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On some variational problems set on domains tending to infinity

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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.
    Mathematics Subject Classification: Primary: 35B40, 35J20, 35J25.


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  • [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.


    P. G. Ciarlet, Introduction to Linear Shell Theory, Series in Applied Mathematics, (North-Holland, Amsterdam,) 1, 1998.


    M. Chipot, $l$ Goes to Plus Infinity, Birkhäuser, 2002.doi: 10.1007/978-3-0348-8173-9.


    M. Chipot, to appear.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    I. Chowdhury and P. Roy, On the asymptotic analysis of problems involving fractional laplacian in cylindrical domains tending to infinity, preprint, http://arxiv.org/abs/1509.06697


    I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland Publishing Co., New York, 1976.


    L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, AMS, 1998.


    L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992.


    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983.doi: 10.1007/978-3-642-61798-0.


    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.


    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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