Almost flat angles in surface superconductivity

Abstract

Type-II superconductivity is known to persist close to the sample surface in presence of a strong magnetic field. As a consequence, the ground state energy in the Ginzburg–Landau theory is approximated by an effective one-dimensional model. As shown by Correggi and Giacomelli (2021 Calc. Var. Partial Differential Equations in press), the presence of corners on the surface affects the energy of the sample with a non-trivial contribution. In (Correggi and Giacomelli 2021 Calc. Var. Partial Differential Equations in press), the two-dimensional model problem providing the corner energy is implicitly identified and, although no explicit dependence of the energy on the corner opening angle is derived, a conjecture about its form is proposed. We study here such a conjecture and confirm it, at least to leading order, for corners with almost flat opening angle.

Effects of Corners in Surface Superconductivity

“…As already anticipated in a companion paper [CG2], we prove that in a wedge with opening angle π − δ, 0 < δ ≪ 1, the corner energy is given by…”

“…It is however important to notice that at this stage E corner,β might as well be zero. In a companion paper [CG2] however we prove that, when β is close to π, this is not the case (see also below).…”

Surface Effects in Superconductors with Corners

“…Remark 3.2 (Almost flat angles). Despite a lack of a proof of Conjecture 1, an interesting result which makes it stronger is proven in the companion paper [CG3]: we show that in a domain with a corner of opening angle π ± δ, with 0 < δ ≪ 1, the corner energy can be expanded as follows:…”

Discrete spectrum of the magnetic Laplacian on perturbed half‐planes

“…In both situations, the proof follows by construction of an appropriate trial state. In the corner case, the phase of the trial state (see (3.5)) is reminiscent of the construction appearing in the nonlinear setting of [4], while the amplitude (see (3.11)) is obtained by minimizing a new energy functional in (3.16). In the regular case, we use a tensorized trial state in curvilinear coordinates (see (4.1)) involving the bound state of a 1D model operator studied in [10] and revisited in Appendix A.…”