# 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 and 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. [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), 27-95. [3] E. Berchio, D. Cassani and F. Gazzola, Hardy-Rellich inequalities with boundary remainder terms and applications, Manuscripta Mathematica, 131 (2010), 427-458. doi: 10.1007/s00229-009-0328-6. [4] S. Campanato, On the condition of nearness between operators, Ann. Mat. Pura Appl., 167 (1994), 243-256. doi: 10.1007/BF01760335. [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), 177-197. [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, Progress in Nonlinear Differential Equations and their Applications, 85, Birkhäuser, (2014), 187-206. [7] D. Cassani, L. Fattorusso and A. Tarsia, Global existence for nonlocal MEMS, Nonlinear Analysis, 74 (2011), 5722-5726. doi: 10.1016/j.na.2011.05.060. [8] D. Cassani, B. Kaltenbacher and A. Lorenzi, Direct and inverse problems related to MEMS, Inverse Problems, 25 (2009), 105002 (22pp). doi: 10.1088/0266-5611/25/10/105002. [9] D. Cassani and A. Tarsia, Maximum principle for higher order operators in general domains and applications, preprint, 2015. [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, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3. [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), 725-734. doi: 10.1090/S0033-569X-09-01159-5. [12] K. M. Hui, The existence and dynamic properties of a parabolic nonlocal MEMS equation, Nonlinear Anal., 74 (2011), 298-316. doi: 10.1016/j.na.2010.08.045. [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), 505-534. doi: 10.1216/RMJ-2011-41-2-505. [14] F. H. Lin and Y. S. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. London, Ser. A, 463 (2007), 1323-1337. doi: 10.1098/rspa.2007.1816. [15] J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control, J. Engrg. Math., 41 (2001), 345-366. doi: 10.1023/A:1012292311304. [16] A. Tarsia, Differential equations and implicit functions: A generalization of the near operator theorem, Topol. Methods Nonlinear Anal., 11 (1998), 115-133.

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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. [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), 27-95. [3] E. Berchio, D. Cassani and F. Gazzola, Hardy-Rellich inequalities with boundary remainder terms and applications, Manuscripta Mathematica, 131 (2010), 427-458. doi: 10.1007/s00229-009-0328-6. [4] S. Campanato, On the condition of nearness between operators, Ann. Mat. Pura Appl., 167 (1994), 243-256. doi: 10.1007/BF01760335. [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), 177-197. [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, Progress in Nonlinear Differential Equations and their Applications, 85, Birkhäuser, (2014), 187-206. [7] D. Cassani, L. Fattorusso and A. Tarsia, Global existence for nonlocal MEMS, Nonlinear Analysis, 74 (2011), 5722-5726. doi: 10.1016/j.na.2011.05.060. [8] D. Cassani, B. Kaltenbacher and A. Lorenzi, Direct and inverse problems related to MEMS, Inverse Problems, 25 (2009), 105002 (22pp). doi: 10.1088/0266-5611/25/10/105002. [9] D. Cassani and A. Tarsia, Maximum principle for higher order operators in general domains and applications, preprint, 2015. [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, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3. [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), 725-734. doi: 10.1090/S0033-569X-09-01159-5. [12] K. M. Hui, The existence and dynamic properties of a parabolic nonlocal MEMS equation, Nonlinear Anal., 74 (2011), 298-316. doi: 10.1016/j.na.2010.08.045. [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), 505-534. doi: 10.1216/RMJ-2011-41-2-505. [14] F. H. Lin and Y. S. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. London, Ser. A, 463 (2007), 1323-1337. doi: 10.1098/rspa.2007.1816. [15] J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control, J. Engrg. Math., 41 (2001), 345-366. doi: 10.1023/A:1012292311304. [16] A. Tarsia, Differential equations and implicit functions: A generalization of the near operator theorem, Topol. Methods Nonlinear Anal., 11 (1998), 115-133.
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