On finite energy solutions of 4-harmonic and ES-4-harmonic maps

TL;DR

This paper proves that finite energy solutions to 4-harmonic and ES-4-harmonic maps from Euclidean space are trivial, highlighting a key difference in their energy conditions despite similar qualitative behaviors.

Contribution

It demonstrates that finite energy solutions for both types of maps must be trivial, with a novel distinction in the energy thresholds required for each.

Findings

Finite energy solutions are trivial for both maps.

Different energy finiteness conditions distinguish the two problems.

Highlights a fundamental difference in their variational structures.

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.