# zbMATH — the first resource for mathematics

On the energy of type-II superconductors in the mixed phase. (English) Zbl 0964.49006
The paper deals with the asymptotic study of minimizers of the Ginzburg-Landau energy functional with applied magnetic field. The framework corresponds to high values of the Ginzburg-Landau parameter $$\kappa$$ and to a prescribed external constant magnetic field $$h_{ex}$$, with $$h_{ex}$$ varying between the critical fields $$H_{c_1}$$ and $$H_{c_3}$$. The main result describes the vortex structure of minimizers of the Ginzburg-Landau energy as $$\kappa\rightarrow +\infty$$. More precisely, the authors give the leading term in the asymptotic expansion of the minimal energy and show that energy minimizers have vortices whose density tends to be uniform and equal to $$h_{ex}$$. The proofs are based on refined elliptic estimates and they combine various upper and lower bounds of the energy on “good” and “bad” subdomains.

##### MSC:
 49J35 Existence of solutions for minimax problems 58E50 Applications of variational problems in infinite-dimensional spaces to the sciences 82D55 Statistical mechanics of superconductors 35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text:
##### References:
 [1] Abrikosov A., Soviet Phys. JETP 5 pp 1174– (1957) [2] DOI: 10.1016/S0021-7824(98)80064-0 · Zbl 0904.35023 [3] DOI: 10.1137/0524073 · Zbl 0783.35018 [4] DOI: 10.1103/PhysRev.108.1175 · Zbl 0090.45401 [5] Bethuel F., Annales IHP, Analyse non linéaire 12 pp 243– (1995) [6] Comte M., C. R. Acad. Sci., Paris, Ser. I 320 (3) pp 289– (1995) [7] DOI: 10.1007/BF01190822 · Zbl 0869.35036 [8] DOI: 10.1137/S0036141097323163 · Zbl 0920.35058 [9] DOI: 10.1137/S0036141097300581 · Zbl 0928.35045 [10] Mironescu P., C. R. Acad. Sci., Paris, Ser. I 323 (6) pp 593– (1996) [11] DOI: 10.1006/jfan.1997.3170 · Zbl 0908.58004 [12] DOI: 10.1016/S0764-4442(98)80120-1 · Zbl 0913.35134 [13] DOI: 10.1142/S0219199799000109 · Zbl 0944.49007 [14] DOI: 10.1142/S0219199799000134 · Zbl 0964.49005 [15] DOI: 10.1007/s002050050177 · Zbl 0959.35154
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.