Abstract
We analyze the Ginzburg–Landau energy in the presence of an applied magnetic field when the superconducting sample occupies a thin neighborhood of a bounded, closed manifold in \({\mathbb R^3}\). We establish Γ-convergence to a reduced Ginzburg–Landau model posed on the manifold in which the magnetic potential is replaced in the limit by the tangential component of the applied magnetic potential. We then study the limiting problem, constructing two-vortex critical points when the manifold \({\mathcal{M}}\) is a simply connected surface of revolution and the applied field is constant and vertical. Finally, we calculate that the exact asymptotic value of the first critical field H c1 is simply (4π/(area of \({\mathcal{M}}\))) ln κ for large values of the Ginzburg–Landau parameter κ. Merging this with the Γ-convergence result, we also obtain the same asymptotic value for H c1 in 3d valid for large κ and sufficiently thin shells.
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Contreras, A., Sternberg, P. Gamma-convergence and the emergence of vortices for Ginzburg–Landau on thin shells and manifolds. Calc. Var. 38, 243–274 (2010). https://doi.org/10.1007/s00526-009-0285-7
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DOI: https://doi.org/10.1007/s00526-009-0285-7