On some variational problems set on domains tending to infinity

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

On some variational problems set on domains tending to infinity

Michel Chipot ^1, , Aleksandar Mojsic ^2, and Prosenjit Roy ^3,

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

Abstract
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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.

Keywords: cylindrical domains., asymptotic analysis, Variational methods.

Mathematics Subject Classification: Primary: 35B40, 35J20, 35J2.

Citation: Michel Chipot, Aleksandar Mojsic, Prosenjit Roy. On some variational problems set on domains tending to infinity. Discrete and Continuous Dynamical Systems, 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), 1081-1091. doi: 10.1090/S0002-9939-08-09630-5.
[2]	P. G. Ciarlet, Introduction to Linear Shell Theory, Series in Applied Mathematics, (North-Holland, Amsterdam,) 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), 133-159. 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), 15-44. 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), 3579-3602. 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), 3-11. 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), 199-227.
[10]	M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique, C. R. Math. Acad. Sci., Paris, 346 (2008), 21-26. 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), 4875-4890. 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), 137-154. 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, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkotosho, Tokyo, 20 (2004), 16-27.
[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., New York, 1976.
[16]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, AMS, 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), 3320-3331. 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), 1190-1212. doi: 10.1016/j.jmaa.2007.11.001.

show all references

References:

[1]	J. Borwein, A. J. Guirao, P. Hajek and J. Vanderwerff, Uniformly convex functions on Banach spaces, Proc. Amer. Math. Soc., 137 (2009), 1081-1091. doi: 10.1090/S0002-9939-08-09630-5.
[2]	P. G. Ciarlet, Introduction to Linear Shell Theory, Series in Applied Mathematics, (North-Holland, Amsterdam,) 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), 133-159. 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), 15-44. 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), 3579-3602. 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), 3-11. 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), 199-227.
[10]	M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique, C. R. Math. Acad. Sci., Paris, 346 (2008), 21-26. 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), 4875-4890. 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), 137-154. 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, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkotosho, Tokyo, 20 (2004), 16-27.
[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., New York, 1976.
[16]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, AMS, 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), 3320-3331. 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), 1190-1212. doi: 10.1016/j.jmaa.2007.11.001.

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