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Two-parameter homogenization for a nonlinear periodic Robin problem for the Poisson equation: a functional analytic approach

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 \(\delta \). The relative size of each periodic perforation is instead determined by a positive parameter \(\epsilon \). We prove the existence of a family of solutions which depends on \(\epsilon \) and \(\delta \) and we analyze the behavior of such a family as \((\epsilon ,\delta )\) tends to (0, 0) by an approach which is alternative to that of asymptotic expansions and of classical homogenization theory.

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References

  1. Ammari, H., Kang, H.: Polarization and Moment Tensors. Applied Mathematical Sciences, vol. 162. Springer, New York (2007)

    Google Scholar 

  2. Ammari, H., Kang, H., Lee, H.: Layer Potential Techniques in Spectral Analysis. American Mathematical Society, Providence (2009)

    Book  MATH  Google Scholar 

  3. Böhme, R., Tomi, F.: Zur Struktur der Lösungsmenge des Plateauproblems. Math. Z. 133, 1–29 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  4. Bonnaillie-Noël, V., Dambrine, M., Tordeux, S., Vial, G.: Interactions between moderately close inclusions for the Laplace equation. Math. Models Methods Appl. Sci. 19, 1853–1882 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  5. Bonnaillie-Noël, V., Dambrine, M., Lacave, C.: Interactions between moderately close inclusions for the 2D Dirichlet-Laplacian. Appl. Math. Res. Express AMRX 2016, 1–23 (2016)

    Article  MATH  Google Scholar 

  6. Braides, A., Defranceschi, A.: Homogenization of Multiple Integrals. Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, New York (1998)

    MATH  Google Scholar 

  7. Cabarrubias, B., Donato, P.: Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions. Appl. Anal. 91, 1111–1127 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  8. Castro, L.P., Pesetskaya, E., Rogosin, S.V.: Effective conductivity of a composite material with non-ideal contact conditions. Complex Var. Elliptic Equ. 54, 1085–1100 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  9. Cioranescu, D., Murat, F.: Un terme étrange venu d’ailleurs. II. In: Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. III (Paris, 1980/1981), Volume 70 of Research Notes in Mathematics, pp. 154–178, 425–426. Pitman, Boston (1982)

  10. Cioranescu, D., Murat, F.: Un terme étrange venu d’ailleurs. In: Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. II (Paris, 1979/1980), Volume 60 of Research Notes in Mathematics, pp. 98–138, 389–390. Pitman, Boston (1982)

  11. Dalla Riva, M., Lanza de Cristoforis, M., Musolino, P.: A local uniqueness result for a quasi-linear heat transmission problem in a periodic two-phase dilute composite. Recent Trends in Operator Theory and Partial Differential Equations-The Roland Duduchava Anniversary Volume. Operator Theory: Advances and Applications, vol. 258, pp. 193–227. Birkhäuser Verlag, Basel (2017)

  12. Dalla Riva, M.: Energy integral of the Stokes flow in a singularly perturbed exterior domain. Opusc. Math. 32, 647–659 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  13. Dalla Riva, M.: Stokes flow in a singularly perturbed exterior domain. Complex Var. Elliptic Equ. 58, 231–257 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  14. Dalla Riva, M., 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, 771–794 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  15. Dalla Riva, M., Lanza de Cristoforis, M.: A singularly perturbed nonlinear traction boundary value problem for linearized elastostatics: a functional analytic approach. Analysis (Munich) 30, 67–92 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Dalla Riva, M., Musolino, P., Rogosin, S.V.: Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole. Asymptot. Anal. 92, 339–361 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Dauge, M., Tordeux, S., Vial, G.: Selfsimilar perturbation near a corner: matching versus multiscale expansions for a model problem. In: Around the Research of Vladimir Maz’ya. II, 95–134, International Mathematical Series (N. Y.), 12. Springer, New York (2010)

  18. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  19. Drygas, P., Mityushev, V.: Effective conductivity of unidirectional cylinders with interfacial resistance. Q. J. Mech. Appl. Math. 62, 235–262 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  20. Folland, G.B.: Introduction to Partial Differential Equations, 2nd edn. Princeton University Press, Princeton (1995)

    MATH  Google Scholar 

  21. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  22. Henry, D.: Topics in Nonlinear Analysis, 192. Trabalho de Matemática, Brasilia (1982)

    Google Scholar 

  23. Kapanadze, D., Mishuris, G., Pesetskaya, E.: Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials. Complex Var. Elliptic Equ. 60, 1–23 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  24. Khruslov, E.J.: The method of orthogonal projections and the Dirichlet boundary value problem in domains with a fine-grained boundary (Russian). Mat. Sb. (N.S.) 88 (130), 38–60 (1972)

  25. Kozlov, V., Maz’ya, V., Movchan, A.: Asymptotic Analysis of Fields in Multi-structures. Oxford Mathematical Monographs. The Clarendon Press, New York (1999)

    MATH  Google Scholar 

  26. Lanza de Cristoforis, M.: A singular domain perturbation problem for the Poisson equation. More progresses in analysis. In: Proceedings of the 5th International ISAAC Congress, Catania, Italy, July 25–30, 2005, pp. 955–965. World Scientific, Hackensack(2009)

  27. Lanza de Cristoforis, M., Musolino, P.: A functional analytic approach to homogenization problems. In: Integral Methods in Science and Engineering: Theoretical and Computational Advances, Proceedings of the 13th International Conference on Integral Methods in Science and Engineering, IMSE 2014, Karlsruhe, Germany 21–25 July 2014, pp. 353–359. Birkhauser Verlag, Basel (2015)

  28. Lanza de Cristoforis, M.: Properties and pathologies of the composition and inversion operators in Schauder spaces. Acc. Naz. delle Sci. detta dei XL 15, 93–109 (1991)

  29. Lanza 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, 1–27 (2002)

  30. Lanza 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, 945–977 (2007)

  31. Lanza de Cristoforis, M.: Asymptotic behavior of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole: a functional analytic approach. Analysis (Munich) 28, 63–93 (2008)

  32. Lanza de Cristoforis, M., Musolino, P.: A perturbation result for periodic layer potentials of general second order differential operators with constant coefficients. Far East J. Math. Sci. (FJMS) 52, 75–120 (2011)

  33. Lanza de Cristoforis, M., Musolino, P.: A singularly perturbed nonlinear Robin problem in a periodically perforated domain: a functional analytic approach. Complex Var. Elliptic Equ. 58, 511–536 (2013)

  34. Lanza de Cristoforis, M., Musolino, P.: A real analyticity result for a nonlinear integral operator. J. Integral Equ. Appl. 25, 21–46 (2013)

  35. Lanza de Cristoforis, M., 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, 253–272 (2016)

  36. Lanza de Cristoforis, M., Rossi, L.: Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density. J. Integral Equ. Appl. 16, 137–174 (2004)

  37. Lanza de Cristoforis, M., Musolino, P.: Two-parameter anisotropic homogenization for a Dirichlet problem for the Poisson equation in an unbounded periodically perforated domain: a functional analytic approach (Submitted)

  38. Marčenko, V.A., Khruslov, E.Y.: Boundary value problems in domains with a fine-grained boundary. Izdat. “Naukova Dumka”. Kiev (1974) (in Russian)

  39. Maz’ya, V., Nazarov, S., Plamenevskij, B.: Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. In: Operator Theory: Advances and Applications, vols. I, II, pp. 111, 112. Birkhäuser Verlag, Basel (2000)

  40. Maz’ya, V., Movchan, A.: Asymptotic treatment of perforated domains without homogenization. Math. Nachr. 283, 104–125 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  41. Maz’ya, V., Movchan, A., Nieves, M.: Green’s Kernels and Meso-Scale Approximations in Perforated Domains. Lecture Notes in Mathematics, vol. 2077. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  42. Miranda, C.: Sulle proprietà di regolarità di certe trasformazioni integrali. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. Sez. I 7, 303–336 (1965)

    MATH  Google Scholar 

  43. Musolino, P.: A singularly perturbed Dirichlet problem for the Poisson equation in a periodically perforated domain: a functional analytic approach. In: Advances in Harmonic Analysis and Operator Theory. The Stefan Samko Anniversary Volume, Operator Theory: Advances and Applications, vol. 229, pp. 269–289. Birkhäuser Verlag, Basel (2013)

  44. Musolino, P.: A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain: a functional analytic approach. Math. Methods Appl. Sci. 35, 334–349 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  45. Novotny, A.A., Sokołowski, J.: Topological Derivatives in Shape Optimization. Interaction of Mechanics and Mathematics. Springer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  46. Ozawa, S.: Electrostatic capacity and eigenvalues of the Laplacian. J. Fac. Sci. Univ. Tokyo Sect. IA Math 30, 53–62 (1983)

    MathSciNet  MATH  Google Scholar 

  47. Preciso, P.: 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, 99–104 (2000)

    MATH  Google Scholar 

  48. Valent, T.: Boundary Value Problems of Finite Elasticity. Local Theorems on Existence, Uniqueness and Analytic Dependence on Data. Springer, New York (1988)

    Book  MATH  Google Scholar 

  49. Ward, M.J., Keller, J.B.: Strong localized perturbations of eigenvalue problems. SIAM J. Appl. Math. 53, 770–798 (1993)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Massimo Lanza de Cristoforis.

Additional information

The authors acknowledge the support of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). M. Lanza de Cristoforis acknowledges the support of the project BIRD168373/16 “Singular perturbation problems for the heat equation in a perforated domain” of the University of Padua and of the Grant EP/M013545/1: “Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs” from the EPSRC, UK. P. Musolino acknowledges the support of an ‘assegno di ricerca INdAM’. P. Musolino is a Sêr CYMRU II COFUND fellow, also supported by the ‘Sêr Cymru National Research Network for Low Carbon, Energy and Environment’.

A Appendix

A Appendix

We first introduce the following variant of a result of Preciso [47, Prop. 1.1, p. 101].

Proposition A.1

Let \(n_{1}\), \(n_{2}\in {\mathbb {N}}{\setminus }\{0\}\), \(\rho \in ]0,+\infty [\), \(m\in {\mathbb {N}}\), \(\alpha \in ]0,1]\). Let \({\varOmega }_{1}\) be a bounded open subset of \({\mathbb {R}}^{n_{1}}\). Let \({\varOmega }_{2}\) be a bounded open connected subset of \({\mathbb {R}}^{n_{2}}\) of class \(C^1\). Then the composition operator T from \(C^{0}_{\omega ,\rho }({\mathrm {cl}}{\varOmega }_{1})\times C^{m,\alpha }( {\mathrm {cl}}{\varOmega }_{2},{\varOmega }_{1}) \) to \(C^{m,\alpha }({\mathrm {cl}}{\varOmega }_{2})\) defined by

$$\begin{aligned} T[u, v]\equiv u\circ v \quad \forall (u,v)\in C^{0}_{\omega ,\rho }({\mathrm {cl}}{\varOmega }_{1})\times C^{m,\alpha }( {\mathrm {cl}}{\varOmega }_{2},{\varOmega }_{1}), \end{aligned}$$

is real analytic.

Then we introduce the following statement of [44, Lem. 3.8, Prop. 3.14, Rmk. 3.15].

Theorem A.2

Let \(m \in {\mathbb {N}}{\setminus }\{0\}\), \(\alpha \in ]0,1[\). Let \(p\in Q\). Let \({\varOmega }\) be as in (1.1). Let \(\epsilon _{0}\) be as in (1.2). Let \(\tilde{g}\in C^{m,\alpha }(\partial {\varOmega })\). Then there exist \(\epsilon _{1}\in ]0,\epsilon _{0}[\) and an open neighborhood \(\tilde{{\varGamma }}\) of \(\tilde{g}\) in \(C^{m,\alpha }(\partial {\varOmega })\) and a real analytic map \((\hat{\eta }[\cdot ,\cdot ], \hat{\xi }[\cdot ,\cdot ])\) from \(]-\epsilon _{1},\epsilon _{1}[\times \tilde{{\varGamma }}\) to \( C^{m,\alpha }(\partial {\varOmega })_{0}\times {\mathbb {R}}\) such that the only solution \(\varsigma [\epsilon ,g]\in C^{m,\alpha }_{q}({\mathrm {cl}}{\mathbb {S}}[{\varOmega }_{p,\epsilon }]^{-})\) of the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{ll} {\varDelta }u (x)=0 &{} \forall x\in {\mathbb {S}}(\epsilon ,1 )^{-}, \\ u\ {\mathrm {is}}\ q-{\mathrm {periodic\ in }}\ {\mathbb {S}}(\epsilon ,1 )^{-}, \\ u(p+t\epsilon )= g(t) &{} \forall t\in \partial {\varOmega }, \end{array} \right. \end{aligned}$$

is delivered by the formula

$$\begin{aligned} \varsigma [\epsilon ,g](x)=w^{-}_{q}[\partial {\varOmega }_{p,\epsilon }, \hat{\eta }[\epsilon ,g] (\epsilon ^{-1}(\cdot - p))](x)+\hat{\xi }[\epsilon ,g]\qquad \forall x\in {\mathrm {cl}} {\mathbb {S}}(\epsilon ,1 )^{-}, \end{aligned}$$

for all \((\epsilon ,g)\in ]0,\epsilon _{1}[\times \tilde{{\varGamma }}\). Moreover,

$$\begin{aligned} (\hat{\eta }[0,\tilde{g}], \hat{\xi }[0,\tilde{g}])=(\tilde{\eta },\tilde{\xi }), \end{aligned}$$

where \((\tilde{\eta },\tilde{\xi })\in C^{m,\alpha }(\partial {\varOmega })_{0}\times {\mathbb {R}}\) is the only solution of the equation

$$\begin{aligned} -\frac{1}{2}\tilde{\eta } + w[\partial {\varOmega },\tilde{\eta }]+\tilde{\xi }=\tilde{g}\qquad {\mathrm {on}}\ \partial {\varOmega }\,. \end{aligned}$$

Also,

$$\begin{aligned} \tilde{\xi }=\int _{\partial {\varOmega }}\tilde{g}\tilde{\tau }\,d\sigma , \end{aligned}$$

where \(\tilde{\tau }\in C^{m-1,\alpha }(\partial {\varOmega })\) is the only solution of the problem

$$\begin{aligned} -\frac{1}{2}\tau +w_{*}[\partial {\varOmega },\tau ]=0\qquad {\mathrm {on}}\ \partial {\varOmega },\qquad \int _{\partial {\varOmega }}\tau \,d\sigma =1\,. \end{aligned}$$
(A.1)

In order to compute \(\tilde{\xi }\), the following lemma is sometimes useful.

Lemma A.3

Let the same assumptions of Theorem A.2 hold. Then

$$\begin{aligned} \lim _{]0,\epsilon _{1}[\times \tilde{{\varGamma }}\ni (\epsilon ,g)\rightarrow (0,\tilde{g})}\varsigma [\epsilon ,g](x)=\tilde{\xi }\qquad \forall x\in {\mathbb {R}}^{n}{\setminus } (p+q{\mathbb {Z}}^{n})\,. \end{aligned}$$

Proof

Since

$$\begin{aligned} \varsigma [\epsilon ,g](x)= & {} -\epsilon ^{n-1}\int _{\partial {\varOmega }}\nu _{{\varOmega }}(s)DS_{q,n}(x-p-\epsilon s)\hat{\eta }[\epsilon ,g](s)\,d\sigma _{s}+\hat{\xi }[\epsilon ,g]\\&\forall x\in {\mathrm {cl}}{\mathbb {S}}[{\varOmega }_{p,\epsilon }]^{-}, \end{aligned}$$

for all \( (\epsilon ,\gamma ) \in ]0,\epsilon _{1}[\times \tilde{{\varGamma }} \), the statement follows by the continuity of \(\hat{\eta }\) and \(\hat{\xi }\) at \((0,\tilde{g})\), and by the continuity of \(DS_{q,n}\) in \({\mathbb {R}}^{n}{\setminus } (p+q{\mathbb {Z}}^{n})\). \(\square \)

Then we deduce the validity of the following corollary.

Corollary A.4

Let the same assumptions of Theorem A.2 hold. Then there exist \(\epsilon _{1}\in ]0,\epsilon _{0}[\), and an open neighborhood \(\tilde{{\varGamma }}\) of \(\tilde{g}\) in \(C^{m,\alpha }(\partial {\varOmega })\), and an analytic map \(J_{1}\) from \(]-\epsilon _{1},\epsilon _{1}[\times \tilde{{\varGamma }}\) to \({\mathbb {R}}\) such that

$$\begin{aligned} \int _{Q{\setminus }{\varOmega }_{p,\epsilon }}\varsigma [\epsilon ,g]\,dx=J_{1}[\epsilon ,g]\qquad \forall (\epsilon ,g)\in ]0,\epsilon _{1}[\times \tilde{{\varGamma }}\,. \end{aligned}$$

Moreover, \(J_{1}[0,\tilde{g}]=m_{n}(Q)\int _{\partial {\varOmega }}\tilde{g}\tilde{\tau }\,d\sigma \), where \(\tilde{\tau }\) is the only solution in \(C^{m-1,\alpha }(\partial {\varOmega })\) of problem (A.1).

Proof

We first observe that

$$\begin{aligned} \int _{Q{\setminus }{\mathrm {cl}}{\varOmega }_{p,\epsilon }}\varsigma [\epsilon ,g]\,d\sigma= & {} \int _{Q{\setminus }{\mathrm {cl}}{\varOmega }_{p,\epsilon }} w_{q}^{-}[\partial {\varOmega }_{p,\epsilon }, \hat{\eta }[\epsilon ,g](\epsilon ^{-1}(\cdot -p))](x)\,dx\nonumber \\&+\,\hat{\xi }[\epsilon ,g]m_{n}(Q{\setminus } {\varOmega }_{p,\epsilon }) \end{aligned}$$
(A.2)

for all \((\epsilon ,g)\in ]0,\epsilon _{1}[\times \tilde{{\varGamma }}\). Next we note that

$$\begin{aligned}&\int _{Q{\setminus }{\mathrm {cl}}{\varOmega }_{p,\epsilon }}w_{q}^{-}[\partial {\varOmega }_{p,\epsilon },\hat{\eta }[\epsilon ,g](\epsilon ^{-1}(\cdot -p))](x)\,dx\nonumber \\&\quad =-\int _{Q{\setminus }{\mathrm {cl}}{\varOmega }_{p,\epsilon }} \int _{\partial {\varOmega }_{p,\epsilon }}\nu _{ {\varOmega }_{p,\epsilon } }(y)DS_{q,n}(x-y) \hat{\eta }[\epsilon ,g](\epsilon ^{-1}(y-p))\,d\sigma _{y}\,dx \nonumber \\&\quad =-\int _{Q{\setminus }{\mathrm {cl}}{\varOmega }_{p,\epsilon }} \sum _{j=1}^{n}\frac{\partial }{\partial x_{j}}\int _{\partial {\varOmega }_{p,\epsilon }} S_{q,n}(x-y)\hat{\eta }[\epsilon ,g](\epsilon ^{-1}(y-p))(\nu _{{\varOmega }_{p,\epsilon }}(y))_{j}\,d\sigma _{y}\,dx\nonumber \\&\quad =\int _{\partial {\varOmega }_{p,\epsilon }}\sum _{j=1}^{n} (\nu _{{\varOmega }_{p,\epsilon }}(x))_{j} \int _{\partial {\varOmega }_{p,\epsilon }} S_{q,n}(x-y)\hat{\eta }[\epsilon ,g](\epsilon ^{-1}(y-p))(\nu _{{\varOmega }_{p,\epsilon }}(y))_{j}\,d\sigma _{y}\,d\sigma _{x} \nonumber \\&\quad =\sum _{j=1}^{n} \int _{\partial {\varOmega }}(\nu _{{\varOmega }}(t))_{j}\int _{\partial {\varOmega }}S_{q,n}(\epsilon (t-s))\hat{\eta }[\epsilon ,g](s)(\nu _{{\varOmega }}(s))_{j} \,d\sigma _{s}d\sigma _{t}\epsilon ^{2n-2} \nonumber \\&\quad =\sum _{j=1}^{n} \int _{\partial {\varOmega }}(\nu _{{\varOmega }}(t))_{j}\int _{\partial {\varOmega }}S_{ n} (t-s)\hat{\eta }[\epsilon ,g](s)(\nu _{{\varOmega }}(s))_{j} \,d\sigma _{s}d\sigma _{t}\epsilon ^{n} \nonumber \\&\qquad +\,\frac{\delta _{2,n}}{2\pi }\epsilon (\epsilon \log \epsilon ) \sum _{j=1}^{n} \int _{\partial {\varOmega }}(\nu _{{\varOmega }}(t))_{j}\,d\sigma _{t} \int _{\partial {\varOmega }}\hat{\eta }[\epsilon ,g](s)(\nu _{{\varOmega }}(s))_{j} \,d\sigma _{s} \nonumber \\&\qquad +\,\sum _{j=1}^{n} \int _{\partial {\varOmega }}(\nu _{{\varOmega }}(t))_{j}\int _{\partial {\varOmega }}R_{q, n}(\epsilon (t-s))\hat{\eta }[\epsilon ,g](s)(\nu _{{\varOmega }}(s))_{j} \,d\sigma _{s}d\sigma _{t}\epsilon ^{2n-2} \nonumber \\&\quad =\sum _{j=1}^{n} \int _{\partial {\varOmega }}(\nu _{{\varOmega }}(t))_{j}\int _{\partial {\varOmega }}S_{ n} (t-s)\hat{\eta }[\epsilon ,g](s)(\nu _{{\varOmega }}(s))_{j} \,d\sigma _{s}d\sigma _{t}\epsilon ^{n} \nonumber \\&\qquad +\,\sum _{j=1}^{n} \int _{\partial {\varOmega }}(\nu _{{\varOmega }}(t))_{j}\int _{\partial {\varOmega }}R_{q, n}(\epsilon (t-s))\hat{\eta }[\epsilon ,g](s)(\nu _{{\varOmega }}(s))_{j} \,d\sigma _{s}d\sigma _{t}\epsilon ^{2n-2}, \end{aligned}$$
(A.3)

for all \((\epsilon ,g)\in ]0,\epsilon _{1}[\times \tilde{{\varGamma }}\). Thus it is natural to define \(J_{1}\) as the map from \(]-\epsilon _{1},\epsilon _{1}[\times \tilde{{\varGamma }}\) to \({\mathbb {R}}\) which takes \((\epsilon ,g)\) to the sum of the right hand side of (A.3) and of the term \(\hat{\xi }[\epsilon ,g]m_{n}(Q{\setminus } {\varOmega }_{p,\epsilon }) =\hat{\xi }[\epsilon ,g](m_{n}(Q)-\epsilon ^{n}m_{n}({\varOmega }))\) in the right hand side of equality (A.2). By classical potential theory, the operator \(v[\partial {\varOmega },\cdot ]_{|\partial {\varOmega }}\) is linear and continuous from \(C^{m-1,\alpha }(\partial {\varOmega })\) to \(C^{m,\alpha }(\partial {\varOmega })\). Then the continuity of the pointwise product in \(C^{m-1,\alpha }(\partial {\varOmega })\) and the analyticity of \(\hat{\eta }[\cdot ,\cdot ]\) imply the analyticity of the first sum in the right hand side of (A.3). Then the analyticity of the map in (6.9), and the continuity of the product in \(C^{m-1,\alpha }(\partial {\varOmega })\) and the analyticity of \(\hat{\eta }[\cdot ,\cdot ]\) imply the analyticity of the second sum in the right hand side of (A.3) in the variable \((\epsilon ,g)\). The analyticity of \(\hat{\xi }[\cdot ,\cdot ]\) implies the analyticity of the term \(\hat{\xi }[\epsilon ,g](m_{n}(Q)-\epsilon ^{n}m_{n}({\varOmega }))\) upon the variable \((\epsilon ,g)\). Hence, \(J_{1}[\cdot ,\cdot ]\) is real analytic from \(]-\epsilon _{1},\epsilon _{1}[\times \tilde{{\varGamma }}\) to \({\mathbb {R}}\). Finally,

$$\begin{aligned} J_{1}[0,\tilde{g}]=m_{n}(Q)\hat{\xi }[0,\tilde{g}]=m_{n}(Q)\tilde{\xi } =m_{n}(Q)\int _{\partial {\varOmega }}\tilde{\tau }\tilde{g}\,d\sigma , \end{aligned}$$

where \(\tilde{\tau }\) is the unique solution of problem (A.1). \(\square \)

Next we introduce the following technical statement.

Proposition A.5

Let \(m \in {\mathbb {N}}{\setminus }\{0\}\), \(\alpha \in ]0,1[\). Let \(p\in Q\). Let \({\varOmega }\) be as in (1.1). Let \(\epsilon _{0}\) be as in (1.2).

  1. (i)

    Let \(\rho \in ]0,+\infty [\). Then there exists a real analytic map G from \(]-\epsilon _{0},\epsilon _{0}[\times C^{0}_{\omega ,\rho }({\mathrm {cl}}Q)\) to \({\mathbb {R}}\) such that

    $$\begin{aligned} \int _{Q{\setminus }{\varOmega }_{p,\epsilon } }h\,dx=G[\epsilon ,h] \qquad \forall (\epsilon ,h)\in ]0,\epsilon _{0}[\times C^{0}_{\omega ,\rho }({\mathrm {cl}}Q),\\ G[0,h]=\int _{Q }h\,dx \qquad \forall h\in C^{0}_{\omega ,\rho }({\mathrm {cl}}Q) \,. \end{aligned}$$
  2. (ii)

    There exists a real analytic function \(G_{1}\) from \(]-\epsilon _{0},\epsilon _{0}[\) to \({\mathbb {R}}\) such that

    $$\begin{aligned} \int _{Q{\setminus }{\varOmega }_{p,\epsilon }}S_{q,n}(x-p)\,dx= G_{1}(\epsilon )-\delta _{2,n}\frac{\epsilon ^{2}\log \epsilon }{2\pi }m_{n}({\varOmega })\qquad \forall \epsilon \in ]0,\epsilon _{0}[\,. \end{aligned}$$

    Moreover,

    $$\begin{aligned} G_{1}(0)=\int _{Q}S_{q,n}(x-p)\,dx\,. \end{aligned}$$

Proof

For the existence of G, we follow the proof of Lemma 2.2 of [26] and we note that \(\int _{Q{\setminus }{\varOmega }_{p,\epsilon }}h\,dx=\int _{Q}h\,dx-\epsilon ^{n}\int _{{\varOmega }}h(p+\epsilon s)\,ds\) for all \((\epsilon ,h)\in ]0,\epsilon _{1}[\times C^{0}_{\omega ,\rho }({\mathrm {cl}}Q)\), and we define G as the map from \(]-\epsilon _{0},\epsilon _{0}[\times C^{0}_{\omega ,\rho }({\mathrm {cl}}Q)\) to \({\mathbb {R}}\) which takes \((\epsilon ,h)\) to the right hand side of such an equality. The analyticity of G follows by Proposition A.1. The formula for G[0, h] follows by the definition of G. Next we turn to prove statement (ii). By identity (2.3) and by the rule of change of variables, we have

$$\begin{aligned}&\int _{Q{\setminus }{\varOmega }_{p,\epsilon }}S_{q,n}(x-p)\,dx= \int _{Q}S_{q,n}(x-p)\,dx\\&\qquad -\,\epsilon ^{2}\int _{{\varOmega }}S_{n}(t)\,dt -\delta _{2,n}\frac{\epsilon ^{2}\log \epsilon }{2\pi }m_{n}({\varOmega }) -\epsilon ^{n}\int _{{\varOmega }}R_{q,n}(\epsilon t)\,dt \qquad \forall \epsilon \in ]0,\epsilon _{0}[\,. \end{aligned}$$

Then we can set

$$\begin{aligned} G_{1}(\epsilon )\equiv \int _{Q}S_{q,n}(x-p)\,dx-\epsilon ^{2}\int _{{\varOmega }}S_{n}(t)\,dt -\epsilon ^{n}\int _{{\varOmega }}R_{q,n}(\epsilon t)\,dt \qquad \forall \epsilon \in ]-\epsilon _{0},\epsilon _{0}[\,. \end{aligned}$$

By the analyticity of \(R_{q,n}\) in \(({\mathbb {R}}^{n}{\setminus } q{\mathbb {Z}}^{n})\cup \{0\}\) and by analyticity results on the composition operator (cf. Böhme and Tomi [3, p. 10], Henry [22, p. 29], Valent [48, Thm. 5.2, p. 44]), we deduce that the map from \(]-\epsilon _{0},\epsilon _{0}[\) to \(C^{m,\alpha }({\mathrm {cl}}{\varOmega })\), which takes \(\epsilon \) to the function \(R_{q,n}(\epsilon t)\) of the variable \(t\in {\mathrm {cl}}{\varOmega }\) is real analytic. Then by the continuity of the linear operator from \(C^{m,\alpha }({\mathrm {cl}}{\varOmega })\) to \({\mathbb {R}}\) which takes a map to its integral, the function \(G_{1}\) is analytic from \(]-\epsilon _{0},\epsilon _{0}[\) to \({\mathbb {R}}\). Then we obviously have \(G_{1}(0)=\int _{Q}S_{q,n}(x-p)\,dx\).

\(\square \)

Next we introduce the following inequality for dilated q-periodic functions, which we prove by arguments akin to those of Braides and De Franceschi [6, ex. 27, p. 20]. We denote by \(u_{\delta }\) the function from \({\mathbb {R}}^{n}\) to \({\mathbb {C}}\) defined by

$$\begin{aligned} u_{\delta }(x)\equiv u(x/\delta )\qquad \forall x \in {\mathbb {R}}^{n}, \end{aligned}$$
(A.4)

for all \(\delta \in ]0,+\infty [\) and for all q-periodic functions \(u\in L^{1}_{{\mathrm {loc}}}({\mathbb {R}}^{n})\). Then we have the following.

Lemma A.6

Let \(r\in [1,+\infty [\), \(\delta _{0}\in ]0,+\infty [\). Let V be a bounded open subset of \({\mathbb {R}}^{n}\). Then there exists \(C\in ]0,+\infty [\) such that

$$\begin{aligned} \Vert u_{\delta }\Vert _{L^{r}(V)}\le C \Vert u \Vert _{L^{r}(Q)}\qquad \forall \delta \in ]0,\delta _{0}[, \end{aligned}$$

for all q-periodic \(u\in L^{1}_{{\mathrm {loc}}}({\mathbb {R}}^{n})\).

Proof

Since V is bounded, there exists a family \(\{z_{l}\}_{l=1}^{s}\) of points of \({\mathbb {Z}}^{n}\) such that

$$\begin{aligned} V\subseteq \bigcup _{l=1}^{s} (qz_{l}+{\mathrm {cl}}Q) \,. \end{aligned}$$

Then the q-periodicity of u implies that

$$\begin{aligned} \int _{V}|u_{\delta }(y)|^{r}\,dy\le & {} \sum _{l=1}^{s}\int _{ qz_{l}+{\mathrm {cl}}Q }|u_{\delta }(y)|^{r}\,dy =\sum _{l=1}^{s}\int _{\delta ^{-1}qz_{l}+\delta ^{-1}{\mathrm {cl}}Q }|u (x)|^{r}\,dx\delta ^{n} \\\le & {} \sum _{l=1}^{s}\int _{\delta ^{-1}qz_{l}+([\delta ^{-1}]+1){\mathrm {cl}}Q }|u (x)|^{r}\,dx\delta ^{n} =s\int _{ ([\delta ^{-1}]+1){\mathrm {cl}}Q }|u (x)|^{r}\,dx\delta ^{n} \\\le & {} C^{r}\int _{ Q }|u(x)|^{r}\,dx \qquad \forall \delta \in ]0,\delta _{0}[, \end{aligned}$$

for all q-periodic \(u\in L^{1}_{{\mathrm {loc}}}({\mathbb {R}}^{n})\), where

$$\begin{aligned} C\equiv s^{1/r}\left\{ \sup _{ \delta \in ]0,\delta _{0}[ }([\delta ^{-1}]+1)^{n}\delta ^{n} \right\} ^{1/r}<+\infty , \end{aligned}$$

and where \([\delta ^{-1}]\) denotes the integer part of \(\delta ^{-1}\). \(\square \)

Next we introduce the following lemma for dilated q-periodic functions.

Lemma A.7

Let \(u\in L^{1}_{{\mathrm {loc}}}({\mathbb {R}}^{n})\) be a q-periodic function. Let \(\tilde{y}\in {\mathbb {R}}^{n}\), \(s\in ]0,+\infty [\), \(l\in {\mathbb {N}}{\setminus }\{0\}\). Then the following equality holds

$$\begin{aligned} \int _{{\mathbb {R}}^{n}}u_{s/l}(x)\chi _{\tilde{y}+sQ}(x)\,dx=s^{n}\int _{Q}u\,dx, \end{aligned}$$

[see (A.4)].

Proof

Since \(u_{s/l}\) is \(l^{-1}sq\)-periodic, it is also sq-periodic and accordingly,

$$\begin{aligned} \int _{{\mathbb {R}}^{n}}u_{s/l}(x)\chi _{\tilde{y}+sQ}(x)\,dx =\int _{\tilde{y}+sQ}u_{s/l}(x)\,dx=\int _{sQ}u_{s/l}(x)\,dx\,. \end{aligned}$$

Next we observe that

$$\begin{aligned} \bigcup _{ 0\le z_{j}\le l -1 }(qz+l^{-1} Q) \subseteq Q,\qquad m_{n}\left( Q{\setminus }\bigcup _{0\le z_{j}\le l-1 }(qz+l^{-1} Q) \right) =0\,. \end{aligned}$$

Accordingly, the \(l^{-1}s q\)-periodicity of \(u_{s/l}(\cdot )\) implies that

$$\begin{aligned} \int _{sQ}u_{s/l}(x)\,dx= & {} \int _{sl^{-1}Q}u_{s/l}(x)\,dxl^{n}\\= & {} \int _{sl^{-1}Q}u( x/(s/l))\,dxl^{n}=\int _{Q}u(y)\,dyl^{n}(s/l)^{n}= s^{n}\int _{Q}u\,dx\,. \end{aligned}$$

\(\square \)

Finally, we introduce the following elementary lemma of [33, Lem. A.5].

Lemma A.8

Let \(m\in {\mathbb {N}}{\setminus }\{0\}\), \(\alpha \in ]0,1[\). Let \(p\in Q\). Let \({\varOmega }\) be as in (1.1). Let \(\epsilon _{0}\in ]0,+\infty [\) be as in (1.2). Let \(\epsilon _{1}\in ]0,\epsilon _{0}[\).

  1. (i)

    Let \(\tilde{{\varOmega }}\) be an open subset of \({\mathbb {R}}^{n}\) with a nonzero distance from \( p+q{\mathbb {Z}}^{n} \). Then there exist \(\epsilon _{ \tilde{{\varOmega }} }^{*}\in ]0, \epsilon _{1}[\) such that

    $$\begin{aligned} {\mathrm {cl}} \tilde{{\varOmega }} \subseteq {\mathbb {S}}[{\varOmega }_{p,\epsilon }]^{-} \qquad \forall \epsilon \in [-\epsilon _{ \tilde{{\varOmega }} }^{*},\epsilon _{ \tilde{{\varOmega }} }^{*}] , \end{aligned}$$

    and \(\epsilon _{ \tilde{{\varOmega }} }\in ]0,\epsilon _{ \tilde{{\varOmega }} }^{*}[\) such that

    $$\begin{aligned} {\mathrm {cl}}{\mathbb {S}}[{\varOmega }_{p,\epsilon _{ \tilde{{\varOmega }} }^{*}}]^{-} \subseteq {\mathbb {S}}[{\varOmega }_{p,\epsilon }]^{-} \qquad \forall \epsilon \in [-\epsilon _{ \tilde{{\varOmega }} } ,\epsilon _{ \tilde{{\varOmega }} } ] \,. \end{aligned}$$
  2. (ii)

    Let \( {\varOmega }^{\sharp }\) be a bounded open subset of \({\mathbb {R}}^{n}\) such that \({\varOmega }^{\sharp }\subseteq {\mathbb {R}}^{n}{\setminus } {\mathrm {cl}}{\varOmega }\). Then there exists \(\epsilon _{ {\varOmega }^{\sharp },r }\in ]0,\epsilon _{1}[\) such that

    $$\begin{aligned} p+\epsilon {\mathrm {cl}}{\varOmega }^{\sharp } \subseteq Q, \qquad p+\epsilon {\varOmega }^{\sharp }\subseteq {\mathbb {S}}[{\varOmega }_{p,\epsilon }]^{-}\qquad \forall \epsilon \in [-\epsilon _{ {\varOmega }^{\sharp },r } ,\epsilon _{ {\varOmega }^{\sharp } ,r} ]{\setminus }\{0\}\,. \end{aligned}$$

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Lanza de Cristoforis, M., Musolino, P. Two-parameter homogenization for a nonlinear periodic Robin problem for the Poisson equation: a functional analytic approach. Rev Mat Complut 31, 63–110 (2018). https://doi.org/10.1007/s13163-017-0242-5

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Keywords

  • Nonlinear Robin problem
  • Singularly perturbed domain
  • Poisson equation
  • Periodically perforated domain
  • Homogenization
  • Real analytic continuation in Banach space

Mathematics Subject Classification

  • 35J25
  • 31B10
  • 45A05
  • 47H30