American Institute of Mathematical Sciences

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
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.  doi: 10.1016/0362-546X(92)90015-7.  Google Scholar [2] R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975).   Google Scholar [3] G. Allaire and F. Malige, Analyse asymptotique spectrale d’un problème de diffusion neutronique,, C. R. Acad. Sci. Paris, 324 (1997), 939.   Google Scholar [4] G. Allaire and A. Piatnitski, Uniform spectral asymptotics for singularly perturbed locally periodic operators,, Comm. in PDE, 27 (2002), 705.  doi: 10.1081/PDE-120002871.  Google Scholar [5] H. Attouch, "Variational Convergence for Functions and Operators,", Applicable Mathematics Series, (1984).   Google Scholar [6] T. Ya. Azizov and I. S. Iokhvidov, "Linear Operators in Spaces with an Indefinite Metric,", Pure and Applied Mathematics (New York), (1989).   Google Scholar [7] A. Belyaev, A. Pyatnitskiĭ and G. Chechkin, Averaging in a perforated domain with an oscillating third boundary condition,, Sbornik Math., 192 (2001), 933.  doi: 10.1070/SM2001v192n07ABEH000576.  Google Scholar [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.  doi: 10.1051/cocv:2008073.  Google Scholar [9] G. Chechkin, A. Piatnitski and A. Shamaev, "Homogenization. Methods and Applications,", Translations of Mathematical Monographs, 234 (2007).   Google Scholar [10] D. Cioranescu and P. Donato, On a Robin problem in perforated domains,, in, 9 (1995), 123.   Google Scholar [11] D. Cioranescu and F. Murat, A strange term coming from nowhere,, in, 31 (1997), 45.   Google Scholar [12] D. Cioranescu, J. Saint Jean Paulin, Homogenization in open sets with holes,, J. Math. Anal. Appl., 71 (1979), 590.  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] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).   Google Scholar [15] S. Kozlov, Reducibility of quasiperiodic differential operators and averaging,, Trans. Moscow Math. Soc., 2 (1984), 101.   Google Scholar [16] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,", Vol. 1, (1968).   Google Scholar [17] U. Mosco, Convergence of convex sets and of solutions of variational inequalities,, Advances in Mathematics, 3 (1969), 510.  doi: 10.1016/0001-8708(69)90009-7.  Google Scholar [18] S. A. Nazarov, "Asymptotic Analysis of Thin Plates and Rods," (in Russian),, Novosibirsk, (2002).   Google Scholar [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.  doi: 10.1007/s00205-010-0370-2.  Google Scholar [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.   Google Scholar [21] O. Oleĭnik, A. Shamaev and G. Yosifian, "Mathematical Problems in Elasticity and Homogenization,", Studies in Mathematics and its Applications, 26 (1992).   Google Scholar [22] S. E. Pastukhova, On the error of averaging for the Steklov problem in a punctured domain,, Differential Equations, 31 (1995), 975.   Google Scholar [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.  doi: 10.3934/dcdsb.2007.7.859.  Google Scholar [24] M. Vanninathan, Homogenization of eigenvalue problems in perforated domains,, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239.  doi: 10.1007/BF02838079.  Google Scholar [25] W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation,", Graduate Texts in Mathematics, 120 (1989).   Google Scholar

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.  doi: 10.1016/0362-546X(92)90015-7.  Google Scholar [2] R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975).   Google Scholar [3] G. Allaire and F. Malige, Analyse asymptotique spectrale d’un problème de diffusion neutronique,, C. R. Acad. Sci. Paris, 324 (1997), 939.   Google Scholar [4] G. Allaire and A. Piatnitski, Uniform spectral asymptotics for singularly perturbed locally periodic operators,, Comm. in PDE, 27 (2002), 705.  doi: 10.1081/PDE-120002871.  Google Scholar [5] H. Attouch, "Variational Convergence for Functions and Operators,", Applicable Mathematics Series, (1984).   Google Scholar [6] T. Ya. Azizov and I. S. Iokhvidov, "Linear Operators in Spaces with an Indefinite Metric,", Pure and Applied Mathematics (New York), (1989).   Google Scholar [7] A. Belyaev, A. Pyatnitskiĭ and G. Chechkin, Averaging in a perforated domain with an oscillating third boundary condition,, Sbornik Math., 192 (2001), 933.  doi: 10.1070/SM2001v192n07ABEH000576.  Google Scholar [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.  doi: 10.1051/cocv:2008073.  Google Scholar [9] G. Chechkin, A. Piatnitski and A. Shamaev, "Homogenization. Methods and Applications,", Translations of Mathematical Monographs, 234 (2007).   Google Scholar [10] D. Cioranescu and P. Donato, On a Robin problem in perforated domains,, in, 9 (1995), 123.   Google Scholar [11] D. Cioranescu and F. Murat, A strange term coming from nowhere,, in, 31 (1997), 45.   Google Scholar [12] D. Cioranescu, J. Saint Jean Paulin, Homogenization in open sets with holes,, J. Math. Anal. Appl., 71 (1979), 590.  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] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition, (1998).   Google Scholar [15] S. Kozlov, Reducibility of quasiperiodic differential operators and averaging,, Trans. Moscow Math. Soc., 2 (1984), 101.   Google Scholar [16] J.-L. Lions and E. Magenes, "Problèmes aux Limites Non Homogènes et Applications,", Vol. 1, (1968).   Google Scholar [17] U. Mosco, Convergence of convex sets and of solutions of variational inequalities,, Advances in Mathematics, 3 (1969), 510.  doi: 10.1016/0001-8708(69)90009-7.  Google Scholar [18] S. A. Nazarov, "Asymptotic Analysis of Thin Plates and Rods," (in Russian),, Novosibirsk, (2002).   Google Scholar [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.  doi: 10.1007/s00205-010-0370-2.  Google Scholar [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.   Google Scholar [21] O. Oleĭnik, A. Shamaev and G. Yosifian, "Mathematical Problems in Elasticity and Homogenization,", Studies in Mathematics and its Applications, 26 (1992).   Google Scholar [22] S. E. Pastukhova, On the error of averaging for the Steklov problem in a punctured domain,, Differential Equations, 31 (1995), 975.   Google Scholar [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.  doi: 10.3934/dcdsb.2007.7.859.  Google Scholar [24] M. Vanninathan, Homogenization of eigenvalue problems in perforated domains,, Proc. Indian Acad. Sci. Math. Sci., 90 (1981), 239.  doi: 10.1007/BF02838079.  Google Scholar [25] W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation,", Graduate Texts in Mathematics, 120 (1989).   Google Scholar
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