Hostname: page-component-686fd747b7-n7dl9 Total loading time: 0 Render date: 2025-02-13T13:49:35.230Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Behaviour of the Energy Integral of a Two-Parameter Homogenization Problem with Nonlinear Periodic Robin Boundary Conditions

Part of: Higher-dimensional theory Linear integral equations Elliptic equations and systems Nonlinear operators and their properties

Published online by Cambridge University Press:  22 March 2019

Massimo Lanza de Cristoforis  and
Paolo Musolino
Massimo Lanza de Cristoforis
Affiliation:
Dipartimento di Matematica ‘Tullio Levi-Civita’, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy (mldc@math.unipd.it; musolino@math.unipd.it)
Paolo Musolino
Affiliation:
Dipartimento di Matematica ‘Tullio Levi-Civita’, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy (mldc@math.unipd.it; musolino@math.unipd.it)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a nonlinear Robin problem for the Poisson equation in an unbounded periodically perforated domain. The domain has a periodic structure, and the size of each cell is determined by a positive parameter δ. The relative size of each periodic perforation is determined by a positive parameter ε. Under suitable assumptions, such a problem admits a family of solutions which depends on ε and δ. We analyse the behaviour the energy integral of such a family as (ε, δ) tends to (0, 0) by an approach that represents an alternative to asymptotic expansions and classical homogenization theory.

Keywords

Nonlinear Robin problemsingularly perturbed domainPoisson equationperiodically perforated domainhomogenizationenergy integralreal analytic continuation in Banach space

MSC classification

Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 31B10: Integral representations, integral operators, integral equations methods 45A05: Linear integral equations 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 4 , November 2019 , pp. 985 - 1016
Copyright
Copyright © Edinburgh Mathematical Society 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Ammari, H. and Kang, H., Polarization and moment tensors, Applied Mathematical Sciences, Volume 162 (Springer, New York, 2007).Google Scholar
2Ammari, H., Kang, H. and Lee, H., Layer potential techniques in spectral analysis (American Mathematical Society, Providence, RI, 2009).Google Scholar
3Bonnaillie-Noël, V., Dambrine, M., Tordeux, S. and Vial, G., Interactions between moderately close inclusions for the Laplace equation, Math. Models Methods Appl. Sci. 19 (2009), 1853–1882.Google Scholar
4Cabarrubias, B. and Donato, P., Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions, Appl. Anal. 91 (2012), 1111–1127.Google Scholar
5Castro, L. P., Pesetskaya, E. and Rogosin, S. V., Effective conductivity of a composite material with non-ideal contact conditions, Complex Var. Elliptic Equ. 54 (2009), 1085–1100.Google Scholar
6Cioranescu, D. and Murat, F., Un terme étrange venu d'ailleurs, In Nonlinear partial differential equations and their applications. Collège de France Seminar, Volume II (Paris, 1979/1980), Pitman Research Notes in Mathematics, Volume 60 (Pitman, Boston, MA, 1982)pp. 98–138, 389–390.Google Scholar
7Cioranescu, D. and Murat, F., Un terme étrange venu d'ailleurs. II, In Nonlinear partial differential equations and their applications. Collège de France Seminar, Volume III (Paris, 1980/1981), Pitman Research Notes in Mathematics, Volume 70 (Pitman, Boston, MA, 1982)pp. 154–178, 425–426.Google Scholar
8Dalla Riva, M., Stokes flow in a singularly perturbed exterior domain, Complex Var. Elliptic Equ. 58 (2013), 231–257.Google Scholar
9Dalla Riva, M. and Lanza de Cristoforis, M., Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem: a functional analytic approach, Complex Var. Elliptic Equ. 55 (2010), 771–794.Google Scholar
10Dalla Riva, M., Musolino, P. and Rogosin, S. V., Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole, Asymptot. Anal. 92 (2015), 339–361.Google Scholar
11Dauge, M., Tordeux, S. and Vial, G., Selfsimilar perturbation near a corner: matching versus multiscale expansions for a model problem, in Around the research of Vladimir Maz'ya. II, International Mathematical Series, Volume 12 (Springer, New York, 2010)pp. 95–134.Google Scholar
12Deimling, K., Nonlinear functional analysis (Springer-Verlag, Berlin, 1985).Google Scholar
13Folland, G. B., Introduction to partial differential equations, 2nd edn (Princeton University Press, Princeton, NJ, 1995).Google Scholar
14Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order (Springer Verlag, Berlin, 1983).Google Scholar
15John, F., Partial differential equations, 4th edn, Applied Mathematical Sciences,Volume 1 (Springer-Verlag, New York, 1982).Google Scholar
16Kapanadze, D., Mishuris, G. and Pesetskaya, E., Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials, Complex Var. Elliptic Equ. 60 (2015), 1–23.Google Scholar
17Kozlov, V., Maz'ya, V. and Movchan, A., Asymptotic analysis of fields in multi-structures, Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, New York, 1999).Google Scholar
18Lanza de Cristoforis, M., Properties and pathologies of the composition and inversion operators in Schauder spaces, Acc. Naz. delle Sci. detta dei XL 15 (1991), 93–109.Google Scholar
19Lanza de Cristoforis, M., Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces, Comput. Methods Funct. Theory 2 (2002), 1–27.Google Scholar
20Lanza de Cristoforis, M., Asymptotic behavior of the solutions of a nonlinear Robin problem for the Laplace operator in a domain with a small hole: a functional analytic approach, Complex Var. Elliptic Equ. 52 (2007), 945–977.Google Scholar
21Lanza de Cristoforis, M. and Musolino, P., A perturbation result for periodic layer potentials of general second order differential operators with constant coefficients, Far East J. Math. Sci. 52 (2011), 75–120.Google Scholar
22Lanza de Cristoforis, M. and Musolino, P., A singularly perturbed nonlinear Robin problem in a periodically perforated domain: a functional analytic approach, Complex Var. Elliptic Equ. 58 (2013), 511–536.Google Scholar
23Lanza de Cristoforis, M. and Musolino, P., A singularly perturbed Neumann problem for the Poisson equation in a periodically perforated domain. A functional analytic approach, Z. Angew. Math. Mech. 92 (2016), 253–272.Google Scholar
24Lanza de Cristoforis, M. and Musolino, P., Two-parameter anisotropic homogenization for a Dirichlet problem for the Poisson equation in an unbounded periodically perforated domain. A functional analytic approach, Math. Nachr. 291 (2018), 1310–1341.Google Scholar
25Lanza de Cristoforis, M. and Musolino, P., Two-parameter homogenization for a nonlinear periodic Robin problem for the Poisson equation. A functional analytic approach, Rev. Mat. Complut. 31 (2018), 63–110.Google Scholar
26Lanza de Cristoforis, M. and Rossi, L., Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density, J. Integral Equations Appl. 16 (2004), 137–174.Google Scholar
27Marčenko, V. A. and Khruslov, E. Ya., Boundary value problems in domains with a fine-grained boundary. Izdat. (Naukova Dumka, Kiev, 1974, in Russian).Google Scholar
28Maz'ya, V. and Movchan, A., Asymptotic treatment of perforated domains without homogenization, Math. Nachr. 283 (2010), 104–125.Google Scholar
29Maz'ya, V., Nazarov, S. and Plamenevskij, B., Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, Volumes I, II, Operator Theory: Advances and Applications, Volumes 111, 112 (Birkhäuser Verlag, Basel, 2000).Google Scholar
30Maz'ya, V., Movchan, A. and Nieves, M., Green's kernels and meso-scale approximations in perforated domains, Lecture Notes in Mathematics, Volume 2077 (Springer, Berlin, 2013).Google Scholar
31Miranda, C., Sulle proprietà di regolarità di certe trasformazioni integrali, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I 7 (1965), 303–336.Google Scholar
32Novotny, A. A. and Sokołowski, J., Topological derivatives in shape optimization, Interaction of Mechanics and Mathematics (Springer, Heidelberg, 2013).Google Scholar
33Preciso, L., Regularity of the composition and of the inversion operator and perturbation analysis of the conformal sewing problem in Romieu type spaces, Tr. Inst. Mat. Minsk 5 (2000), 99–104.Google Scholar