Large solutions for harmonic maps in two dimensions

AbstractWe seek critical points of the functionalE(u)= $$\mathop \smallint \limits_\Omega$$ |βu|2, where Ω is the unit disk in ℝ2 andu:Ω→S2 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)=(Rx,Ry, $$\sqrt {1 - R^2 }$$ ) andR<1.