikov, Vasiliĭ V. On an extension and application of the two-scale convergence method, Mat. Sb., Volume 191 (2000) no. 7, pp. 973-1014 | DOI | MR

[25] Zhikov, Vasiliĭ V.; Kozlov, Sergeĭ M.; Oleĭnik, Ol’ga A. Homogenization of differential operators and integral functionals, Springer, 1994 (translated from the Russian by G.A. Yosifian)