A bifurcation result for harmonic maps from an annulus to \(S^2\) with not symmetric boundary data (Q2701601)

The author considers the problem of minimizing the energy of the maps \(u(r,\theta)\) from the annulus \(\Omega_{\rho }=\{(x,y)\in {\mathbb R}^2\mid \rho^2<x^2+y^2<1\}\) to \(S^2\) such that \(u(r,\theta)\) is equal to \((\cos\theta ,\sin\theta ,0)\) for \(r=\rho \), and to \((\cos(\theta +\theta_0), \sin(\theta +\theta_0),0)\) for \(r=1\), where \(\theta_0\in [0,\pi ]\) is a fixed angle. NEWLINENEWLINENEWLINEIt is proved that the minimum is attained at a unique harmonic map \(u_{\rho}\) which is a planar map if \(\log^2\rho +3\theta^2_0\leq\pi^2\), while it is not planar in the case \(\log^2\rho +\theta^2_0>\pi^2\). Moreover, it is shown that \(u_{\rho}\) tends to \(\bar v\) as \(\rho\to 0\), where \(\bar v\) minimizes the energy of the maps \(v(r,\theta)\) from \(B_1=\{(x,y)\in {\mathbb R}^2\mid x^2+y^2<1\}\) to \(S^2\), with the boundary condition \(v(1,\theta)=(\sin(\theta +\theta_0), \sin(\theta +\theta_0),0)\).