Advances in Differential Equations

Homogenization of harmonic maps with large number of vortices and applications in superconductivity and superfluidity

Leonid Berlyand and Evgenii Khruslov

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We study a nonlinear homogenization problem of harmonic maps which describes an ideal superconducting or an ideal superfluid medium with a large number of vortices and the degree conditions prescribed at the external insulating boundary. We derive the homogenized problem which describes the limiting behavior of the fluxes when the total number of vortices tends to infinity. The homogenized problem is described in terms of the effective vorticity and the effective anisotropy tensor. The calculation of this tensor amounts to solving a linear cell problem, which is well studied in the homogenization literature and can be solved by using existing numerical packages. The convergence of the fluxes is rigorously proved. We also discuss unusual features of the homogenized limit for the wave functions. The proofs are based on a variational approach which does not require periodicity and can be used in more general situations.

Article information

Adv. Differential Equations, Volume 6, Number 2 (2001), 229-256.

First available in Project Euclid: 2 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50]
Secondary: 35J60: Nonlinear elliptic equations 58E20: Harmonic maps [See also 53C43], etc. 82D50: Superfluids 82D55: Superconductors


Berlyand, Leonid; Khruslov, Evgenii. Homogenization of harmonic maps with large number of vortices and applications in superconductivity and superfluidity. Adv. Differential Equations 6 (2001), no. 2, 229--256.

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