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Transactions of the American Mathematical Society

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Gauge Invariant Eigenvalue Problems
in ${\mathbb{R}}^{2}$ and in ${\mathbb{R}}^{2}_{+}$

Authors: Kening Lu and Xing-Bin Pan
Journal: Trans. Amer. Math. Soc. 352 (2000), 1247-1276
MSC (1991): Primary 82D55
Published electronically: October 6, 1999
MathSciNet review: 1675206
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Abstract: This paper is devoted to the study of the eigenvalue problems for the Ginzburg-Landau operator in the entire plane ${\mathbb{R}}^{2}$ and in the half plane ${\mathbb{R}}^{2}_{+}$. The estimates for the eigenvalues are obtained and the existence of the associate eigenfunctions is proved when $curl\ A$ is a non-zero constant. These results are very useful for estimating the first eigenvalue of the Ginzburg-Landau operator with a gauge-invariant boundary condition in a bounded domain, which is closely related to estimates of the upper critical field in the theory of superconductivity.

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Additional Information

Kening Lu
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

Xing-Bin Pan
Affiliation: Center for Mathematical Sciences, Zhejiang University, Hangzhou 310027, P.R. China; Department of Mathematics, National University of Singapore, Singapore

Keywords: Superconductivity, Ginzburg-Landau operator, eigenvalue
Received by editor(s): November 1, 1996
Received by editor(s) in revised form: December 18, 1997
Published electronically: October 6, 1999
Article copyright: © Copyright 1999 American Mathematical Society

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