Leonid Berlyand, Mathematics and Materials Research Institute, Penn State University

"Ginzburg-Landau minimizers with prescribed degrees Capacity of the domain and emergence of vortices"

Abstract

Let $\Omega$ be a 2D domain with holes $\omega_0,\omega_1, \ldots, \omega_j, j=1... k$ . In domain $A=\Omega\setminus\big(\cup_{j=0}^k\omega_j\big)$ consider class ${\mathcal J}$ of complex valued maps having degrees and on $\partial \Omega$ , $\partial \omega_0$ respectively and degree on $\partial \omega_j, j=1... k$ .

We show that if cap $(A)\ge\pi$ , minimizers of the Ginzburg-Landau energy $E_\kappa$ exist for each $\mathcal \kappa$ . They are vortexless and converge in to a minimizing -valued harmonic map as the coherency length $\mathcal \kappa ^{-1}$ tends to . When cap $(A)<\pi$ , we establish existence of quasi-minimizers, which exhibit a different qualitative behavior: they have exactly two zeroes (vortices) rapidly converging to $\partial A$ .

This is a joint work with P. Mironescu (Orsay, France).

TUESDAY, May 3, 2005
Time: 1:30 P.M.
Location: DH 4303

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