A bifurcation result for harmonic maps from an annulus to $S^2$ with not symmetric boundary data

We consider the problem of minimizing the energy of the maps $u(r,\theta)$ from the annulus $\Omega_\rho=B_1\backslash\bar B_\rho$ 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.

We prove that the minimum is attained at a unique harmonic map $u_\rho$which is a planar map if $\log^2\rho+3\theta_0^2\le\pi^2$, while it is not planar in the case $\log^2\rho+\theta_0^2>\pi^2$.

Moreover, we show that $u_\rho$ tends to $\bar v$ as $\rho\to 0$, where $\bar v$ minimizes the energy of the maps $v(r,\theta)$ from $B_1$ to $S^2$, with the boundary condition $v(1,\theta)=(\cos(\theta+\theta_0)$, $\sin(\theta+\theta_0),0)$.