Global Jacobian and Γ-convergence in a two-dimensional Ginzburg-Landau model for boundary vortices

Global Uniform Estimate for the Modulus of Two-Dimensional Ginzburg-Landau Vortexless Solutions with Asymptotically Infinite Boundary Energy

“…grid R ε , each cell of the grid having the size ∼ ε β with β ∈ (0, 1) (i.e., much larger than the length-scale of a vortex) such that the energy E ε (u ε ; R ε ) on the 1-dimensional grid R ε is of order E ε (u ε ; Ω)/ε β . Then Theorem 1 implies that |u ε | − 1 behaves as a positive power of ε in Ω, and v ε = u ε /|u ε | is a "good" approximation of u ε (in terms of the L 2 norm, their global Jacobian etc., see [8]). In that context, we give the following consequence of Theorem 1 for a cell of the grid leading to a key estimate needed in [8] (only a weaker version of this key estimate was needed in [9,5]):…”

“…Then Theorem 1 implies that |u ε | − 1 behaves as a positive power of ε in Ω, and v ε = u ε /|u ε | is a "good" approximation of u ε (in terms of the L 2 norm, their global Jacobian etc., see [8]). In that context, we give the following consequence of Theorem 1 for a cell of the grid leading to a key estimate needed in [8] (only a weaker version of this key estimate was needed in [9,5]):…”

“…We simply mention here that |u ε | − 1 measures how close the system is to a superconducting state, and refer the interested reader to the monographs [2,14]. Another motivation comes from several studies of the pattern formation in thin ferromagnetic films [9,5,8], where one wishes to approximate u ε by S 1 -valued maps away from the vortices. In a vortexless region Ω (assume e.g.…”

“…As already hinted at, the regime ( 4) is motivated by the study of boundary vortices (see e.g. [12,8]). The typical example is given by the formation of a dipole of two boundary vortices (in the absence of interior vortices).…”

“…Thus, the maximum principle 6 implies that ρε ≥ 1−(1+δ ε ) 2 in Ω, i.e., |u ε | ≤ 1 + δ ε in Ω yielding (7) by (8).…”