# American Institute of Mathematical Sciences

2016, 36(7): 3603-3621. doi: 10.3934/dcds.2016.36.3603

## On some variational problems set on domains tending to infinity

 1 Institute of Mathematics, University of Zürich, Winterthurerstrasse 190, CH-8057 Zürich 2 Helmut Schmidt University / University of the Federal Armed Forces Hamburg, Department of Mechanical Engineering, Holstenhofweg 85, 22043, Hamburg, Germany 3 Tata Institute of Fundamental Research- CAM, Sharadanagar, GKVK Campus, Postbox - 560065, Bangalore, India

Received  April 2015 Revised  January 2016 Published  March 2016

Let $\Omega_\ell = \ell\omega_1 \times \omega_2$ where $\omega_1 \subset \mathbb{R}^p$ and $\omega_2 \subset \mathbb{R}^{n-p}$ are assumed to be open and bounded. We consider the following minimization problem: $$E_{\Omega_\ell}(u_\ell) = \min_{u\in W_0^{1,q}(\Omega_\ell)}E_{\Omega_\ell}(u)$$ where $E_{\Omega_\ell}(u) = \int_{\Omega_\ell}F(\nabla u)-fu$, $F$ is a convex function and $f\in L^{q'}(\omega_2)$. We are interested in studying the asymptotic behavior of the solution $u_\ell$ as $\ell$ tends to infinity.
Citation: Michel Chipot, Aleksandar Mojsic, Prosenjit Roy. On some variational problems set on domains tending to infinity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3603-3621. doi: 10.3934/dcds.2016.36.3603
##### References:
 [1] J. Borwein, A. J. Guirao, P. Hajek and J. Vanderwerff, Uniformly convex functions on Banach spaces,, Proc. Amer. Math. Soc., 137 (2009), 1081. doi: 10.1090/S0002-9939-08-09630-5. [2] P. G. Ciarlet, Introduction to Linear Shell Theory,, Series in Applied Mathematics, 1 (1998). [3] M. Chipot, $l$ Goes to Plus Infinity,, Birkhäuser, (2002). doi: 10.1007/978-3-0348-8173-9. [4] M. Chipot, to, appear., (). [5] M. Chipot and S. Mardare, Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction,, J. Math. Pures Appl., 90 (2013), 133. doi: 10.1016/j.matpur.2008.04.002. [6] M. Chipot and A. Rougirel, On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded,, Commun. Contemp. Math., 4 (2002), 15. doi: 10.1142/S0219199702000555. [7] M. Chipot and A. Rougirel, On the asymptotic behaviour of the eigenmodes for elliptic problems in domain becoming unbounded,, Trans. AMS, 360 (2008), 3579. doi: 10.1090/S0002-9947-08-04361-4. [8] M. Chipot and A. Rougirel, Remarks on the asymptotic behaviour of the solution to parabolic problems in domains becoming unbounded,, Nonlinear Analysis, 47 (2001), 3. doi: 10.1016/S0362-546X(01)00151-1. [9] M. Chipot, P. Roy and I. Shafrir, Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity,, Asymptotic Analysis, 85 (2013), 199. [10] M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique,, C. R. Math. Acad. Sci., 346 (2008), 21. doi: 10.1016/j.crma.2007.12.004. [11] M. Chipot and K. Yeressian, On the asymptotic behavior of variational inequalities set in cylinders,, Discrete Contin. Dyn. Syst., 33 (2013), 4875. doi: 10.3934/dcds.2013.33.4875. [12] M. Chipot and K. Yeressian, Asymptotic behaviour of the solution to variational inequalities with joint constraints on its value and its gradient,, Contemporary Mathematics, 594 (2013), 137. doi: 10.1090/conm/594/11797. [13] M. Chipot and Y. Xie, On the asymptotic behaviour of the p-Laplace equation in cylinders becoming unbounded,, Nonlinear partial differential equations and their applications, 20 (2004), 16. [14] I. Chowdhury and P. Roy, On the asymptotic analysis of problems involving fractional laplacian in cylindrical domains tending to infinity,, preprint, (). [15] I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, North-Holland Publishing Co., (1976). [16] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, 19 (1998). [17] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992). [18] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer, (1983). doi: 10.1007/978-3-642-61798-0. [19] S. Guesmia, Some convergence results for quasilinear parabolic boundary value problems in cylindrical domains of large size,, Nonlinear Anal., 70 (2009), 3320. doi: 10.1016/j.na.2008.04.036. [20] S. Guesmia, Some results on the asymptotic behaviour for hyperbolic problems in cylindrical domains becoming unbounded,, J. Math. Anal. Appl., 341 (2008), 1190. doi: 10.1016/j.jmaa.2007.11.001.

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##### References:
 [1] J. Borwein, A. J. Guirao, P. Hajek and J. Vanderwerff, Uniformly convex functions on Banach spaces,, Proc. Amer. Math. Soc., 137 (2009), 1081. doi: 10.1090/S0002-9939-08-09630-5. [2] P. G. Ciarlet, Introduction to Linear Shell Theory,, Series in Applied Mathematics, 1 (1998). [3] M. Chipot, $l$ Goes to Plus Infinity,, Birkhäuser, (2002). doi: 10.1007/978-3-0348-8173-9. [4] M. Chipot, to, appear., (). [5] M. Chipot and S. Mardare, Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction,, J. Math. Pures Appl., 90 (2013), 133. doi: 10.1016/j.matpur.2008.04.002. [6] M. Chipot and A. Rougirel, On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded,, Commun. Contemp. Math., 4 (2002), 15. doi: 10.1142/S0219199702000555. [7] M. Chipot and A. Rougirel, On the asymptotic behaviour of the eigenmodes for elliptic problems in domain becoming unbounded,, Trans. AMS, 360 (2008), 3579. doi: 10.1090/S0002-9947-08-04361-4. [8] M. Chipot and A. Rougirel, Remarks on the asymptotic behaviour of the solution to parabolic problems in domains becoming unbounded,, Nonlinear Analysis, 47 (2001), 3. doi: 10.1016/S0362-546X(01)00151-1. [9] M. Chipot, P. Roy and I. Shafrir, Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity,, Asymptotic Analysis, 85 (2013), 199. [10] M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique,, C. R. Math. Acad. Sci., 346 (2008), 21. doi: 10.1016/j.crma.2007.12.004. [11] M. Chipot and K. Yeressian, On the asymptotic behavior of variational inequalities set in cylinders,, Discrete Contin. Dyn. Syst., 33 (2013), 4875. doi: 10.3934/dcds.2013.33.4875. [12] M. Chipot and K. Yeressian, Asymptotic behaviour of the solution to variational inequalities with joint constraints on its value and its gradient,, Contemporary Mathematics, 594 (2013), 137. doi: 10.1090/conm/594/11797. [13] M. Chipot and Y. Xie, On the asymptotic behaviour of the p-Laplace equation in cylinders becoming unbounded,, Nonlinear partial differential equations and their applications, 20 (2004), 16. [14] I. Chowdhury and P. Roy, On the asymptotic analysis of problems involving fractional laplacian in cylindrical domains tending to infinity,, preprint, (). [15] I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, North-Holland Publishing Co., (1976). [16] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, 19 (1998). [17] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992). [18] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer, (1983). doi: 10.1007/978-3-642-61798-0. [19] S. Guesmia, Some convergence results for quasilinear parabolic boundary value problems in cylindrical domains of large size,, Nonlinear Anal., 70 (2009), 3320. doi: 10.1016/j.na.2008.04.036. [20] S. Guesmia, Some results on the asymptotic behaviour for hyperbolic problems in cylindrical domains becoming unbounded,, J. Math. Anal. Appl., 341 (2008), 1190. doi: 10.1016/j.jmaa.2007.11.001.
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