March  2012, 7(1): 151-178. doi: 10.3934/nhm.2012.7.151

Steklov problems in perforated domains with a coefficient of indefinite sign

1. 

Department of mathematical sciences, Politecnico, Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

2. 

Institute for Problems in Mechanical Engineering RAS, Bolshoi ave., 61, 199178, St-Petersburg, Russian Federation

3. 

Narvik University College, Postbox 385, 8505 Narvik

Received  September 2011 Revised  January 2012 Published  February 2012

We consider homogenization of Steklov spectral problem for a divergence form elliptic operator in periodically perforated domain under the assumption that the spectral weight function changes sign. We show that the limit behaviour of the spectrum depends essentially on wether the average of the weight function over the boundary of holes is positive, or negative or equal to zero. In all these cases we construct the asymptotics of the eigenpairs.
Citation: Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks and Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151
References:
[1]

E. Acerbi, V. Chiadò Piat, G. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal., 18 (1992), 481-496. doi: 10.1016/0362-546X(92)90015-7.

[2]

R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[3]

G. Allaire and F. Malige, Analyse asymptotique spectrale d’un problème de diffusion neutronique, C. R. Acad. Sci. Paris, Série I, 324 (1997), 939-944.

[4]

G. Allaire and A. Piatnitski, Uniform spectral asymptotics for singularly perturbed locally periodic operators, Comm. in PDE, 27 (2002), 705-725. doi: 10.1081/PDE-120002871.

[5]

H. Attouch, "Variational Convergence for Functions and Operators," Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984.

[6]

T. Ya. Azizov and I. S. Iokhvidov, "Linear Operators in Spaces with an Indefinite Metric," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1989.

[7]

A. Belyaev, A. Pyatnitskiĭ and G. Chechkin, Averaging in a perforated domain with an oscillating third boundary condition, Sbornik Math., 192 (2001), 933-949. doi: 10.1070/SM2001v192n07ABEH000576.

[8]

V. Chiadò Piat and A. Piatnitski, $\Gamma$-convergence approach to variational problems in perforated domains with Fourier boundary conditions, ESAIM: COCV, 16 (2010), 148-175. doi: 10.1051/cocv:2008073.

[9]

G. Chechkin, A. Piatnitski and A. Shamaev, "Homogenization. Methods and Applications," Translations of Mathematical Monographs, 234, American Mathematical Society, Providence, RI, 2007.

[10]

D. Cioranescu and P. Donato, On a Robin problem in perforated domains, in "Homogenization and Applications to Material Sciences" (Nice, 1995), GAKUTO Internat. Ser. Math. Sci. Appl., 9, Gakkōtosho, Tokyo, (1995), 123-135.

[11]

D. Cioranescu and F. Murat, A strange term coming from nowhere, in "Topics in the Mathematical Modelling of Composite Materials," Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, (1997), 45-93.

[12]

D. Cioranescu, J. Saint Jean Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl., 71 (1979), 590-607. doi: 10.1016/0022-247X(79)90211-7.

[13]

H. Douanla, Homogenization of Steklov spectral problems with indefinite density function in perforated domains,, preprint, (). 

[14]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[15]

S. Kozlov, Reducibility of quasiperiodic differential operators and averaging, Trans. Moscow Math. Soc., 2 (1984), 101-126.

[16]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications," Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968.

[17]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Mathematics, 3 (1969), 510-585. doi: 10.1016/0001-8708(69)90009-7.

[18]

S. A. Nazarov, "Asymptotic Analysis of Thin Plates and Rods," (in Russian), Novosibirsk, 2002.

[19]

S. Nazarov, I. Pankratova and A. Piatnitski, Homogenization of the spectral problem for periodic elliptic operators with sign-changing density function, Arch. Rational Mech. Anal., 200 (2011), 747-788. doi: 10.1007/s00205-010-0370-2.

[20]

S. Nazarov and A. Piatnitski, Homogenization of the spectral Dirichlet problem for a system of differential equations with rapidly oscillating coefficients and changing sign sensity, Journal of Mathematical Sciences, 169 (2010), 212-248.

[21]

O. Oleĭnik, A. Shamaev and G. Yosifian, "Mathematical Problems in Elasticity and Homogenization," Studies in Mathematics and its Applications, 26, North-Holland Publishing Co., Amsterdam, 1992.

[22]

S. E. Pastukhova, On the error of averaging for the Steklov problem in a punctured domain, Differential Equations, 31 (1995), 975-986.

[23]

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. doi: 10.3934/dcdsb.2007.7.859.

[24]

M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239-271. doi: 10.1007/BF02838079.

[25]

W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation," Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989.

show all references

References:
[1]

E. Acerbi, V. Chiadò Piat, G. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal., 18 (1992), 481-496. doi: 10.1016/0362-546X(92)90015-7.

[2]

R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[3]

G. Allaire and F. Malige, Analyse asymptotique spectrale d’un problème de diffusion neutronique, C. R. Acad. Sci. Paris, Série I, 324 (1997), 939-944.

[4]

G. Allaire and A. Piatnitski, Uniform spectral asymptotics for singularly perturbed locally periodic operators, Comm. in PDE, 27 (2002), 705-725. doi: 10.1081/PDE-120002871.

[5]

H. Attouch, "Variational Convergence for Functions and Operators," Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984.

[6]

T. Ya. Azizov and I. S. Iokhvidov, "Linear Operators in Spaces with an Indefinite Metric," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1989.

[7]

A. Belyaev, A. Pyatnitskiĭ and G. Chechkin, Averaging in a perforated domain with an oscillating third boundary condition, Sbornik Math., 192 (2001), 933-949. doi: 10.1070/SM2001v192n07ABEH000576.

[8]

V. Chiadò Piat and A. Piatnitski, $\Gamma$-convergence approach to variational problems in perforated domains with Fourier boundary conditions, ESAIM: COCV, 16 (2010), 148-175. doi: 10.1051/cocv:2008073.

[9]

G. Chechkin, A. Piatnitski and A. Shamaev, "Homogenization. Methods and Applications," Translations of Mathematical Monographs, 234, American Mathematical Society, Providence, RI, 2007.

[10]

D. Cioranescu and P. Donato, On a Robin problem in perforated domains, in "Homogenization and Applications to Material Sciences" (Nice, 1995), GAKUTO Internat. Ser. Math. Sci. Appl., 9, Gakkōtosho, Tokyo, (1995), 123-135.

[11]

D. Cioranescu and F. Murat, A strange term coming from nowhere, in "Topics in the Mathematical Modelling of Composite Materials," Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, (1997), 45-93.

[12]

D. Cioranescu, J. Saint Jean Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl., 71 (1979), 590-607. doi: 10.1016/0022-247X(79)90211-7.

[13]

H. Douanla, Homogenization of Steklov spectral problems with indefinite density function in perforated domains,, preprint, (). 

[14]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[15]

S. Kozlov, Reducibility of quasiperiodic differential operators and averaging, Trans. Moscow Math. Soc., 2 (1984), 101-126.

[16]

J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications," Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968.

[17]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Mathematics, 3 (1969), 510-585. doi: 10.1016/0001-8708(69)90009-7.

[18]

S. A. Nazarov, "Asymptotic Analysis of Thin Plates and Rods," (in Russian), Novosibirsk, 2002.

[19]

S. Nazarov, I. Pankratova and A. Piatnitski, Homogenization of the spectral problem for periodic elliptic operators with sign-changing density function, Arch. Rational Mech. Anal., 200 (2011), 747-788. doi: 10.1007/s00205-010-0370-2.

[20]

S. Nazarov and A. Piatnitski, Homogenization of the spectral Dirichlet problem for a system of differential equations with rapidly oscillating coefficients and changing sign sensity, Journal of Mathematical Sciences, 169 (2010), 212-248.

[21]

O. Oleĭnik, A. Shamaev and G. Yosifian, "Mathematical Problems in Elasticity and Homogenization," Studies in Mathematics and its Applications, 26, North-Holland Publishing Co., Amsterdam, 1992.

[22]

S. E. Pastukhova, On the error of averaging for the Steklov problem in a punctured domain, Differential Equations, 31 (1995), 975-986.

[23]

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. doi: 10.3934/dcdsb.2007.7.859.

[24]

M. Vanninathan, Homogenization of eigenvalue problems in perforated domains, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239-271. doi: 10.1007/BF02838079.

[25]

W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation," Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989.

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