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Abstract

When hexH c1 0 i.e., \( \frac{{h_{ex} }} {{\left| {log \varepsilon } \right|}} \to \lambda = \frac{1} {{2\left| {\underline {\xi _0 } } \right|}} \) , then from Theorem 7.2, we get that the limiting minimizer is h0 hence μ* = 0. Moreover, comparing the lower bounds (7.58) and (7.59) to the upper bound of Theorem 7.1, we find \( \frac{{\sum\nolimits_i {\left| {d_i } \right|} }} {{h_{ex} }} \to 0 \) , which means that the number of vortices is o(hex). In other words, for energy-minimizers, vortices first appear for \( \frac{{h_{ex} }} {{\left| {log \varepsilon } \right|}} \to \frac{1} {{2\left| {\underline {\xi _0 } } \right|}} \) , or hexH c1 0 , and next to Λ (defined in (7.5)), and the vorticity mass is much smaller than hex. The analysis of Chapter 7 does not give us the optimal number n of vortices nor the full asymptotic expansion of the first critical field. Thus, a more detailed study will be necessary in this regime \( h_{ex} \sim \frac{{\left| {\left. {log \varepsilon } \right|} \right.}} {{2\left| {\left. {\xi _0 } \right|} \right.}} \) , in which nhex. We will prove that the vortices, even though their number may be diverging, all concentrate around Λ (generically a single point) but that after a suitable blow-up, they tend to arrange in a uniform density on a subdomain of ℝ2, in order to minimize a limiting interaction energy I defined on probability measures.

Keywords

Probability Measure Small Ball Landau Equation Intermediate Regime Large Ball 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2007

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