Abstract
4-harmonic and ES-4-harmonic maps are two generalizations of the well-studied harmonic map equation which are both given by a nonlinear elliptic partial differential equation of order eight. Due to the large number of derivatives it is very difficult to find any difference in the qualitative behavior of these two variational problems. In this article we prove that finite energy solutions of both 4-harmonic and ES-4-harmonic maps from Euclidean space must be trivial. However, the energy that we require to be finite is different for 4-harmonic and ES-4-harmonic maps pointing out a first difference between these two variational problems.
Highlights
Introduction and ResultsAt the heart of the geometric calculus of variations is the aim to find interesting maps between Riemannian manifolds
One of the best studied energy functionals for maps between Riemannian manifolds is the energy of a map φ : (M, g) → (N, h) which is
A possible higher order generalization of harmonic maps is given by the so-called polyharmonic maps of order k or just k-harmonic maps
Summary
At the heart of the geometric calculus of variations is the aim to find interesting maps between Riemannian manifolds. This can be achieved by extremizing a given energy functional. A possible higher order generalization of harmonic maps is given by the so-called polyharmonic maps of order k or just k-harmonic maps These are critical points of the following energy functionals, where we need to distinguish between polyharmonic maps of even and odd order. 2. In the odd case (k = 2s + 1) the critical points of (1.4) are given by 0 = τ2s+1(φ) := ̄ 2s τ (φ) − R N ( ̄ 2s−1τ (φ), dφ(e j ))dφ(e j ). − R N (∇ ̄ e js−1τ (φ), ̄ s−1τ (φ))dφ(e j )
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