Ginzburg-Landau model with small pinning domains

  • Received: 01 March 2011 Revised: 01 October 2011
  • Primary: 49K20, 35J66, 35J50; Secondary: 47H11.

  • We consider a Ginzburg-Landau type energy with a piecewise constant pinning term $a$ in the potential $(a^2 - |u|^2)^2$. The function $a$ is different from 1 only on finitely many disjoint domains, called the pinning domains. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit $\epsilon\to0$; here, $\epsilon$ is the inverse of the Ginzburg-Landau parameter. We study the energy minimization in a smooth simply connected domain $\Omega \subset \mathbb{C}$ with Dirichlet boundary condition $g$ on $\partial \Omega$, with topological degree ${\rm deg}_{\partial \Omega} (g) = d >0$. Our main result is that, for small $\epsilon$, minimizers have $d$ distinct zeros (vortices) which are inside the pinning domains and they have a degree equal to $1$. The question of finding the locations of the pinning domains with vortices is reduced to a discrete minimization problem for a finite-dimensional functional of renormalized energy. We also find the position of the vortices inside the pinning domains and show that, asymptotically, this position is determined by local renormalized energy which does not depend on the external boundary conditions.

    Citation: Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains[J]. Networks and Heterogeneous Media, 2011, 6(4): 715-753. doi: 10.3934/nhm.2011.6.715

    Related Papers:

    [1] Mickaël Dos Santos, Oleksandr Misiats . Ginzburg-Landau model with small pinning domains. Networks and Heterogeneous Media, 2011, 6(4): 715-753. doi: 10.3934/nhm.2011.6.715
    [2] Leonid Berlyand, Volodymyr Rybalko . Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes. Networks and Heterogeneous Media, 2013, 8(1): 115-130. doi: 10.3934/nhm.2013.8.115
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  • We consider a Ginzburg-Landau type energy with a piecewise constant pinning term $a$ in the potential $(a^2 - |u|^2)^2$. The function $a$ is different from 1 only on finitely many disjoint domains, called the pinning domains. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit $\epsilon\to0$; here, $\epsilon$ is the inverse of the Ginzburg-Landau parameter. We study the energy minimization in a smooth simply connected domain $\Omega \subset \mathbb{C}$ with Dirichlet boundary condition $g$ on $\partial \Omega$, with topological degree ${\rm deg}_{\partial \Omega} (g) = d >0$. Our main result is that, for small $\epsilon$, minimizers have $d$ distinct zeros (vortices) which are inside the pinning domains and they have a degree equal to $1$. The question of finding the locations of the pinning domains with vortices is reduced to a discrete minimization problem for a finite-dimensional functional of renormalized energy. We also find the position of the vortices inside the pinning domains and show that, asymptotically, this position is determined by local renormalized energy which does not depend on the external boundary conditions.


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    7. Leonid Berlyand, Vladimir Mityushev, Shawn D Ryan, Multiple Ginzburg–Landau vortices pinned by randomly distributed small holes, 2018, 0272-4960, 10.1093/imamat/hxy033
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