# American Institute of Mathematical Sciences

June  2016, 9(3): 631-642. doi: 10.3934/dcdss.2016017

## Periodic solutions to nonlocal MEMS equations

 1 Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, Via Valleggio 11, 22100 Como, Italy 2 Dipartimento di Matematica “L. Tonelli”, Università di Pisa, Largo B. Pontecorvo, 5. I-56127 Pisa

Received  March 2015 Revised  February 2016 Published  April 2016

Combining a priori estimates with penalization techniques and an implicit function argument based on Campanato's near operators theory, we obtain the existence of periodic solutions for a fourth order integro-differential equation modelling actuators in MEMS devices.
Citation: Daniele Cassani, Antonio Tarsia. Periodic solutions to nonlocal MEMS equations. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 631-642. doi: 10.3934/dcdss.2016017
##### References:
 [1] P. Acquistapace and A. Tarsia, On periodic solutions of non-autonomous second order differential equations in Hilbert spaces,, preprint, (2015). Google Scholar [2] C. Baiocchi, Soluzioni ordinarie e generalizzate del problema di Cauchy per equazioni differenziali astratte lineari del secondo ordine in spazi di Hilbert,, Ricerche Mat., 16 (1967), 27. Google Scholar [3] E. Berchio, D. Cassani and F. Gazzola, Hardy-Rellich inequalities with boundary remainder terms and applications,, Manuscripta Mathematica, 131 (2010), 427. doi: 10.1007/s00229-009-0328-6. Google Scholar [4] S. Campanato, On the condition of nearness between operators,, Ann. Mat. Pura Appl., 167 (1994), 243. doi: 10.1007/BF01760335. Google Scholar [5] D. Cassani, J. M. do Ó and N. Ghoussoub, On a fourth order elliptic problem with a singular nonlinearity,, Adv. Nonlinear Stud., 9 (2009), 177. Google Scholar [6] D. Cassani, L. Fattorusso and A. Tarsia, Nonlocal dynamic problems with singular nonlinearities and applications to MEMS,, in Analysis and Topology in Nonlinear Differential Equations, 85 (2014), 187. Google Scholar [7] D. Cassani, L. Fattorusso and A. Tarsia, Global existence for nonlocal MEMS,, Nonlinear Analysis, 74 (2011), 5722. doi: 10.1016/j.na.2011.05.060. Google Scholar [8] D. Cassani, B. Kaltenbacher and A. Lorenzi, Direct and inverse problems related to MEMS,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/10/105002. Google Scholar [9] D. Cassani and A. Tarsia, Maximum principle for higher order operators in general domains and applications,, preprint, (2015). Google Scholar [10] F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains,, Lecture Notes in Mathematics, 1991 (2010). doi: 10.1007/978-3-642-12245-3. Google Scholar [11] J.-S. Guo, B. Hu and C.-J. Wang, A nonlocal quenching problem arising in a micro-electro mechanical system,, Quart. Appl. Math., 67 (2009), 725. doi: 10.1090/S0033-569X-09-01159-5. Google Scholar [12] K. M. Hui, The existence and dynamic properties of a parabolic nonlocal MEMS equation,, Nonlinear Anal., 74 (2011), 298. doi: 10.1016/j.na.2010.08.045. Google Scholar [13] N. I. Kavallaris, A. A. Lacey, C. V. Nikolopoulos and D. E. Tzanetis, A hyperbolic non-local problem modelling MEMS technology,, Rocky Mountain J. Math., 41 (2011), 505. doi: 10.1216/RMJ-2011-41-2-505. Google Scholar [14] F. H. Lin and Y. S. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation,, Proc. R. Soc. London, 463 (2007), 1323. doi: 10.1098/rspa.2007.1816. Google Scholar [15] J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control,, J. Engrg. Math., 41 (2001), 345. doi: 10.1023/A:1012292311304. Google Scholar [16] A. Tarsia, Differential equations and implicit functions: A generalization of the near operator theorem,, Topol. Methods Nonlinear Anal., 11 (1998), 115. Google Scholar

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##### References:
 [1] P. Acquistapace and A. Tarsia, On periodic solutions of non-autonomous second order differential equations in Hilbert spaces,, preprint, (2015). Google Scholar [2] C. Baiocchi, Soluzioni ordinarie e generalizzate del problema di Cauchy per equazioni differenziali astratte lineari del secondo ordine in spazi di Hilbert,, Ricerche Mat., 16 (1967), 27. Google Scholar [3] E. Berchio, D. Cassani and F. Gazzola, Hardy-Rellich inequalities with boundary remainder terms and applications,, Manuscripta Mathematica, 131 (2010), 427. doi: 10.1007/s00229-009-0328-6. Google Scholar [4] S. Campanato, On the condition of nearness between operators,, Ann. Mat. Pura Appl., 167 (1994), 243. doi: 10.1007/BF01760335. Google Scholar [5] D. Cassani, J. M. do Ó and N. Ghoussoub, On a fourth order elliptic problem with a singular nonlinearity,, Adv. Nonlinear Stud., 9 (2009), 177. Google Scholar [6] D. Cassani, L. Fattorusso and A. Tarsia, Nonlocal dynamic problems with singular nonlinearities and applications to MEMS,, in Analysis and Topology in Nonlinear Differential Equations, 85 (2014), 187. Google Scholar [7] D. Cassani, L. Fattorusso and A. Tarsia, Global existence for nonlocal MEMS,, Nonlinear Analysis, 74 (2011), 5722. doi: 10.1016/j.na.2011.05.060. Google Scholar [8] D. Cassani, B. Kaltenbacher and A. Lorenzi, Direct and inverse problems related to MEMS,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/10/105002. Google Scholar [9] D. Cassani and A. Tarsia, Maximum principle for higher order operators in general domains and applications,, preprint, (2015). Google Scholar [10] F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains,, Lecture Notes in Mathematics, 1991 (2010). doi: 10.1007/978-3-642-12245-3. Google Scholar [11] J.-S. Guo, B. Hu and C.-J. Wang, A nonlocal quenching problem arising in a micro-electro mechanical system,, Quart. Appl. Math., 67 (2009), 725. doi: 10.1090/S0033-569X-09-01159-5. Google Scholar [12] K. M. Hui, The existence and dynamic properties of a parabolic nonlocal MEMS equation,, Nonlinear Anal., 74 (2011), 298. doi: 10.1016/j.na.2010.08.045. Google Scholar [13] N. I. Kavallaris, A. A. Lacey, C. V. Nikolopoulos and D. E. Tzanetis, A hyperbolic non-local problem modelling MEMS technology,, Rocky Mountain J. Math., 41 (2011), 505. doi: 10.1216/RMJ-2011-41-2-505. Google Scholar [14] F. H. Lin and Y. S. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation,, Proc. R. Soc. London, 463 (2007), 1323. doi: 10.1098/rspa.2007.1816. Google Scholar [15] J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control,, J. Engrg. Math., 41 (2001), 345. doi: 10.1023/A:1012292311304. Google Scholar [16] A. Tarsia, Differential equations and implicit functions: A generalization of the near operator theorem,, Topol. Methods Nonlinear Anal., 11 (1998), 115. Google Scholar
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