Partial Regularity of a minimizer of the relaxed energy for biharmonic maps

TL;DR

This paper investigates the regularity and singularity structure of minimizers for the relaxed Hessian energy of biharmonic maps into spheres, establishing existence, smoothness outside a finite measure singular set, and rectifiability in dimension five.

Contribution

It proves the existence of minimizers for the relaxed energy and characterizes their regularity and singular set structure, including rectifiability in five dimensions.

Findings

Minimizers are biharmonic and smooth outside a finite measure singular set.

The singular set has finite (m-4)-dimensional Hausdorff measure.

In 5D, the singular set is 1-rectifiable.

Abstract

In this paper, we study the relaxed energy for biharmonic maps from a $m$-dimensional domain into spheres. By an approximation method, we prove the existence of a minimizer of the relaxed energy of the Hessian energy, and that the minimizer is biharmonic and smooth outside a singular set $\Sigma$ of finite $(m-4)$-dimensional Hausdorff measure. Moreover, when $m=5$, we prove that the singular set $\Sigma$ is 1-rectifiable.