Carnegie Mellon
Department of Mathematical 

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

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


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 $1$ and $1$ on $\partial
\Omega$, $\partial \omega_0$ respectively and degree $0$ 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 $H^1(A)$ to a minimizing $S^1$-valued harmonic map as the coherency length $\mathcal \kappa ^{-1}$ tends to $0$. 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