A Note on Bifurcation for Harmonic Maps on Annular Domains

Abstract

In this paper we consider harmonic maps u(r, θ) from an annular domain ${\Omega _{\rho} = B_{1}\backslash\bar{B}_{\rho}}$ to S ² with the boundary conditions: ${u(\rho, \theta ) = \left(\cos\theta, \sin \theta, 0\right)}$ , and ${u(1,\theta )= \left( \cos \left( \theta +\theta_{0} \right) ,\sin \left( \theta +\theta _{0} \right) ,0\right)}$ , where ${\theta _{0} \in [0, \pi[}$ is a fixed angle. This problem arises from the theory of liquid crystals. We prove, with elementary time map arguments, a bifurcation result, namely the existence of a not trivial (that is not planar) harmonic map of minimum energy ${u_{\theta_0}}$ , for suitable combination of value of ρ and ${\theta _{0}}$ . This result improves the one in Greco (Proc Am Math Soc 129(4):1199–1206, 2000). In the case ${ \theta _{0}=\pi}$ , so that ${u(1, \theta) = \left(-\cos \theta ,-\sin \theta, 0\right)}$ , no bifurcation occurs, since the minimum of the energy is not trivial, and we study the behavior of the harmonic maps ${u_{\theta_0}}$ as ${\theta_0 \rightarrow \pi}$ .