Homogenization of elliptic and parabolic equations with~periodic coefficients in a~bounded domain under the Neumann condition

T. A. Suslina

Izvestiya. Mathematics , Tome 88 (2024) no. 4, pp. 678-759.

Résumé

Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a selfadjoint matrix elliptic second-order differential operator $B_{N,\varepsilon}$, $0\varepsilon\leqslant1$, with the Neumann boundary condition. The principal part of this operator is given in a factorized form. The operator involves first-order and zero-order terms. The coefficients of $B_{N,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. We study the generalized resolvent $(B_{N,\varepsilon}-\zeta Q_0(\,{\cdot}\,/\varepsilon))^{-1}$, where $Q_0$ is a periodic bounded and positive definite matrix-valued function, and $\zeta$ is a complex-valued parameter. We obtain approximations for the generalized resolvent in the operator norm on $L_2(\mathcal{O};\mathbb{C}^n)$ and in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$ with two-parametric error estimates (with respect to $\varepsilon$ and $\zeta$). The results are applied to study the behaviour of the solutions of the initial boundary value problem with the Neumann condition for the parabolic equation $Q_0(\mathbf{x}/\varepsilon) \, \partial_t \mathbf{u}_\varepsilon(\mathbf{x},t) = -(B_{N,\varepsilon} \mathbf{u}_\varepsilon)(\mathbf{x},t)$ in a cylinder $\mathcal{O} \times (0,T)$, where $0$.

Détail

Keywords: periodic differential operators, elliptic systems, parabolic systems, homogenization, operator error estimates.

Bibliographie

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

[2] N. Bakhvalov and 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 | DOI | MR | Zbl

[3] O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical problems in elasticity and homogenization, Stud. Math. Appl., 26, North-Holland Publishing Co., Amsterdam, 1992 | MR | Zbl

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

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

[6] T. A. Suslina, “Homogenization of elliptic operators with periodic coefficients in dependence of the spectral parameter”, St. Petersburg Math. J., 27:4 (2016), 651–708 | DOI

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

[8] M. Sh. Birman and T. A. Suslina, “Homogenization with corrector term for periodic elliptic differential operators”, St. Petersburg Math. J., 17:6 (2006), 897–973 | DOI

[9] M. Sh. Birman and T. A. Suslina, “Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class $H^1(\mathbb R^d)$”, St. Petersburg Math. J., 18:6 (2007), 857–955 | DOI

[10] T. A. Suslina, “Homogenization in the Sobolev class $H^1(\mathbb{R}^d)$ for second order periodic elliptic operators with the inclusion of first order terms”, St. Petersburg Math. J., 22:1 (2011), 81–162 | DOI

[11] T. A. Suslina, “Homogenization of elliptic systems with periodic coefficients: operator error estimates in $L_2(\mathbb{R}^d)$ with corrector taken into account”, St. Petersburg Math. J., 26:4 (2015), 643–693 | DOI

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

[13] T. A. Suslina, “Homogenization of a periodic parabolic Cauchy problem”, Nonlinear equations and spectral theory, Amer. Math. Soc. Transl. Ser. 2, 220, Adv. Math. Sci., 59, Amer. Math. Soc., Providence, RI, 2007, 201–233 | DOI | MR | Zbl

[14] T. Suslina, “Homogenization of a periodic parabolic Cauchy problem in the Sobolev space $H^1(\mathbb{R}^d)$”, Math. Model. Nat. Phenom., 5:4 (2010), 390–447 | DOI | MR | Zbl

[15] Yu. M. Meshkova, “Homogenization of the Cauchy problem for parabolic systems with periodic coefficients”, St. Petersburg Math. J., 25:6 (2014), 981–1019 | DOI

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

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

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

[19] V. V. Zhikov and 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

[20] V. V. Zhikov and S. E. Pastukhova, “Operator estimates in homogenization theory”, Russian Math. Surveys, 71:3 (2016), 417–511 | DOI

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

[22] N. N. Senik, “Homogenization for non-self-adjoint periodic elliptic operators on an infinite cylinder”, SIAM J. Math. Anal., 49:2 (2017), 874–898 | DOI | MR | Zbl

[23] Sh. Moskow and M. Vogelius, “First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof”, Proc. Roy. Soc. Edinburgh Sect. A, 127:6 (1997), 1263–1299 | DOI | MR | Zbl

[24] S. Moskow and M. Vogelius, First order corrections to the homogenized eigenvalues of a periodic composite medium. The case of Neumann boundary conditions, preprint, Rutgers Univ., 1997

[25] G. Griso, “Error estimate and unfolding for periodic homogenization”, Asymptot. Anal., 40:3-4 (2004), 269–286 | MR | Zbl

[26] G. Griso, “Interior error estimate for periodic homogenization”, Anal. Appl. (Singap.), 4:1 (2006), 61–79 | DOI | MR | Zbl

[27] C. E. Kenig, Fanghua Lin, and Zhongwei Shen, “Convergence rates in $L^2$ for elliptic homogenization problems”, Arch. Ration. Mech. Anal., 203:3 (2012), 1009–1036 | DOI | MR | Zbl

[28] M. A. Pakhnin and T. A. Suslina, “Operator error estimates for homogenization of the elliptic Dirichlet problem in a bounded domain”, St. Petersburg Math. J., 24:6 (2013), 949–976 | DOI

[29] 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

[30] Qiang Xu, “Uniform regularity estimates in homogenization theory of elliptic system with lower order terms”, J. Math. Anal. Appl., 438:2 (2016), 1066–1107 | DOI | MR | Zbl

[31] Qiang Xu, “Uniform regularity estimates in homogenization theory of elliptic systems with lower order terms on the Neumann boundary problem”, J. Differential Equations, 261:8 (2016), 4368–4423 | DOI | MR | Zbl

[32] Qiang Xu, “Convergence rates for general elliptic homogenization problems in Lipschitz domains”, SIAM J. Math. Anal., 48:6 (2016), 3742–3788 | DOI | MR | Zbl

[33] Zhongwei Shen, Periodic homogenization of elliptic systems, Oper. Theory Adv. Appl., 269, Adv. Partial Differ. Equ. (Basel), Birkhäuser/Springer, Cham, 2018 | DOI | MR | Zbl

[34] Zhongwei Shen and Jinping Zhuge, “Convergence rates in periodic homogenization of systems of elasticity”, Proc. Amer. Math. Soc., 145:3 (2017), 1187–1202 | DOI | MR | Zbl

[35] Yu. M. Meshkova and T. A. Suslina, “Homogenization of initial boundary value problems for parabolic systems with periodic coefficients”, Appl. Anal., 95:8 (2016), 1736–1775 | DOI | MR | Zbl

[36] Jun Geng and Zhongwei Shen, “Convergence rates in parabolic homogenization with time-dependent periodic coefficients”, J. Funct. Anal., 272:5 (2017), 2092–2113 | DOI | MR | Zbl

[37] Yu. M. Meshkova and T. A. Suslina, “Two-parametric error estimates in homogenization of second order elliptic systems in $\mathbb{R}^d$”, Appl. Anal., 95:7 (2016), 1413–1448 | DOI | MR | Zbl

[38] T. A. Suslina, “Operator-theoretic approach to the homogenization of Schrödinger-type equations with periodic coefficients”, Russian Math. Surveys, 78:6 (2023), 1023–1154 | DOI

[39] O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, Appl. Math. Sci., 49, Springer-Verlag, New York, 1985 | DOI | MR | Zbl

[40] Yu. M. Meshkova and T. A. Suslina, Homogenization of the Dirichlet problem for elliptic systems: two-parametric error estimates, 2017, arXiv: 1702.00550v4

[41] Yu. M. Meshkova and T. A. Suslina, “Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients”, Funct. Anal. Appl., 51:3 (2017), 230–235 | DOI

[42] Yu. M. Meshkova and T. A. Suslina, “Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates”, St. Petersburg Math. J., 29:6 (2018), 935–978 | DOI

[43] Yu. M. Meshkova, “On homogenization of the first initial–boundary value problem for periodic hyperbolic systems”, Appl. Anal., 99:9 (2020), 1528–1563 | DOI | MR | Zbl

[44] J. Nečas, Direct methods in the theory of elliptic equations, Transl. from the French, Springer Monogr. Math., Springer, Heidelberg, 2012 | DOI | MR | Zbl

[45] W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge Univ. Press, Cambridge, 2000 | MR | Zbl

[46] V. A. Kondrat'ev and S. D. Èĭdel'man, “On conditions on the boundary surface in the theory of elliptic boundary value problems”, Soviet Math. Dokl., 20 (1979), 561–563

[47] V. G. Maz'ya and T. O. Shaposhnikova, Theory of multipliers in spaces of differentiable functions, Monogr. Stud. Math., 23, Pitman, Boston, MA, 1985 | MR | Zbl

[48] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Ser., 30, Princeton Univ. Press, Princeton, NJ, 1970 | MR | Zbl

[49] V. S. Rychkov, “On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains”, J. London Math. Soc. (2), 60:1 (1999), 237–257 | DOI | MR | Zbl

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

[51] T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., 132, Springer-Verlag New York, Inc., New York, 1966 | DOI | MR | Zbl