Local minimizers of the Ginzburg–Landau functional with prescribed degrees
We consider, in a smooth bounded multiply connected domain D ⊂ R 2 , the Ginzburg–Landau energy E ε ( u ) = 1 2 ∫ D | ∇ u | 2 + 1 4 ε 2 ∫ D ( 1 − | u | 2 ) 2 subject to prescribed degree conditions on each component of ∂ D . In general, minimal energy maps do not exist [L. Berlyand, P. Mironescu, Gi...
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| Published in | Journal of functional analysis Vol. 257; no. 4; pp. 1053 - 1091 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
15.08.2009
Elsevier |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0022-1236 1096-0783 |
| DOI | 10.1016/j.jfa.2009.02.023 |
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| Abstract | We consider, in a smooth bounded multiply connected domain
D
⊂
R
2
, the Ginzburg–Landau energy
E
ε
(
u
)
=
1
2
∫
D
|
∇
u
|
2
+
1
4
ε
2
∫
D
(
1
−
|
u
|
2
)
2
subject to prescribed degree conditions on each component of
∂
D
. In general, minimal energy maps do not exist [L. Berlyand, P. Mironescu, Ginzburg–Landau minimizers in perforated domains with prescribed degrees, preprint, 2004]. When
D
has a single hole, Berlyand and Rybalko [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008,
http://www.math.psu.edu/berlyand/publications/publications.html] proved that for small
ε local minimizers do exist. We extend the result in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008,
http://www.math.psu.edu/berlyand/publications/publications.html]:
E
ε
(
u
)
has, in domains
D
with
2
,
3
,
…
holes and for small
ε, local minimizers. Our approach is very similar to the one in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008,
http://www.math.psu.edu/berlyand/publications/publications.html]; the main difference stems in the construction of test functions with energy control. |
|---|---|
| AbstractList | We consider, in a smooth bounded multiply connected domain $\dom\subset\R^2$, the Ginzburg-Landau energy $\d E_\v(u)=\frac{1}{2}\int_\dom{|\n u|^2}+\frac{1}{4\v^2}\int_\dom{(1-|u|^2)^2}$ subject to prescribed degree conditions on each component of $\p\dom$. In general, minimal energy maps do not exist \cite{BeMi1}. When $\dom$ has a single hole, Berlyand and Rybalko \cite{BeRy1} proved that for small $\v$ local minimizers do exist. We extend the result in \cite{BeRy1}: $\d E_\v(u)$ has, in domains $\dom$ with $2,3,...$ holes and for small $\v$, local minimizers. Our approach is very similar to the one in \cite{BeRy1}; the main difference stems in the construction of test functions with energy control. We consider, in a smooth bounded multiply connected domain D ⊂ R 2 , the Ginzburg–Landau energy E ε ( u ) = 1 2 ∫ D | ∇ u | 2 + 1 4 ε 2 ∫ D ( 1 − | u | 2 ) 2 subject to prescribed degree conditions on each component of ∂ D . In general, minimal energy maps do not exist [L. Berlyand, P. Mironescu, Ginzburg–Landau minimizers in perforated domains with prescribed degrees, preprint, 2004]. When D has a single hole, Berlyand and Rybalko [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html] proved that for small ε local minimizers do exist. We extend the result in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html]: E ε ( u ) has, in domains D with 2 , 3 , … holes and for small ε, local minimizers. Our approach is very similar to the one in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html]; the main difference stems in the construction of test functions with energy control. |
| Author | Dos Santos, Mickaël |
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| Cites_doi | 10.1007/s005260050106 10.1016/j.crma.2006.05.013 10.57262/ade/1355867927 |
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| Keywords | Ginzburg–Landau functional Prescribed degrees Local minimizers Ginzburg-Landau functional local minimizers prescribed degrees |
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| References | André, Shafrir (bib001) 1998; 7 André, Shafrir (bib002) 2004; 9 L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008 Brezis, Bethuel, Hélein (bib007) 1994 Berlyand, Golovaty, Rybalko (bib003) 2006; 343 L. Berlyand, P. Mironescu, Ginzburg–Landau minimizers in perforated domains with prescribed degrees, preprint, 2004 Sandier, Serfaty (bib009) 2007 Tinkham (bib010) 1996 Mironescu (bib008) 1996; 41 Brezis (bib006) 2006; vol. 244 Berlyand (10.1016/j.jfa.2009.02.023_bib003) 2006; 343 André (10.1016/j.jfa.2009.02.023_bib001) 1998; 7 10.1016/j.jfa.2009.02.023_bib005 10.1016/j.jfa.2009.02.023_bib004 Brezis (10.1016/j.jfa.2009.02.023_bib006) 2006; vol. 244 Sandier (10.1016/j.jfa.2009.02.023_bib009) 2007 Mironescu (10.1016/j.jfa.2009.02.023_bib008) 1996; 41 Tinkham (10.1016/j.jfa.2009.02.023_bib010) 1996 Brezis (10.1016/j.jfa.2009.02.023_bib007) 1994 André (10.1016/j.jfa.2009.02.023_bib002) 2004; 9 |
| References_xml | – year: 1996 ident: bib010 article-title: Introduction to Superconductivity – year: 1994 ident: bib007 article-title: Ginzburg–Landau Vortices – volume: 7 start-page: 191 year: 1998 end-page: 217 ident: bib001 article-title: Minimisation of a Ginzburg–Landau type functional with boundary condition which is not publication-title: Calc. Var. Partial Differential Equations – volume: 9 start-page: 891 year: 2004 end-page: 960 ident: bib002 article-title: On the minimizers of a Ginzburg–Landau type energy when the boundary condition has zeros publication-title: Adv. Differential Equations – volume: vol. 244 year: 2006 ident: bib006 article-title: New questions related to the topological degree publication-title: The Unity of Mathematics – volume: 41 start-page: 263 year: 1996 end-page: 271 ident: bib008 article-title: Explicit bounds for solutions to a Ginzburg–Landau type equation publication-title: Rev. Roumaine Math. Pures Appl. – reference: L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, – volume: 343 start-page: 63 year: 2006 end-page: 68 ident: bib003 article-title: Nonexistence of Ginzburg–Landau minimizers with prescribed degree on the boundary of a doubly connected domain publication-title: C. R. Math. Acad. Sci. Paris – year: 2007 ident: bib009 article-title: Vortices in the Magnetic Ginzburg–Landau Model – reference: L. Berlyand, P. Mironescu, Ginzburg–Landau minimizers in perforated domains with prescribed degrees, preprint, 2004 – volume: 7 start-page: 191 year: 1998 ident: 10.1016/j.jfa.2009.02.023_bib001 article-title: Minimisation of a Ginzburg–Landau type functional with boundary condition which is not S1-valued publication-title: Calc. Var. Partial Differential Equations doi: 10.1007/s005260050106 – ident: 10.1016/j.jfa.2009.02.023_bib005 – year: 1996 ident: 10.1016/j.jfa.2009.02.023_bib010 – volume: vol. 244 year: 2006 ident: 10.1016/j.jfa.2009.02.023_bib006 article-title: New questions related to the topological degree – year: 2007 ident: 10.1016/j.jfa.2009.02.023_bib009 – volume: 41 start-page: 263 year: 1996 ident: 10.1016/j.jfa.2009.02.023_bib008 article-title: Explicit bounds for solutions to a Ginzburg–Landau type equation publication-title: Rev. Roumaine Math. Pures Appl. – volume: 343 start-page: 63 year: 2006 ident: 10.1016/j.jfa.2009.02.023_bib003 article-title: Nonexistence of Ginzburg–Landau minimizers with prescribed degree on the boundary of a doubly connected domain publication-title: C. R. Math. Acad. Sci. Paris doi: 10.1016/j.crma.2006.05.013 – volume: 9 start-page: 891 year: 2004 ident: 10.1016/j.jfa.2009.02.023_bib002 article-title: On the minimizers of a Ginzburg–Landau type energy when the boundary condition has zeros publication-title: Adv. Differential Equations doi: 10.57262/ade/1355867927 – ident: 10.1016/j.jfa.2009.02.023_bib004 – year: 1994 ident: 10.1016/j.jfa.2009.02.023_bib007 |
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| Snippet | We consider, in a smooth bounded multiply connected domain
D
⊂
R
2
, the Ginzburg–Landau energy
E
ε
(
u
)
=
1
2
∫
D
|
∇
u
|
2
+
1
4
ε
2
∫
D
(
1
−
|
u
|
2
)
2... We consider, in a smooth bounded multiply connected domain $\dom\subset\R^2$, the Ginzburg-Landau energy $\d E_\v(u)=\frac{1}{2}\int_\dom{|\n... |
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| StartPage | 1053 |
| SubjectTerms | Analysis of PDEs Ginzburg–Landau functional Local minimizers Mathematics Prescribed degrees |
| Title | Local minimizers of the Ginzburg–Landau functional with prescribed degrees |
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