Existence of Biharmonic Curves and Symmetric Biharmonic Maps

where n is the exterior normal direction of ∂Ω. In other words, we look for a “best” way to extend the boundary value φ with the prescribed normal derivative ψ. Typical examples of Ω and N are the unit ball and the unit sphere, respectively. In this case, ψ : ∂Ω → TφN means φ (x) · ψ (x) = 0 for all |x| = 1. With the given Dirichlet data φ, the most natural extension is perhaps the harmonic map. Recall that a map u : Ω → N is harmonic if and only if its tension field T (u) vanishes. In terms of the second fundamental form A of N ⊂ R, T (u) can be expressed as T (u) ≡ ∆u−A (u) (∇u,∇u) , (2)