Abstract.
We consider weakly p-harmonic maps (p≥2) from a compact connected Riemannian manifold Mm(m≥2) to the the standard sphere Sn with values in the closed hemisphere Sn+ = {x∈ Sn : xn+1 ≥ 0 } (n ≥ 2). We first prove that if u=(u1,...,u n +1):M→Sn 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 ϕ : ∂ 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.
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Mathematics Subject Classification (2000): 58E20; 35J70
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Fardoun, A. On weakly p-harmonic maps to a closed hemisphere. manuscripta math. 116, 57–69 (2005). https://doi.org/10.1007/s00229-004-0516-3
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DOI: https://doi.org/10.1007/s00229-004-0516-3