The Dirichlet Problem for Harmonic Maps from the Disk into the Euclidean n-Sphere.

Abstract Let Ω = { ( x , y ) ∈ ℝ 2 | x 2 + y 2 1 } , S n = { v ∈ ℝ n + 1 | | v | = 1 } (n ⩾ 2), and let γ ∈ C2,δ(∂Ω; Sn). We study the following problem (*) { u ∈ C 2 ( Ω ; S n ) ∩ C 0 ( Ω ¯ ; S n ) − Δ u = u | ∇ u | 2 u = γ o n ∂ Ω . Problem (*) is the « Dirichlet » problem for a harmonic function u which takes its values in Sn. We prove that, if γ is not constant, then (*) has at least two distinct solutions.

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