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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium
Luisa Mascarenhas
Univ. Nova de Lisboa
Title: Waveguides with Robin conditions

Abstract: In a previous paper the authors analyzed the 3D-1D asymptotic behavior of the spectral problem for the Laplace operator, with homogeneous Dirichlet boundary conditions. A possible physical motivation for this problem is the understanding of the behavior of the probability density associated with the wave function of a particle confined in a thin waveguide. The study revealed very interesting effects on the energy levels, caused by the geometrical characteristics of the thin domain, such as the curvature and the torsion. In the present paper we analyze the Laplace operator under more general boundary conditions (Robin conditions) more precisely,

$$ \left\{ \begin{array}{ll} -\Delta u_{\epsilon} = \lambda_{\epsilon}u_{\epsilon}, & \textrm{ in } \Omega_{\epsilon} \\ \frac{\partial u_{\epsilon}}{\partial n_{\epsilon}} +\gamma_{\epsilon}u_{\epsilon}=0, & \textrm{ on } \partial \Omega_{\epsilon}, \end{array} \right. $$

where $\epsilon$ is a small positive parameter, $\Omega_{\epsilon} \subset \mathbb{R}^3$ is a thin and long domain generated by a cross section $\omega_{\epsilon} (\omega \subset \mathbb{R}^2)$ which rotates along a curve through an angle $\alpha(s)$ with respect to the Frenet frame, and $\gamma_{\epsilon} \in L^{\infty}(\partial \omega)$.

Two rather distinct situations may emerge: if some geometric conditions are satisfied, then we will obtain a 1D limit problem, with torsion and curvature effects generalizing, for the Robin conditions, the results previously obtained. If, on the contrary, those geometric conditions are not satisfied, then a localization phenomena appears around the minimum point of a certain function depending on geometric parameters and on both the curvature and the rotation angle of the waveguide's cross section.

Recording: http://vnc.math.cmu.edu/cna/CNA-Mascarenhas-Apr-21-2011.avi
Date: Thursday, April 21, 2011
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  David Kinderlehrer