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On the Ginzburg‐Landau critical field in three dimensions
Author(s) -
Fournais S.,
Helffer B.
Publication year - 2009
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20247
Subject(s) - monotonic function , mathematics , ground state , critical field , laplace operator , superconductivity , field (mathematics) , magnetic field , state (computer science) , neumann boundary condition , conjecture , energy (signal processing) , constant (computer programming) , function (biology) , boundary value problem , condensed matter physics , mathematical physics , mathematical analysis , pure mathematics , physics , quantum mechanics , statistics , algorithm , evolutionary biology , computer science , programming language , biology
We study the three‐dimensional Ginzburg‐Landau model of superconductivity. Several “natural” definitions of the (third) critical field, H C 3, governing the transition from the superconducting state to the normal state, are considered. We analyze the relation between these fields and give conditions as to when they coincide. An interesting part of the analysis is the study of the monotonicity of the ground state energy of the Laplacian with constant magnetic field and with Neumann (magnetic) boundary condition in a domain Ω. It is proved that the ground state energy is a strictly increasing function of the field strength for sufficiently large fields. As a consequence of our analysis, we give an affirmative answer to a conjecture by Pan. © 2008 Wiley Periodicals, Inc.