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] 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

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
 [1] Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 [2] Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2777-2808. doi: 10.3934/dcds.2020385 [3] Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 [4] Micol Amar, Daniele Andreucci, Claudia Timofte. Homogenization of a modified bidomain model involving imperfect transmission. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021040 [5] Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533 [6] Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067 [7] Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190 [8] Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022 [9] Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021049 [10] Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 [11] Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 [12] Giulio Ciraolo, Antonio Greco. An overdetermined problem associated to the Finsler Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1025-1038. doi: 10.3934/cpaa.2021004 [13] Yang Zhang. A free boundary problem of the cancer invasion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021092 [14] Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463 [15] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384 [16] Sergei Avdonin, Julian Edward. An inverse problem for quantum trees with observations at interior vertices. Networks & Heterogeneous Media, 2021, 16 (2) : 317-339. doi: 10.3934/nhm.2021008 [17] Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151 [18] Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205 [19] Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453 [20] Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021031

2019 Impact Factor: 1.053

Metrics

• PDF downloads (90)
• HTML views (0)
• Cited by (11)

Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]