The Dirichlet problem for harmonic maps from the disk into the euclidean n-sphere
1985; Elsevier BV; Volume: 2; Issue: 2 Linguagem: Inglês
10.1016/s0294-1449(16)30406-1
ISSN1873-1430
AutoresVieri Benci, Jean‐Michel Coron,
Tópico(s)Numerical methods in inverse problems
ResumoAbstract 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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