Abstract
We study the Ginzburg–Landau functional describing an extreme type-II superconductor wire with cross section with finitely many corners at the boundary. We derive the ground state energy asymptotics up to o(1) errors in the surface superconductivity regime, i.e., between the second and third critical fields. We show that, compared to the case of smooth domains, each corner provides an additional contribution of order {mathcal {O}}(1) depending on the corner opening angle. The corner energy is in turn obtained from an implicit model problem in an infinite wedge-like domain with fixed magnetic field. We also prove that such an auxiliary problem is well-posed and its ground state energy bounded and, finally, state a conjecture about its explicit dependence on the opening angle of the sector.
Highlights
The phenomenon of conventional superconductivity is nowadays very well understood at the microscopic level thanks to the Bardeen–Cooper–Schrieffer (BCS) theory [8]: a collective behavior of the current carriers in the material is responsible for a sudden drop of the resistivity below a certain critical temperature
It is astonishing how a phenomenological model as the Ginzburg– Landau (GL) theory [42] is capable of predicting most of the key equilibrium features of the phenomenon, in particular concerning the response of the superconducting material to an external field
We identify the model problem which yields such a new contribution in terms of a genuine 2D model
Summary
The phenomenon of conventional superconductivity (see, e.g., [55] for a review of the physics of superconductors) is nowadays very well understood at the microscopic level thanks to the Bardeen–Cooper–Schrieffer (BCS) theory [8]: a collective behavior of the current carriers in the material is responsible for a sudden drop of the resistivity below a certain critical temperature It is astonishing how a phenomenological model as the Ginzburg– Landau (GL) theory [42] is capable of predicting most of the key equilibrium features of the phenomenon, in particular concerning the response of the superconducting material to an external field. Under this idealization, one can identify the mathematical counterparts of the critical values of the external magnetic field described above in terms of properties of the minimizing configuration (ψGL, AGL) and it is possible to precisely identify the behavior of such thresholds (see, e.g., [53] for an extensive discussion of the first phase transition). Based on sharp estimates (Agmon estimates) of the decay of ψGL in the distance from the boundary (see “Appendix B.3”); the third critical field marking the transition to the normal
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