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

  • Radu Ignat

    Université de Toulouse, CNRS, UPS IMT, Toulouse, France
  • Luc Nguyen

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

We study the existence, uniqueness and minimality of critical points of the form mε,η(x)=(fε,η(x)xx,gε,η(x))m_{\varepsilon,\eta}(x) = (f_{\varepsilon,\eta}(|x|)\smash{\frac{x}{|x|}}, g_{\varepsilon,\eta}(|x|)) of the functional Eε,η[m]=BN[12m2+14ε2(1m2)2+12η2mN+12]dxE_{\varepsilon,\eta}[m] = \int_{B^N} [\frac{1}{2} |\nabla m|^2 + \smash{\frac{1}{4\varepsilon^2}} (1 - |m|^2)^2 + \frac{1}{2\eta^2} m_{N+1}^2]\,dx for m=(m1,,mN,mN+1)H1(BN,RN+1)m=(m_1, \dots, m_N, m_{N+1}) \in H^1(B^N,\mathbb{R}^{N+1}) with m(x)=(x,0)m(x) = (x,0) on BN\partial B^N. We establish a necessary and sufficient condition on the dimension NN and the parameters ε\varepsilon and η\eta for the existence of an escaping vortex solution (fε,η,gε,η)(f_{\varepsilon,\eta}, g_{\varepsilon,\eta}) with gε,η>0g_{\varepsilon,\eta}> 0. We also establish its uniqueness and local minimality. In particular, when η=0\eta = 0, we prove the local minimality of the degree-one vortex solution for the Ginzburg–Landau (GL) energy for every ε>0\varepsilon > 0 and N2N \geq 2. Similarly, when ε=0\varepsilon = 0, we prove the local minimality of the degree-one escaping vortex solution to an SN\mathbb{S}^N-valued GL model in micromagnetics for all η>0\eta > 0 and 2N62 \leq N \leq 6.

Cite this article

Radu Ignat, Luc Nguyen, Local minimality of RN\mathbb{R}^N-valued and SN\mathbb{S}^N-valued Ginzburg–Landau vortex solutions in the unit ball BNB^N. Ann. Inst. H. Poincaré Anal. Non Linéaire (2023),

DOI 10.4171/AIHPC/84