Local minimality of ${\mathbb{R}}^N$-valued and ${\mathbb{S}}^N$-valued Ginzburg–Landau vortex solutions in the unit ball $B^N$

Abstract

We study the existence, uniqueness and minimality of critical points of the form $m_{\epsilon ,\eta }(x)=(f_{\epsilon ,\eta }(|x|)\frac{x}{|x|},g_{\epsilon ,\eta }(|x|))$ of the functional $E_{\epsilon ,\eta }[m]={\int }_{B^N}[\frac{1}{2}|\nabla m{|}^2+\frac{1}{4{\epsilon }^2}(1-|m{|}^2{)}^2+\frac{1}{2{\eta }^2}{m}_{N+1}^2]dx$ for $m=(m_1,\dots ,m_N,m_{N+1})\in H^1(B^N,{\mathbb{R}}^{N+1})$ with $m(x)=(x,0)$ on $\partial B^N$. We establish a necessary and sufficient condition on the dimension $N$ and the parameters $\epsilon$ and $\eta$ for the existence of an escaping vortex solution $(f_{\epsilon ,\eta },g_{\epsilon ,\eta })$ with $g_{\epsilon ,\eta }>0$. We also establish its uniqueness and local minimality. In particular, when $\eta =0$, we prove the local minimality of the degree-one vortex solution for the Ginzburg–Landau (GL) energy for every $\epsilon >0$ and $N\ge 2$. Similarly, when $\epsilon =0$, we prove the local minimality of the degree-one escaping vortex solution to an ${\mathbb{S}}^N$-valued GL model in micromagnetics for all $\eta >0$ and $2\le N\le 6$.