# Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation

Leonid Berlyand; Volodymyr Rybalko

Journal of the European Mathematical Society (2010)

- Volume: 012, Issue: 6, page 1497-1531
- ISSN: 1435-9855

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topBerlyand, Leonid, and Rybalko, Volodymyr. "Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation." Journal of the European Mathematical Society 012.6 (2010): 1497-1531. <http://eudml.org/doc/277301>.

@article{Berlyand2010,

abstract = {We study solutions of the 2D Ginzburg–Landau equation $-\Delta u+\varepsilon ^\{-2\}u(|u|^2-1)=0$ 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 $\varepsilon $. For the Dirichlet boundary condition (“stiff” problem), the existence of stable solutions with vortices, whose energy blows up as $0$, 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 $H^1$-limit.},

author = {Berlyand, Leonid, Rybalko, Volodymyr},

journal = {Journal of the European Mathematical Society},

keywords = {Ginzburg-Landau equation; Ginzburg-Landau energy functional; Dirichlet condition; Neumann condition; stable solution; Ginzburg-Landau equation; Ginzburg-Landau energy functional; Dirichlet condition; Neumann condition; stable solution},

language = {eng},

number = {6},

pages = {1497-1531},

publisher = {European Mathematical Society Publishing House},

title = {Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation},

url = {http://eudml.org/doc/277301},

volume = {012},

year = {2010},

}

TY - JOUR

AU - Berlyand, Leonid

AU - Rybalko, Volodymyr

TI - Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation

JO - Journal of the European Mathematical Society

PY - 2010

PB - European Mathematical Society Publishing House

VL - 012

IS - 6

SP - 1497

EP - 1531

AB - We study solutions of the 2D Ginzburg–Landau equation $-\Delta u+\varepsilon ^{-2}u(|u|^2-1)=0$ 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 $\varepsilon $. For the Dirichlet boundary condition (“stiff” problem), the existence of stable solutions with vortices, whose energy blows up as $0$, 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 $H^1$-limit.

LA - eng

KW - Ginzburg-Landau equation; Ginzburg-Landau energy functional; Dirichlet condition; Neumann condition; stable solution; Ginzburg-Landau equation; Ginzburg-Landau energy functional; Dirichlet condition; Neumann condition; stable solution

UR - http://eudml.org/doc/277301

ER -

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