Ginzburg-Landau Vortices

The original motivation of this study comes from the following questions that were mentioned to one ofus by H. Matano. Let 2 2 G= B = {x=(X1lX2) E 2 ; x~ + x~ = Ixl 1}. 1 Consider the Ginzburg-Landau functional 2 2 (1) E~(u) = ~ LIVul + 4~2 L(lu1 _1)2 which is defined for maps u E H1(G;C) also identified with Hl(G;R2). Fix the boundary condition 9(X) =X on 8G and set H; = {u E H1(G;C); u = 9 on 8G}. It is easy to see that (2) is achieved by some u~ that is smooth and satisfies the Euler equation in G, -~u~ = :2 u~(1 _lu~12) (3) { on aGo u~ =9 Themaximum principleeasily implies (see e.g., F. Bethuel, H. Brezisand F. Helein (2]) that any solution u~ of (3) satisfies lu~1 ~ 1 in G. In particular, a subsequence (u~,.) converges in the w - LOO(G) topology to a limit u .

Résumé
The mathematics in this book apply directly to classical problems in superconductors, superfluids and liquid crystals. It should be of interest to mathematicians, physicists and engineers working on modern materials research.

Contenu
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 $$ \frac{1}{{{\varepsilon^2}}}{\smallintG}{\left( {{{\left| {{u{\varepsilon }}} \right|}^2} - 1} \right)^2} $$.- 4. $$ \left| {{ue}} \right| \geqslant \frac{1}{2} $$ 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 $$ {\smallintG}{\left| {\nabla {v{\varepsilon }}} \right|^p} \leqslant {C_p} $$ for p