Higher Values of the Applied Field

Abstract

The previous chapter dealt with minimizers of the Ginzburg-Landau functional when the applied field was O(|log ε|). The applied field behaving asymptotically like λ|log ε|, letting λ → ∞ in Theorem 7.2 indicates that for energy-minimizers for applied fields h_ex ≫ |log ε|, we must have $\frac{{\mu \left( {u_\varepsilon ,A_\varepsilon } \right)}} {{h_{ex} }} \to 1, and \frac{{h_\varepsilon }} {{h_{ex} }} \to 1$ . But in this regime, $\frac{{G_\varepsilon \left( {u_\varepsilon ,A_\varepsilon } \right)}} {{h_{ex} ^2 }} \to 0$ and the arguments of Chapter 7 do not give, even formally, the leading order term of the minimal energy. Moreover, the tools which were at the heart of the result, namely the vortex balls construction of Theorem 4.1 and the Jacobian estimate of Theorem 6.1 break down for higher values of h_ex.