Large solutions for harmonic maps in two dimensions

We seek critical points of the functionalE(u)= \(\mathop \smallint \limits_\Omega\) |βu|², where Ω is the unit disk in ?² andu:Ω→S ² satisfies the boundary conditionu=γ on ?Ω. We prove that if γ is not a constant, thenE has a local minimum which is different from the absolute minimum. We discuss in more details the case where γ(x, y)=(R _x,R _y, \(\sqrt {1 - R^2 }\) ) andR<1.