Published in:Progress in Nonlinear Differential Equations and Their Applications ; 13 SpringerLink ; Bücher Springer eBook Collection ; Mathematics and Statistics
Description:
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.