Study of a 3D-Ginzburg–Landau functional with a discontinuous pinning term
In a convex domain Ω⊂R3, we consider the minimization of a 3D-Ginzburg–Landau type energy Eε(u)=12∫Ω|∇u|2+12ε2(a2−|u|2)2 with a discontinuous pinning term a among H1(Ω,C)-maps subject to a Dirichlet boundary condition g∈H1/2(∂Ω,S1). The pinning term a:R3→R+∗ takes a constant value b∈(0,1) in ω, an i...
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| Published in | Nonlinear analysis Vol. 75; no. 17; pp. 6275 - 6296 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
01.11.2012
Elsevier |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0362-546X 1873-5215 |
| DOI | 10.1016/j.na.2012.07.004 |
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| Abstract | In a convex domain Ω⊂R3, we consider the minimization of a 3D-Ginzburg–Landau type energy Eε(u)=12∫Ω|∇u|2+12ε2(a2−|u|2)2 with a discontinuous pinning term a among H1(Ω,C)-maps subject to a Dirichlet boundary condition g∈H1/2(∂Ω,S1). The pinning term a:R3→R+∗ takes a constant value b∈(0,1) in ω, an inner strictly convex subdomain of Ω, and 1 outside ω. We prove energy estimates with various error terms depending on assumptions on Ω,ω and g. In some special cases, we identify the vorticity defects via the concentration of the energy. Under hypotheses on the singularities of g (the singularities are polarized and quantified by their degrees which are ±1), vorticity defects are geodesics (computed w.r.t. a geodesic metric da2 depending only on a) joining two paired singularities of gpi&nσ(i) where σ is a minimal connection (computed w.r.t. a metric da2) of the singularities of g and p1,…,pk are the positive (resp. n1,…,nk are the negative) singularities. |
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| AbstractList | In a convex domain $\O\subset\R^3$, we consider the minimization of a $3D$-Ginzburg-Landau type energy $E_\v(u)=\frac{1}{2}\int_\O|\n u|^2+\frac{1}{2\v^2}(a^2-|u|^2)^2$ with a discontinuous pinning term $a$ among $H^1(\O,\C)$-maps subject to a Dirichlet boundary condition $g\in H^{1/2}(\p\O,\S^1)$. The pinning term $a:\R^3\to\R^*_+$ takes a constant value $b\in(0,1)$ in $\o$, an inner strictly convex subdomain of $\O$, and $1$ outside $\o$. We prove energy estimates with various error terms depending on assumptions on $\O,\o$ and $g$. In some special cases, we identify the vorticity defects via the concentration of the energy. Under hypotheses on the singularities of $g$ (the singularities are polarized and quantified by their degrees which are $\pm 1$), vorticity defects are geodesics (computed w.r.t. a geodesic metric $d_{a^2}$ depending only on $a$) joining two paired singularities of $g$ $p_i\& n_{\sigma(i)}$ where $\sigma$ is a minimal connection (computed w.r.t. a metric $d_{a^2}$) of the singularities of $g$ and $p_1,...p_k$ are the positive (resp. $n_1,...,n_k$ the negative) singularities. In a convex domain Ω⊂R3, we consider the minimization of a 3D-Ginzburg–Landau type energy Eε(u)=12∫Ω|∇u|2+12ε2(a2−|u|2)2 with a discontinuous pinning term a among H1(Ω,C)-maps subject to a Dirichlet boundary condition g∈H1/2(∂Ω,S1). The pinning term a:R3→R+∗ takes a constant value b∈(0,1) in ω, an inner strictly convex subdomain of Ω, and 1 outside ω. We prove energy estimates with various error terms depending on assumptions on Ω,ω and g. In some special cases, we identify the vorticity defects via the concentration of the energy. Under hypotheses on the singularities of g (the singularities are polarized and quantified by their degrees which are ±1), vorticity defects are geodesics (computed w.r.t. a geodesic metric da2 depending only on a) joining two paired singularities of gpi&nσ(i) where σ is a minimal connection (computed w.r.t. a metric da2) of the singularities of g and p1,…,pk are the positive (resp. n1,…,nk are the negative) singularities. |
| Author | Dos Santos, Mickaël |
| Author_xml | – sequence: 1 givenname: Mickaël surname: Dos Santos fullname: Dos Santos, Mickaël email: mickael.dos-santos@u-pec.fr organization: Université Paris-Est, LAMA–CNRS UMR 8050, 61, Avenue du Général de Gaulle, 94010 Créteil, France |
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| Keywords | 35J50 Vorticity defects Pinning 35B40 Energy minimization Ginzburg–Landau energy 35J55 |
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| References | Bethuel, Brezis, Orlandi (br000035) 2000; 331 Dos Santos, Misiats (br000045) 2011; 6 Lassoued, Mironescu (br000005) 1999; 77 Lin, Rivière (br000015) 1999; 1 Brezis, Coron, Lieb (br000025) 1986; 107 Brezis (br000050) 2006; vol. 244 Sandier (br000010) 2001; 50 Brezis (br000055) 1987; vol. 5 Bourgain, Brezis, Mironescu (br000020) 2004; 99 Lin, Rivière (br000030) 2001; 54 Liu, Zhou (br000040) 2010; 11 Demazure (br000060) 2000 |
| References_xml | – volume: 77 start-page: 1 year: 1999 end-page: 26 ident: br000005 article-title: Ginzburg–Landau type energy with discontinuous constraint publication-title: J. Anal. Math. – volume: 99 start-page: 1 year: 2004 end-page: 115 ident: br000020 article-title: maps with values into the circle: minimal connections, lifting, and the Ginzburg–Landau equation publication-title: Publ. Math. Inst. Hautes Études Sci. – volume: 50 start-page: 1807 year: 2001 end-page: 1844 ident: br000010 article-title: Ginzburg–Landau minimizers from publication-title: Indiana Univ. Math. J. – volume: 1 start-page: 237 year: 1999 end-page: 311 ident: br000015 article-title: Complex Ginzburg–Landau equations in high dimensions and codimension two area minimizing currents publication-title: J. Eur. Math. Soc. (JEMS) – volume: 107 start-page: 649 year: 1986 end-page: 705 ident: br000025 article-title: Harmonic maps with defects publication-title: Comm. Math. Phys. – volume: 11 start-page: 1046 year: 2010 end-page: 1060 ident: br000040 article-title: Vortex-filaments for inhomogeneous superconductors in three dimensions publication-title: Nonlinear Anal. RWA – volume: 54 start-page: 206 year: 2001 end-page: 228 ident: br000030 article-title: A quantization property for static Ginzburg–Landau vortices publication-title: Comm. Pure Appl. Math. – year: 2000 ident: br000060 publication-title: Bifurcations and Catastrophes – volume: 6 year: 2011 ident: br000045 article-title: Ginzburg–Landau model with small pinning domains publication-title: Netw. Heterog. Media – volume: vol. 5 start-page: 31 year: 1987 end-page: 52 ident: br000055 article-title: Liquid crystals and energy estimates for publication-title: Theory and Applications of Liquid Crystals (Minneapolis, Minn., 1985) – volume: 331 start-page: 763 year: 2000 end-page: 770 ident: br000035 article-title: Small energy solutions to the Ginzburg–Landau equation publication-title: C. R. Acad. Sci. Paris Sér. I Math. – volume: vol. 244 start-page: 137 year: 2006 end-page: 154 ident: br000050 article-title: New questions related to the topological degree publication-title: The Unity of Mathematics |
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| Snippet | In a convex domain Ω⊂R3, we consider the minimization of a 3D-Ginzburg–Landau type energy Eε(u)=12∫Ω|∇u|2+12ε2(a2−|u|2)2 with a discontinuous pinning term a... In a convex domain $\O\subset\R^3$, we consider the minimization of a $3D$-Ginzburg-Landau type energy $E_\v(u)=\frac{1}{2}\int_\O|\n... |
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| SubjectTerms | Analysis of PDEs Energy minimization Ginzburg–Landau energy Mathematics Pinning Vorticity defects |
| Title | Study of a 3D-Ginzburg–Landau functional with a discontinuous pinning term |
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