Ginzburg Landau minimizers in perforated domains with prescribed degrees

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Niveau: Supérieur, Doctorat, Bac+8
Ginzburg-Landau minimizers in perforated domains with prescribed degrees Leonid Berlyand(1), Petru Mironescu(2) October 2004, with an update in June 2008 Abstract. Suppose that ? is a 2D domain with holes ?0, ?1, . . . , ?j, j = 1...k. In the perforated domain A = ? \ ( ?kj=0 ?j) we consider the class J of complex valued maps having degrees 1 and ?1 on the boundaries ∂?, ∂?0 respectively and degree 0 on the boundaries of other holes. We investigate whether the minimum of the Ginzburg-Landau energy E? is attained in J , as well as the asymptotic behavior of minimizers as the coherency length ??1 tends to 0. We show that the answer to these questions is determined by the value of the H1-capacity cap(A) of the domain. If cap(A) ≥ pi (domain A is ”thin”), minimizers exist for each ?. Moreover they are vortexless and converge in H1(A) (and even better) to a minimizing S1-valued harmonic map as ? ? ∞. When cap(A) < pi (domain A is ”thick”), we establish existence of quasi-minimizers (maps with “almost minimal energy”), which exhibit a different qualitative behavior : they have exactly two zeroes (vortices) rapidly converging to ∂A as ? ? ∞ .

  • has has

  • asymptotic behavior

  • complex-valued maps

  • loops around

  • dirichlet boundary

  • ∂v ∂?

  • physical insulating

  • ginzburg-landau energy

  • maps


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