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A bifurcation result for harmonic maps from an annulus to with not symmetric boundary data
Authors:C. Greco
Affiliation:Dipartimento di Matematica, Università degli Studi di Bari, Via Orabona 4, 70125 Bari, Italy
Abstract:

We consider the problem of minimizing the energy of the maps $u(r,theta)$ from the annulus $Omega_rho=B_1backslashbar B_rho$ to $S^2$ such that $u(r,theta)$ is equal to $(costheta,sintheta,0)$ for $r=rho$, and to $(cos(theta+theta_0)$, $sin(theta+theta_0),0)$ for $r=1$, where $theta_0in[0,pi]$ is a fixed angle.

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

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

Keywords:Harmonic maps   Dirichlet problem
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