Ginzburg-Landau Vortices

Abstract

Bethuel, Fabrice, 1%3-Ginzburg-Landau vortices I Fabrice Bethuel, Halm Brezis, Frederic Helein.p. cm. --(Progress in nonIinear differential equations and their applications ; v. 13) Included bibliographical references and index.

Uniqueness of vortexless Ginzburg-Landau type minimizers in two dimensions

“…Uniqueness of a u-minimizer follows now from the strict convexity of the ϕ-energy. 4 In contrast, there is no similar trick transforming the GinzburgLandau energy E ε into a strictly convex one. However, it is possible to obtain a sort of logconvexity of E ε in a neighborhood of a minimizer of modulus close to 1.…”

“…When the energy is not supposed to be small, we complete Step 1. under the assumption that ε is small. In this case we use a clearing out technique; this well-known approach [4,23,17,18,5] is revisited in the special case we are interested in in Section 7. Unlike the approach in [26], where Step 2. is obtained via pointwise estimates, 7 our approach relies on global estimates.…”

“…The references for this section are: for the degree, [10], [11], [8], [9]; for lifting, [7], [9]; for GL, [4], [16], [1].…”

Singularities of harmonic maps

“…Such phenomena was first discovered with the Ginzburg-Landau functional in 2 dimensions by F. Bethuel, H. Brezis, and F. Hélein [9], [10], [111]. There is now a vast growing literature on Ginzburg-Landau.…”

“…See e.g. the book [10] and recent survey [80]. Higher dimensional singularities occur in a Ginzburg-Landau problem in [94,82] and in the n-harmonic problem in [63].…”