×

Asymptotic behaviour of the energy integral of a two-parameter homogenization problem with nonlinear periodic Robin boundary conditions. (English) Zbl 1439.35154

The focus of this paper is on a periodic Poisson problem for the Laplace operator in \(\mathbb R^n \) where the periodic cell is actually perforated, and on the boundary of the hole they impose a (periodic) non-linear Robin condition, of the type \[ \begin{cases} \Delta u =f, & \text{in the cell,}\\ \frac{\partial u}{\partial \nu}+G(x,u)=0, & \text{on the boundary of the hole,}\\ u \text{ is periodic.} & \end{cases} \] Here also \(f\) is periodic, and \(G\) is suitably rescaled.
The authors introduced this problem in [Rev. Mat. Complut. 31, No. 1, 63–110 (2018; Zbl 1393.35031)], where they studied the behavior of the function in dependence of two parameters: the dimension of the hole \(\epsilon\) and the dimension of the periodic cell \(\delta\). In particular, they considered the singular limit as \((\epsilon,\delta)\to(0,0)\), showing that there are several factors that have an impact on the result, such as the analytic properties of \(f\) in the cell.
In this paper they continue the analysis started in [loc. cit.] by looking at the singular limit \((\epsilon,\delta)\to(0,0)\) of the energy functional (in the periodicity cell) of the problem above. The central result of the paper (Theorem 5.1) is that the energy functional can be written as a combination of analytic functions (singular and non-singular in \((0,0)\)), and the form of this combinantion depends on the properties of \(f\). This is achieved by exploiting in a suitable way the analogous result for the behavior at \((0,0)\) of the solution \(u\).

MSC:

35J25 Boundary value problems for second-order elliptic equations
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
45A05 Linear integral equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)

Citations:

Zbl 1393.35031
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 1H.Ammari and H.Kang, Polarization and moment tensors, Applied Mathematical Sciences, Volume 162 (Springer, New York, 2007). · Zbl 1220.35001
[2] 2H.Ammari, H.Kang and H.Lee, Layer potential techniques in spectral analysis (American Mathematical Society, Providence, RI, 2009). · Zbl 1167.47001
[3] 3V.Bonnaillie-Noël, M.Dambrine, S.Tordeux and G.Vial, Interactions between moderately close inclusions for the Laplace equation, Math. Models Methods Appl. Sci.19 (2009), 1853-1882. · Zbl 1191.35112
[4] 4B.Cabarrubias and P.Donato, Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions, Appl. Anal.91 (2012), 1111-1127. · Zbl 1254.35017
[5] 5L. P.Castro, E.Pesetskaya and S. V.Rogosin, Effective conductivity of a composite material with non-ideal contact conditions, Complex Var. Elliptic Equ.54 (2009), 1085-1100. · Zbl 1184.30029
[6] 6D.Cioranescu and F.Murat, 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.
[7] 7D.Cioranescu and F.Murat, 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.
[8] 8M.Dalla Riva, Stokes flow in a singularly perturbed exterior domain, Complex Var. Elliptic Equ.58 (2013), 231-257. · Zbl 1263.76024
[9] 9M.Dalla Riva and M.Lanza de Cristoforis, Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem: a functional analytic approach, Complex Var. Elliptic Equ.55 (2010), 771-794. · Zbl 1200.35146
[10] 10M.Dalla Riva, P.Musolino and S. V.Rogosin, Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole, Asymptot. Anal.92 (2015), 339-361. · Zbl 1327.35079
[11] 11M.Dauge, S.Tordeux and G.Vial, 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. · Zbl 1190.35069
[12] 12K.Deimling, Nonlinear functional analysis (Springer-Verlag, Berlin, 1985). · Zbl 0559.47040
[13] 13G. B.Folland, Introduction to partial differential equations, 2nd edn (Princeton University Press, Princeton, NJ, 1995). · Zbl 0841.35001
[14] 14D.Gilbarg and N. S.Trudinger, Elliptic partial differential equations of second order (Springer Verlag, Berlin, 1983). · Zbl 0562.35001
[15] 15F.John, Partial differential equations, 4th edn, Applied Mathematical Sciences,Volume 1 (Springer-Verlag, New York, 1982). · Zbl 0472.35001
[16] 16D.Kapanadze, G.Mishuris and E.Pesetskaya, Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials, Complex Var. Elliptic Equ.60 (2015), 1-23. · Zbl 1316.30041
[17] 17V.Kozlov, V.Maz’ya and A.Movchan, Asymptotic analysis of fields in multi-structures, Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, New York, 1999). · Zbl 0951.35004
[18] 18M.Lanza de Cristoforis, Properties and pathologies of the composition and inversion operators in Schauder spaces, Acc. Naz. delle Sci. detta dei XL15 (1991), 93-109. · Zbl 0829.47059
[19] 19M.Lanza de Cristoforis, Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces, Comput. Methods Funct. Theory2 (2002), 1-27. · Zbl 1026.30009
[20] 20M.Lanza de Cristoforis, 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. · Zbl 1143.35057
[21] 21M.Lanza de Cristoforis and P.Musolino, 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. · Zbl 1239.31003
[22] 22M.Lanza de Cristoforis and P.Musolino, A singularly perturbed nonlinear Robin problem in a periodically perforated domain: a functional analytic approach, Complex Var. Elliptic Equ.58 (2013), 511-536. · Zbl 1270.31001
[23] 23M.Lanza de Cristoforis and P.Musolino, 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.
[24] 24M.Lanza de Cristoforis and P.Musolino, 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. · Zbl 1404.35119
[25] 25M.Lanza de Cristoforis and P.Musolino, Two-parameter homogenization for a nonlinear periodic Robin problem for the Poisson equation. A functional analytic approach, Rev. Mat. Complut.31 (2018), 63-110. · Zbl 1393.35031
[26] 26M.Lanza de Cristoforis and L.Rossi, 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. · Zbl 1094.31001
[27] 27V. A.Marčenko and E. Ya.Khruslov, Boundary value problems in domains with a fine-grained boundary. Izdat. (Naukova Dumka, Kiev, 1974, in Russian). · Zbl 0289.35002
[28] 28V.Maz’ya and A.Movchan, Asymptotic treatment of perforated domains without homogenization, Math. Nachr.283 (2010), 104-125. · Zbl 1185.35060
[29] 29V.Maz’ya, S.Nazarov and B.Plamenevskij, 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). · Zbl 1127.35300
[30] 30V.Maz’ya, A.Movchan and M.Nieves, Green’s kernels and meso-scale approximations in perforated domains, Lecture Notes in Mathematics, Volume 2077 (Springer, Berlin, 2013). · Zbl 1273.35007
[31] 31C.Miranda, Sulle proprietà di regolarità di certe trasformazioni integrali, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I7 (1965), 303-336. · Zbl 0183.12701
[32] 32A. A.Novotny and J.Sokołowski, Topological derivatives in shape optimization, Interaction of Mechanics and Mathematics (Springer, Heidelberg, 2013). · Zbl 1276.35002
[33] 33L.Preciso, Regularity of the composition and of the inversion operator and perturbation analysis of the conformal sewing problem in Romieu type spaces, Tr. Inst. Mat. Minsk5 (2000), 99-104. · Zbl 0955.47039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.