Ginzburg-Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices
Résumé
Let $\Omega\subset {\mathbb R}^2$ be a simply connected domain, let $\omega$ be a simply connected subdomain of $\Omega$, and set $A=\Omega\setminus\overline\omega$. Let $J$ be the class of complex-valued maps on the annular domain $A$ with degree $1$ both on $\partial\Omega$ and on $\partial\omega$. We consider the variational problem of minimizing the Ginzburg- Landau energy $E_\lambda$ among all maps in $J$. Because only the degrees of the map are prescribed on the boundary, the set $J$ is not closed under weak $H^1$-convergence. We show that the attainability of the minimum of $E_\lambda$ over $J$ is determined by the value of cap$\,(A)$, where cap$\,(A)$ is the $H^1$-capacity of the domain $A$. In contrast, it is known that the existence of minimizers of $E_\lambda$ among the maps with a prescribed Dirichlet boundary data does not depend on this geometric characteristic. When cap$\,(A)\ge \pi$ (such an $A$ is denoted as either subcritical - if cap$\,(A)$>$\pi$ or critical - if cap$\,(A)= \pi$), we show that the global minimizers of $E_\lambda$ exist for each $\lambda>0$ and that they are vortexless when $\lambda$ is large. Assuming that $\lambda\to\infty$, we demonstrate that the minimizers of $E_\lambda$ converge in $H^1(A)$ (and even better) to an ${\mathbb S}^1$-valued harmonic map which we explicitly identify. When cap$\, (A)$<$\pi$ (i.e., $A$ is supercritical), we prove that either (i) there is a critical value $\lambda_0$ such that the global minimizers exist when $\lambda$<$\lambda_0$ and they do not exist when $\lambda$>$\lambda_0$, or (ii) the global minimizers exist for each $\lambda>0$. We conjecture that the second case never occurs. Further, for large $\lambda$, we establish that the minimizing sequences/minimizers in supercritical domains develop exactly two vortices - a vortex of degree $1$ near $\partial\Omega$ and a vortex of degree $-1$ near $\partial\omega$.
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