On Finite Energy Solutions of 4-harmonic and ES-4-harmonic Maps

Abstract

Abstract4-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.

Unique continuation properties for polyharmonic maps between Riemannian manifolds

“…77,Problem (8.7)]. A rigorous mathematical investigation of (1.9) has recently been initiated by the authors in [6] and was further developed in [5,20]. We point out that, in general, the functional (1.9) coincides with E k (φ) introduced above only when k = 1, 2, 3.…”

“…For the sake of completeness, we point out that another interesting generalization of both harmonic and biharmonic maps can be obtained by studying the critical points of the following higher-order energies: The study of these functionals was proposed by Eells and Sampson in 1965 (see [11]) and, later, by Eells and Lemaire in 1983 [9, p. 77, Problem (8.7)]. A rigorous mathematical investigation of (1.9) has recently been initiated by the authors in [6] and was further developed in [5, 20]. We point out that, in general, the functional (1.9) coincides with introduced above only when .…”

On polyharmonic helices in space forms

“…However, if r = 1, 2, 3 or in the case of a one-dimensional domain, both functionals (1.3) and (1.4) coincide. For an extensive analysis of (1.4) and a discussion of the differences between (1.3) and (1.4), we refer to the recent articles [3,6,8,27].…”