TY - JOUR
T1 - Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation
AU - Berlyand, L.
AU - Rybalko, V.
N1 - Copyright:
Copyright 2010 Elsevier B.V., All rights reserved.
PY - 2010
Y1 - 2010
N2 - We study solutions of the 2D Ginzburg-Landau equation - Δu +1/ε2u(|u|2 - 1) = 0 subject to "semi- stiff" boundary conditions: Dirichlet conditions for the modulus, |u| = 1, and homogeneous Neumann conditions for the phase. The principal result of this work shows that there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in the limit of small ε. For the Dirichlet boundary condition ("stiff" problem), the existence of stable solutions with vortices, whose energy blows up as ε → 0, is well known. By contrast, stable solutions with vortices are not established in the case of the homogeneous Neumann ("soft") boundary condition. In this work, we develop a variational method which allows one to construct local minimizers of the corresponding Ginzburg-Landau energy functional. We introduce an approximate bulk degree as the key ingredient of this method; unlike the standard degree over the curve, it is preserved in the weak H1-limit.
AB - We study solutions of the 2D Ginzburg-Landau equation - Δu +1/ε2u(|u|2 - 1) = 0 subject to "semi- stiff" boundary conditions: Dirichlet conditions for the modulus, |u| = 1, and homogeneous Neumann conditions for the phase. The principal result of this work shows that there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in the limit of small ε. For the Dirichlet boundary condition ("stiff" problem), the existence of stable solutions with vortices, whose energy blows up as ε → 0, is well known. By contrast, stable solutions with vortices are not established in the case of the homogeneous Neumann ("soft") boundary condition. In this work, we develop a variational method which allows one to construct local minimizers of the corresponding Ginzburg-Landau energy functional. We introduce an approximate bulk degree as the key ingredient of this method; unlike the standard degree over the curve, it is preserved in the weak H1-limit.
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U2 - 10.4171/JEMS/239
DO - 10.4171/JEMS/239
M3 - Article
AN - SCOPUS:77956962963
SN - 1435-9855
VL - 12
SP - 1497
EP - 1531
JO - Journal of the European Mathematical Society
JF - Journal of the European Mathematical Society
IS - 6
ER -