Lower Bounds for the Energy of Unit Vector Fields and Applications

✍ Scribed by Etienne Sandier

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 460 KB
• Volume: 152
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3170

✦ Synopsis

We prove lower bounds for the Dirichlet energy of a unit vector field defined in a perforated domain of R 2 with nonzero degree on the outer boundary in terms of the total diameter of the holes. We use this to derive lower bounds, and then compactness results for sequences (u = ) of minimizers or almost-minimizers of Ginzburg Landau functionals with coupling constant 1Â= 2 tending to + . 1998 Academic Press

Introduction

Let g: S 1 Ä S 1 map the circle to itself, with winding number d. It is well known that, if d{0, then g cannot be extended to a continuous map u: D Ä S 1 defined on the whole disk. In fact this follows from a density result of Schoen and Uhlenbeck (1983, J. Differential Geom. 18, 253 268) no finite energy extension can be defined on the whole disk. However extensions can be defined on the disk less one or several holes. The smaller the holes are, the greater the energy of the extension.

In the following, we will write

for the energy of a map u defined on 0. We will call radius of a compact set A/R 2 and write |A| the infimum over all finite coverings of A by open disks D 1 , ..., D n of the sum r 1 + } } } +r n where r i is the radius of D i . We can define the radius of any compact subset of a metric space in the exact same way. We give a proof of the following result article no. FU973170