On weakly p-harmonic maps to a closed hemisphere

We consider weakly p-harmonic maps (p2) from a compact connected Riemannian manifold M^m(m2) to the the standard sphere Sⁿ with values in the closed hemisphere Sⁿ₊ = {x Sⁿ : x_n+1 0 } (n 2). We first prove that if u=(u₁,...,u_n+1):MSⁿ is a weakly p-harmonic map satisfying u_n+1(x)>0 a.e. on M, then it is a minimizing p-harmonic map. Next, we give a necessary and sufficient condition for the boundary data : M Sⁿ₊ to achieve uniqueness; and when this condition fails, we are able to describe the set of all minimizers. When M is without boundary, we obtain a Liouville type Theorem for weakly p-harmonic maps.Mathematics Subject Classification (2000): 58E20; 35J70