Ginzburg-Landau Vortices

I. Energy estimates for S1-valued maps.- 1. An auxiliary linear problem.- 2. Variants of Theorem I.1.- 3. S1-valued harmonic maps with prescribed isolated singularities. The canonical harmonic map.- 4. Shrinking holes. Renormalized energy.- II. A lower bound for the energy of S1-valued maps on perforated domains.- III. Some basic estimates for u?.- 1. Estimates when G=BR and g(x)=x/|x|.- 2. An upper bound for E? (u?).- 3. An upper bound for $$ \frac{1}{{{\varepsilon^2}}}{\smallint_G}{\left( {{{\left| {{u_{\varepsilon }}} \right|}^2} - 1} \right)^2} $$.- 4. $$ \left| {{u_e}} \right| \geqslant \frac{1}{2} $$ on "good discs".- IV. Towards locating the singularities: bad discs and good discs.- 1. A covering argument.- 2. Modifying the bad discs.- V. An upper bound for the energy of u? away from the singularities.- 1. A lower bound for the energy of u? near aj.- 2. Proof of Theorem V.l.- VI. u?n converges: u? is born!.- 1. Proof of Theorem VI.1.- 2. Further properties of u? : singularities have degree one and they are not on the boundary.- VII. u? coincides with THE canonical harmonic map having singularities (aj).- VIII. The configuration (aj) minimizes the renormalized energy W.- 1. The general case.- 2. The vanishing gradient property and its various forms.- 3. Construction of critical points of the renormalized energy.- 4. The case G=B1 and $$ g\left( \theta \right) = {e^{{i\theta }}} $$.- 5. The case G=B1 and $$ g\left( \theta \right) = {e^{{i\theta }}} $$ with d?.- IX. Some additional properties of u?.- 1. The zeroes of u?.- 2. The limit of $$ \left\{ {{E_{\varepsilon }}\left( {{u_{\varepsilon }}} \right) - \pi d\left| {\log \varepsilon } \right|} \right\} $$ as $$ \varepsilon \to 0 $$.- 3. $$ {\smallint_G}{\left| {\nabla \left| {{u_{\varepsilon }}}\right|} \right|^2} $$ remains bounded as $$ \varepsilon \to 0 $$.- 4. The bad discs revisited.- X. Non minimizing solutions of the Ginzburg-Landau equation.- 1. Preliminary estimates; bad discs and good discs.- 2. Splitting $$ \left| {\nabla {v_{\varepsilon }}} \right| $$.- 3. Study of the associated linear problems.- 4. The basic estimates: $$ {\smallint_G}{\left| {\nabla {v_{\varepsilon }}} \right|^2} \leqslant C\left| {\log \;\varepsilon } \right| $$ and $$ {\smallint_G}{\left| {\nabla {v_{\varepsilon }}} \right|^p} \leqslant {C_p} $$ for p