On the Steklov problem in a domain perforated along a part of the boundary
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1317-1342.

We study the asymptotic behavior of solutions and eigenelements to a 2-dimensional and 3-dimensional boundary value problem for the Laplace equation in a domain perforated along part of the boundary. On the boundary of holes we set the homogeneous Dirichlet boundary condition and the Steklov spectral condition on the mentioned part of the outer boundary of the domain. Assuming that the boundary microstructure is periodic, we construct the limit problem and prove the homogenization theorem.

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DOI : 10.1051/m2an/2016063
Classification : 35B40, 35D05, 35G30, 35Q35
Mots clés : Homogenization, the Steklov spectral problem, asymptotic methods
Chechkin, Gregory A. 1 ; Gadyl’shin, Rustem R. 2 ; D’Apice, Ciro 3 ; De Maio, Umberto 4

1 Department of Differential Equations, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow 119991, Russia.
2 Department of Mathematics and Statistics, Faculty of Physics and Mathematics, Bashkir State Pedagogical University, Ufa 450000, Russia.
3 Dipartimento di Ingegneria dell’Informazione e Matematica Applicata, Università degli Studi di Salerno, via Ponte don Melillo, 1, 84084 Fisciano (SA), Italia.
4 Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II, Complesso Monte S.Angelo – Edificio “T”, via Cintia 80126 Napoli, Italia.
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Chechkin, Gregory A.; Gadyl’shin, Rustem R.; D’Apice, Ciro; De Maio, Umberto. On the Steklov problem in a domain perforated along a part of the boundary. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1317-1342. doi : 10.1051/m2an/2016063. http://www.numdam.org/articles/10.1051/m2an/2016063/

V.A. Steklov, General Methods of Solutions to Basic Problems of Mathematical Physics. Ph. D. thesis, Empirior Kharkov University, Kharkov (1901).

W. Stekloff, Sur les problèmes fondamentaux de la physique mathématique. Ann. Sci. École Norm. Sup. 19 (1902) 455–490. | DOI | JFM | Numdam | MR

R.V. Isakov, Asymptotics of a spectral series of the Steklov problem for the Laplace equation in a “thin” domain with a nonsmooth boundary. Mat. Zametki 44 (1988) 694–696. (English translation: Math. Notes 44 (1988) 833–842.) | MR | Zbl

N.G. Kuznetsov and O.V. Motygin, The Steklov problem in a half-plane: dependence of eigenvalues on a piecewise-constant coefficient. J. Math. Sci. 127 (2005) 2429–2445. | DOI | MR | Zbl

O.V. Motygin and N.G. Kuznetsov, Eigenvalues of the Steklov problem in an infinite cylinder, Mathematical and numerical aspects of wave propagation – WAVES. Springer Verlag, Berlin-New York (2003) 463–468. | MR | Zbl

J.F. Bonder, P. Groisman and J.D. Rossi, Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach. Ann. Math. Pura Appl. 186 (2007) 341–358. | DOI | MR | Zbl

M.E. Pérez, On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem. Discrete Contin. Dyn. Syst. Ser. B 7 (2007) 859–883. | MR | Zbl

M.G. Armentano and C. Padra, A posteriori error estimates for Steklov eigenvalue problem. Appl. Numer. Math. 58 (2008) 593–601. | DOI | MR | Zbl

Deng Shao Gao, Eigenvalues of the p(x)-Laplacian Steklov problem. J. Math. Anal. Appl. 339 (2008) 925–937. | DOI | MR | Zbl

M. Gazzini and E. Serra, The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues. Ann. Inst. Henri Poincaré Anal. Non Linéaire 25 (2008) 281–302. | DOI | Numdam | MR | Zbl

F. Gazzola and G. Sweers, On positivity for the biharmonic operator under Sleklov boundary conditions. Arch. Ration. Mech. Anal. 188 (2008) 399–427. | DOI | MR | Zbl

S.A. Nazarov and J. Taskinen, On the spectrum of the Steklov problem in a domain with a peak. Vestnik St. Petersburg Univ. Math. 41 (2008) 45–52. | DOI | MR | Zbl

E. Berchio, F. Gazzola and D. Pierotti, Nodal solutions to critical growth elliptic problems under Steklov boundary conditions, Commun. Pure Appl. Anal. 8 (2009) 533–557. | DOI | MR | Zbl

A.G. Chechkina, Convergence of Solutions and Eigenelements of Steklov Type Boundary Value Problems with Boundary Conditions of Rapidly Varying Type. Problem. Mat. Anal. 42 (2009) 129–143. (English translation: J. Math. Sci. 162 (2009) 443–458) | MR

A.G. Chechkina, On Singular Perturbation of the Steklov Type Problem with Degenerating Spectrum. Doklady RAN 440 (2011) 603–606. (English translation: Doklady Math. 84 (2011) 695–698.) | MR | Zbl

V.A. Sadovnichii, A.G. Chechkina, On Estimate of Eigenfunctions to the Steklov-type Problem with Small Parameter in Case of Limit Spectrum Degeneration. Ufa Math. Journal 3 (2011) 127–139. (English translation: Ufa Math. Journal 3 (2011) 122–134.) | MR | Zbl

A.G. Belyaev, On Singularly Perturbed Boundary Value Problems. Ph.D. thesis, Cand. Phys.-Math. Sci., Moscow (1990).

A.G. Belyaev, Homogenization of Mixed Boundary Value Problem for the Poisson Equation in a Domain Perforated along the Boundary. Uspekhi Mat. Nauk 45 (1990) 123.

R.R. Gadyl'Shin, Yu.O. Koroleva and G.A. Chechkin, On the Convergence of Solutions and Eigenelements of a Boundary Value Problem in a Domain Perforated along the Boundary. Differ. Uravn. 46 (2010) 665–677. (English translation: Differ. Equations. 46 (2010) 667–680.) | MR | Zbl

R.R. Gadyl'Shin, Yu.O. Koroleva and G.A. Chechkin, On the Eigenvalue of the Laplacian in a Domain perforated Along the Boundary. Doklady Akademii Nauk 432 (2010) 7–11. (English translation: Russian Academy of Sciencies. Doklady Mathematics 81 (2010) 337–341.) | MR | Zbl

R.R. Gadyl'Shin, Yu.O. Koroleva and G.A. Chechkin, On the Asymptotics of a Solution of a Boundary Value problem in a Domain Perforated Along the Boundary. Bulletin of the Chelyabinsk State University. Mathematics Mechanics Computer Sciences 14 (2011) 27–36 (in Russian). | MR

R.R. Gadyl'Shin, Yu.O. Koroleva and G.A. Chechkin, On the Asymptotic Behavior of a Simple Eigenvalue of a Boundary Value Problem in a Domain Perforated along the Boundary. Differ. Uravn. 47 (2011) 819–828. (English translation: Differ. Equations. 47 (2011) 822–831.) | MR | Zbl

R.R. Gadyl'Shin, D.V. Kozhevnikov and G.A. Chechkin, On the Spectral Problem in a Domain Perforated Along the Boundary. Perturbation of Multiple Eigenvalue. Problem. Math. Anal. 73 (2013) 31–45. (English translation: J. Math. Sci. 196 (3, 2014): 276–292.) | MR | Zbl

R.R. Gadyl'Shin and D.V. Kozhevnikov, Homogenization of the Boundary Value Problem in a Domain Perforated Along a Part of the Boundary. Problem Math. Anal. 75 (2014) 41–60. (English translation: J. Math. Sci. 198 (2014) 701–723.) | MR | Zbl

D.V. Larin, A Degenerate Quasilinear Dirichlet Problem for Domains with a Fine-Grained Boundary. The Case of Surface Distribution of “Grains”. Tr. Inst. Prikl. Mat. Mekh. 2 (1998) 104–115. | MR | Zbl

V.A. Marchenko and E.Ya. Khruslov, Kraevye zadachi v oblastyakh s melkozernistoi granitsei (Boundary Value Problems in Domains with Fine-Grained Boundary). Naukova Dumka, Kiev (1974). | MR | Zbl

V.G. Mikhailenko, Boundary Value Problems with Fine-Grained Boundary for Second-Order Elliptic Differential Operators. I, II. Teor. Funktsii Funktsional. Anal. i Prilozh. 6 (1968) 93–110. | MR

V.G. Mikhailenko, Boundary Value Problems with Fine-Grained Boundary for Second-Order Elliptic Differential Operators. I, II. Teor. Funktsii Funktsional. Anal. i Prilozh. 9 (1969) 75–84. | MR

G.A. Chechkin, T.P. Chechkina, C. D’ Apice and U. De Maio, Homogenization in Domains Randomly Perforated along the Boundary. Discrete Contin. Dyn. Syst. Ser. B (DCDS-B) 12 (2009) 713–730. | MR | Zbl

G.A. Chechkin and E.L. Ostrovskaya, On Behaviour of a Body Randomly Perforated along the Boundary, in Book of Abstracts of the International Conference Differential Equations and Related Topics dedicated to the Centenary Anniversary of Ivan G. Petrovskii. XX Joint Session of Petrovskii Seminar and Moscow Mathematical Society, Moscow, Russia (2001) 88.

T. Del Vecchio, The Thick Neumann's Sieve. Ann. Mat. Pura Appl. 147 (1987) 363–402. | DOI | MR | Zbl

M. Lobo, O.A. Oleinik, M.E. Pérez and T.A. Shaposhnikova, On Homogenization of Solutions of Boundary Value Problems in Domains Perforated along Manifolds. Ann. Sc. Norm. Pisa, Cl. Sci. (4) 25 (1997) 611–629. | Numdam | MR | Zbl

S. Ozawa, Approximation of Green’s Function in a Region with Many Obstacles, Geometry and Analysis of Manifolds (Katata/Kyoto, 1987). Vol. 1339 of Lect. Notes Math. Berlin (1988) 212–225. | MR | Zbl

E. Sánchez-Palencia, Boundary Value Problems in Domains Containing Perforated Walls, Nonlinear Partial Differential Equations and Their Applications. Vol. 3 (1980/1981) 309–325. | MR | Zbl

V.P. Mikhailov, Partial differential equations. Mir, Moscow (1978). | MR

G. Pólya, G. Szegö, Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies, No. 27. Princeton University Press, Princeton, N.J. (1951). | MR | Zbl

J. Sanchez-Hubert and E. Sánchez-Palencia, Acoustic Fluid Flow Through Holes and Permeability of Perforated Walls. J. Math. Anal. Appl. 87 (1982) 427–253. | DOI | MR | Zbl

G.A. Chechkin, Averaging of Boundary Value Problems with a Singular Perturbation of the Boundary Conditions. Mat. Sbornik 184 (1993) 99–150. (English translation: Russian Academy of Sciences. Sbornik. Mathematics 79 (1994) 191–222.) | MR | Zbl

A.M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. Vol. 102 of Transl. of Mathematical Monographs. American Mathematical Society, Providence, RI (1992). | Zbl

V.G. Maz’ya, S.A. Nazarov, B.A. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. I., Vol. II., Vol. 111 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel (2000). | MR | Zbl

S.A. Nazarov, The Neumann problem in angular domains with periodic and parabolic perturbations of the boundary. Tr. Mosk. Mat. Obs. 69 (2008) 182–241. (English translation: Trans. Moscow Math. Soc. (2008) 153–208.) | MR | Zbl

V.A. Kozlov, V.G. Maz’ya, J. Rossmann, Elliptic boundary value problems in domains with point singularities. Vol. 52 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (1997). | MR | Zbl

V.S. Vladimirov, Equations of Mathematical Physics. Nauka, Moscow (1976). | MR

M.Sh. Birman and M.Z. Solomyak, Spectral theory of selfadjoint operators in Hilbert space. Translated from Russian. Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht (1987) xvi+301. | MR | Zbl

T. Kato, Perturbation Theory for Linear Operators. Springer, Berlin – Heidelberg (1966). | MR | Zbl

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