doi:
10.3934/dcds.2009.24.933 
Full text:
(388.1K)
Xing-Bin Pan -
Department of Mathematics, East China Normal University, Shanghai 200062, China (email)
Abstract:
This paper concerns the lowest eigenvalue $\mu(b\N^Q)$ of the
Schrödinger operator in three-dimensions with a magnetic potential
$b\N^Q$, where the vector field $\N^Q$ depends on a matrix $Q$
varying in $SO(3)$ and $b$ is a real parameter. The eigenvalue
variation problem is to minimize the lowest eigenvalue among all $Q$
in $SO(3)$. This problem arises in the phase transitions of smectic
liquid crystals. We give an estimate of the minimum value
inf${\mu(b\N^Q):~Q\in SO(3)\}$ for large $b$, and examine its
dependence on geometry of the domain surface.
Keywords: liquid crystal, magnetic Schrödinger operator, lowest
eigenvalue, eigenvalue variation, Landau-de Gennes model, critical
wave number.
Mathematics Subject Classification: Primary: 82D30, 82D55; Secondary: 35J10, 35P15, 35Q55.
Received: August 2007;
Revised:
June 2008;
Published: April 2009.
