On some variational problems set on domains tending to infinity

Michel Chipot; Aleksandar  Mojsic; Prosenjit  Roy

doi:10.3934/dcds.2016.36.3603

Article Contents

2016, Volume 36, Issue 7: 3603-3621. Doi: 10.3934/dcds.2016.36.3603

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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
3.
Tata Institute of Fundamental Research- CAM, Sharadanagar, GKVK Campus, Postbox - 560065, Bangalore

Received: March 31, 2015

Revised: December 31, 2015

Published: May 2017

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Abstract

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.

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

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