Higher Values of the Applied Field

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 70)


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 hex ≫ |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 hex.


Minimal Energy Applied Field Weak Sense Small Ball Landau Equation 
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© Birkhäuser Boston 2007

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