Energy quantization for biharmonic maps

TL;DR

This paper proves an energy quantization result for solutions to certain 4-dimensional variational problems, including biharmonic maps, using angular energy quantization techniques for 4th order elliptic systems.

Contribution

It introduces an energy quantization framework for biharmonic maps and extends angular energy quantization methods to 4th order elliptic systems with antisymmetric potentials.

Findings

Established energy quantization for biharmonic maps in 4D.

Developed angular energy quantization for 4th order elliptic systems.

Extended previous 2nd order methods to 4th order problems.

Abstract

In the present work we establish an energy quantization (or energy identity) result for solutions to scaling invariant variational problems in dimension 4 which includes biharmonic maps (extrinsic and intrinsic). To that aim we first establish an angular energy quantization for solutions to critical linear 4th order elliptic systems with antisymmetric potentials. The method is inspired by the one introduced by the authors previously in arXiv:1109.3599v1 for 2nd order problems.