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 & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151
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show all references

References:
[1]

Nonlinear Anal., 18 (1992), 481-496. doi: 10.1016/0362-546X(92)90015-7.  Google Scholar

[2]

Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar

[3]

C. R. Acad. Sci. Paris, Série I, 324 (1997), 939-944.  Google Scholar

[4]

Comm. in PDE, 27 (2002), 705-725. doi: 10.1081/PDE-120002871.  Google Scholar

[5]

Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984.  Google Scholar

[6]

Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1989.  Google Scholar

[7]

Sbornik Math., 192 (2001), 933-949. doi: 10.1070/SM2001v192n07ABEH000576.  Google Scholar

[8]

ESAIM: COCV, 16 (2010), 148-175. doi: 10.1051/cocv:2008073.  Google Scholar

[9]

Translations of Mathematical Monographs, 234, American Mathematical Society, Providence, RI, 2007.  Google Scholar

[10]

in "Homogenization and Applications to Material Sciences" (Nice, 1995), GAKUTO Internat. Ser. Math. Sci. Appl., 9, Gakkōtosho, Tokyo, (1995), 123-135.  Google Scholar

[11]

in "Topics in the Mathematical Modelling of Composite Materials," Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, (1997), 45-93.  Google Scholar

[12]

J. Math. Anal. Appl., 71 (1979), 590-607. doi: 10.1016/0022-247X(79)90211-7.  Google Scholar

[13]

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

[14]

Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[15]

Trans. Moscow Math. Soc., 2 (1984), 101-126. Google Scholar

[16]

Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968.  Google Scholar

[17]

Advances in Mathematics, 3 (1969), 510-585. doi: 10.1016/0001-8708(69)90009-7.  Google Scholar

[18]

Novosibirsk, 2002. Google Scholar

[19]

Arch. Rational Mech. Anal., 200 (2011), 747-788. doi: 10.1007/s00205-010-0370-2.  Google Scholar

[20]

Journal of Mathematical Sciences, 169 (2010), 212-248. Google Scholar

[21]

Studies in Mathematics and its Applications, 26, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[22]

Differential Equations, 31 (1995), 975-986.  Google Scholar

[23]

Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 859-883. doi: 10.3934/dcdsb.2007.7.859.  Google Scholar

[24]

Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239-271. doi: 10.1007/BF02838079.  Google Scholar

[25]

Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989.  Google Scholar

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