Operator estimates in homogenization theory

V. V. Zhikov   ; S. E. Pastukhova

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 71 (2016) no. 3, pp. 417-511

Résumé

This paper gives a systematic treatment of two methods for obtaining operator estimates: the shift method and the spectral method. Though substantially different in mathematical technique and physical motivation, these methods produce basically the same results. Besides the classical formulation of the homogenization problem, other formulations of the problem are also considered: homogenization in perforated domains, the case of an unbounded diffusion matrix, non-self-adjoint evolution equations, and higher-order elliptic operators. Bibliography: 62 titles.

Détail

Keywords: shift method, integrated estimate, Steklov smoothing, periodicity, problem on the cell, asymptotics of the fundamental solution, spectral method, Bloch representation of an operator, Nash–Aronson estimate.

Bibliographie

[1] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic analysis for periodic structure, Stud. Math. Appl., 5, North-Holland Publishing Co., Amsterdam–New York, 1978, xxiv+700 pp. | MR | Zbl

[2] N. Bakhvalov, G. Panasenko, Homogenisation: averaging processes in periodic media. Mathematical problems in the mechanics of composite materials, Math. Appl. (Soviet Ser.), 36, Kluwer Acad. Publ., Dordrecht, 1989, xxxvi+366 pp. | DOI | MR | MR | Zbl | Zbl

[3] E. Sánchez-Palencia, Non-homogeneous media and vibration theory, Lecture Notes in Phys., 127, Springer-Verlag, Berlin–New York, 1980, ix+398 pp. | DOI | MR | MR | Zbl

[4] V. V. Jikov, S. M. Kozlov, O. A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994, xii+570 pp. | DOI | MR | MR | Zbl | Zbl

[5] V. V. Zhikov, “Spectral approach to asymptotic problems in diffusion”, Differential Equations, 25:1 (1989), 33–39 | MR | Zbl

[6] T. A. Suslina, “On homogenization of periodic parabolic systems”, Funct. Anal. Appl., 38:4 (2004), 309–312 | DOI | DOI | MR | Zbl

[7] V. V. Zhikov, S. E. Pastukhova, “Estimates of homogenization for a parabolic equation with periodic coefficients”, Russ. J. Math. Phys., 13:2 (2006), 224–237 | DOI | MR | Zbl

[8] J. H. Ortega, E. Zuazua, “Large time behavior in $\mathbb{R}^d$ for linear parabolic equations with periodic coefficients”, Asymptot. Anal., 22:1 (2000), 51–85 | MR | Zbl

[9] M. Sh. Birman, T. A. Suslina, “Second order periodic differential operators. Threshold properties and homogenization”, St. Petersburg Math. J., 15:5 (2004), 639–714 | DOI | MR | Zbl

[10] V. V. Zhikov, “On operator estimates in homogenization theory”, Dokl. Math., 72:1 (2005), 534–538 | MR | Zbl

[11] V. V. Zhikov, “Some estimates from homogenization theory”, Dokl. Math., 73:1 (2006), 96–99 | DOI | MR | Zbl

[12] T. A. Suslina, “Homogenization of a stationary periodic Maxwell system”, St. Petersburg Math. J., 16:5 (2005), 863–922 | DOI | MR | Zbl

[13] V. V. Zhikov, S. E. Pastukhova, “On operator estimates for some problems in homogenization theory”, Russ. J. Math. Phys., 12:4 (2005), 515–524 | MR | Zbl

[14] S. E. Pastukhova, “Some estimates from homogenized elasticity problems”, Dokl. Math., 73:1 (2006), 102–106 | DOI | MR | Zbl

[15] R. M. Christensen, Mechanics of composite materials, Wiley-Interscience, New York, 1979, 348 pp.

[16] L. M. Brekhovskikh, Waves in layered media, Appl. Math. Mech., 16, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York–London, 1980, xiii+503 pp. | MR | MR | Zbl | Zbl

[17] L. Brillouin, M. Parodi, Propagation des ondes dans les milieux périodiques, Masson et Cie, Paris; Dunod, Paris, 1956, iv+347 pp. | MR | Zbl

[18] G. K. Batchelor, “Compression waves in a suspension of gas bubbles in liquid”, Fluid Dynamics Transactions (Warszawa), 4 (1967), 425–445

[19] A. Yu. Belyaev, Usrednenie v zadachakh teorii filtratsii, Nauka, M., 2004, 200 pp. | Zbl

[20] E. V. Sevost'yanova, “An asymptotic expansion of the solution of a second order elliptic equation with periodic rapidly oscillating coefficients”, Math. USSR-Sb., 43:2 (1982), 181–198 | DOI | MR | Zbl

[21] A. Yu. Belyaev, “Compression waves in a fluid with gas bubbles”, J. Appl. Math. Mech., 52:3 (1988), 344–348 | DOI | Zbl

[22] I. A. Aleksandrova, “The spectral method in asymptotic diffusion problems with drift”, Math. Notes, 59:5 (1996), 554–556 | DOI | DOI | MR | Zbl

[23] C. Conca, R. Orive, M. Vanninathan, “Bloch approximation in homogenization and applications”, SIAM J. Math. Anal., 33:5 (2002), 1166–1198 | DOI | MR | Zbl

[24] M. Sh. Birman, T. A. Suslina, “Operator error estimates in the homogenization problem for nonstationary periodic equations”, St. Petersburg Math. J., 20:6 (2009), 873–928 | DOI | MR | Zbl

[25] A. A. Kukushkin, T. A. Suslina, “Usrednenie ellipticheskikh operatorov vysokogo poryadka s periodicheskimi koeffitsientami”, Algebra i analiz, 28:1 (2016), 89–149

[26] O. A. Ladyzhenskaya, N. N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York–London, 1968, xviii+495 pp. | MR | MR | Zbl | Zbl

[27] D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Pure Appl. Math., 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York–London, 1980, xiv+313 pp. | MR | MR | Zbl | Zbl

[28] T. A. Suslina, “Homogenization of the Dirichlet problem for elliptic systems: $L_2$-operator error estimates”, Mathematika, 59:2 (2013), 463–476 | DOI | MR | Zbl

[29] T. Suslina, “Homogenization of the Neumann problem for elliptic systems with periodic coefficients”, SIAM J. Math. Anal., 45:6 (2013), 3453–3493 | DOI | MR | Zbl

[30] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Ser., 43, Princeton Univ. Press, Princeton, NJ, 1993, xiv+695 pp. | MR | Zbl

[31] R. Coifman, P. L. Lions, Y. Meyer, S. Semmes, “Compensated compactness and Hardy spaces”, J. Math. Pures Appl. (9), 72:3 (1993), 247–286 | MR | Zbl

[32] V. G. Maz'ya, I. E. Verbitsky, “Form boundedness of the general second-order differential operator”, Comm. Pure Appl. Math., 59:9 (2006), 1286–1329 | DOI | MR | Zbl

[33] V. V. Zhikov, “Remarks on the uniqueness of a solution of the Dirichlet problem for second-order elliptic equations with lower-order terms”, Funct. Anal. Appl., 38:3 (2004), 173–183 | DOI | DOI | MR | Zbl

[34] V. V. Zhikov, “Diffusion in an incompressible random flow”, Funct. Anal. Appl., 31:3 (1997), 156–166 | DOI | DOI | MR | Zbl

[35] V. V. Zhikov, S. V. Tikhomirova, “On operator estimates in nonsymmetric averaging problems”, J. Math. Sci. (N. Y.), 144:1 (2007), 3870–3874 | DOI | Zbl

[36] V. V. Zhikov, S. E. Pastukhova, “Homogenization of degenerate elliptic equations”, Sib. Math. J., 49:1 (2008), 80–101 | DOI | MR | Zbl

[37] G. Allaire, M. Briane, “Multiscale convergence and reiterated homogenisation”, Proc. Roy. Soc. Edinburgh Sect. A, 126:2 (1996), 297–342 | DOI | MR | Zbl

[38] M. L. Kleptsyna, A. L. Pyatnitski, “Homogenization of a random non-stationary convection-diffusion problem”, Russian Math. Surveys, 57:4 (2002), 729–751 | DOI | DOI | MR | Zbl

[39] D. Lukkassen, G. Nguetseng, P. Wall, “Two-scale convergence”, Int. J. Pure Appl. Math., 2:1 (2002), 35–86 | MR | Zbl

[40] A. M. Meirmanov, “Prilozhenie metoda povtornogo usredneniya differentsialnykh uravnenii v teorii filtratsii szhimaemykh vyazkikh zhidkostei v szhimaemykh treschinovato-poristykh sredakh. Chast I: Mikroskopicheskoe opisanie”, Matem. modelirovanie, 23:1 (2011), 100–114 | MR | Zbl

[41] A. M. Meirmanov, “Prilozhenie metoda povtornogo usredneniya differentsialnykh uravnenii v teorii filtratsii szhimaemykh vyazkikh zhidkostei v szhimaemykh treschinovato-poristykh sredakh. Chast II: Makroskopicheskoe opisanie”, Matem. modelirovanie, 23:4 (2011), 3–22 | MR | Zbl

[42] A. Meirmanov, “Equations of liquid filtration in double porosity media as a reiterated homogenization of Stokes equations”, Proc. Steklov Inst. Math., 278 (2012), 152–160 | DOI | MR | Zbl

[43] D. I. Borisov, “Asymptotics for the solutions of elliptic systems with rapidly oscillating coefficients”, St. Petersburg Math. J., 20:2 (2009), 175–191 | DOI | MR | Zbl

[44] D. I. Borisov, R. R. Gadyl'shin, “On the spectrum of a periodic operator with a small localized perturbation”, Izv. Math., 72:4 (2008), 659–688 | DOI | DOI | MR | Zbl

[45] S. E. Pastukhova, R. N. Tikhomirov, “Estimates of locally periodic and reiterated homogenization for parabolic equations”, Dokl. Math., 80:2 (2009), 674–678 | DOI | MR | Zbl

[46] S. E. Pastukhova, “Operator estimates in nonlinear problems of reiterated homogenization”, Proc. Steklov Inst. Math., 261 (2008), 214–228 | DOI | MR | Zbl

[47] S. Pastukhova, “Estimates in homogenization of parabolic equations with locally periodic coefficients”, Asymptot. Anal., 66:3-4 (2010), 207–228 | DOI | MR | Zbl

[48] S. E. Pastukhova, “Approximation of the exponential of a diffusion operator with multiscale coefficients”, Funct. Anal. Appl., 48:3 (2014), 183–197 | DOI | DOI | Zbl

[49] S. E. Pastukhova, “The Dirichlet problem for elliptic equations with multiscale coefficients. Operator estimates for homogenization”, J. Math. Sci. (N. Y.), 193:2 (2013), 283–300 | DOI | MR | Zbl

[50] S. Pastukhova, R. Tikhomirov, On homogenization estimates in Neuman boundary value problem for an elliptic equation with multiscale coefficients, 2015, 19 pp., arXiv: 1512.06396

[51] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Hà Tiên Ngoan, “Averaging and $G$-convergence of differential operators”, Russian Math. Surveys, 34:5 (1979), 69–147 | DOI | MR | Zbl

[52] T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., 132, Springer-Verlag, New York, 1966, xix+592 pp. | MR | MR | Zbl | Zbl

[53] W. Feller, An introduction to probability theory and its applications, v. II, John Wiley Sons, Inc., New York–London–Sydney, 1966, xviii+626 pp. | MR | MR | Zbl | Zbl

[54] V. B. Korotkov, Integralnye operatory, Nauka, Novosibirsk, 1983, 224 pp. | MR | Zbl

[55] V. V. Zhikov, “Asymptotic behavior and stabilization of solutions of a second order parabolic equation with lower order terms”, Trans. Moscow Math. Soc., 1984, No 2, Amer. Math. Soc., Providence, RI, 1984, 69–99 | MR | Zbl

[56] A. L. Piatnitski, “Averaging a singularly perturbed equation with rapidly oscillating coefficients in a layer”, Math. USSR-Sb., 49:1 (1984), 19–40 | DOI | MR | Zbl

[57] P. Donato, A. Piatnitski, “Averaging of nonstationary parabolic operators with large lower order terms”, Multi-scale problems and asymptotic analysis, GAKUTO Internat. Ser. Math. Sci. Appl., 24, Gakkōtosho, Tokyo, 2006, 153–165 | MR | Zbl

[58] G. Allaire, R. Orive, “Homogenization of periodic non self-adjoint problems with large drift and potential”, ESAIM Control Optim. Calc. Var., 13:4 (2007), 735–749 | DOI | MR | Zbl

[59] S. E. Pastukhova, “Approximations of the operator exponential in a periodic diffusion problem with drift”, Sb. Math., 204:2 (2013), 280–306 | DOI | DOI | MR | Zbl

[60] K. Yosida, Functional analysis, Grundlehren Math. Wiss., 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965, xi+458 pp. | MR | MR | Zbl | Zbl

[61] S. Mizohata, Henbidun hôteisiki ron [The theory of partial differential equations], Contemp. Math., 9, Iwanami Shoten, Tokyo, 1965, viii+462 pp. | MR | Zbl

[62] S. E. Pastukhova, “Approximations of the exponential of an operator with periodic coefficients”, J. Math. Sci. (N. Y.), 181:5 (2012), 668–700 | MR | Zbl