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 .

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 $$ gleft( theta right) = {e^{{itheta }}} $$.- 5. The case G=B1 and $$ gleft( theta right) = {e^{{itheta }}} $$ with d?.- IX. Some additional properties of u?.- 1. The zeroes of u?.- 2. The limit of $$ left{ {{E_{varepsilon }}left( {{u_{varepsilon }}} right) - pi dleft| {log varepsilon } right|} right} $$ 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: $$ {smallint_G}{left| {nabla {v_{varepsilon }}} right|^2} leqslant Cleft| {log ;varepsilon } right| $$ and $$ {smallint_G}{left| {nabla {v_{varepsilon }}} right|^p} leqslant {C_p} $$ for p