Existence of Biharmonic Curves and Symmetric Biharmonic Maps
نویسنده
چکیده
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)
منابع مشابه
Stability of F-biharmonic maps
This paper studies some properties of F-biharmonic maps between Riemannian manifolds. By considering the first variation formula of the F-bienergy functional, F-biharmonicity of conformal maps are investigated. Moreover, the second variation formula for F-biharmonic maps is obtained. As an application, instability and nonexistence theorems for F-biharmonic maps are given.
متن کاملMultiple solutions for a perturbed Navier boundary value problem involving the $p$-biharmonic
The aim of this article is to establish the existence of at least three solutions for a perturbed $p$-biharmonic equation depending on two real parameters. The approach is based on variational methods.
متن کاملBiharmonic Maps into Sol and Nil Spaces
In this paper, we study biharmonic maps into Sol and Nil spaces, two model spaces of Thurston's 3-dimensional geometries. We characterize non-geodesic biharmonic curves in Sol space and prove that there exists no non-geodesic biharmonic helix in Sol space. We also show that a linear map from a Eu-clidean space into Sol or Nil space is biharmonic if and only if it is a harmonic map, and give a c...
متن کاملInfinitely many solutions for a class of $p$-biharmonic equation in $mathbb{R}^N$
Using variational arguments, we prove the existence of infinitely many solutions to a class of $p$-biharmonic equation in $mathbb{R}^N$. The existence of nontrivial solution is established under a new set of hypotheses on the potential $V(x)$ and the weight functions $h_1(x), h_2(x)$.
متن کاملB-FOCAL CURVES OF BIHARMONIC B-GENERAL HELICES IN Heis
In this paper, we study B-focal curves of biharmonic B -general helices according to Bishop frame in the Heisenberg group Heis Finally, we characterize the B-focal curves of biharmonic B- general helices in terms of Bishop frame in the Heisenberg group Heis
متن کامل