# Blow-up analysis and boundary regularity for variationally biharmonic maps

## TL;DR

This paper studies the boundary regularity of variationally biharmonic maps, showing that the absence of certain biharmonic spheres in the target guarantees smoothness up to the boundary.

## Contribution

It establishes boundary regularity results for variationally biharmonic maps by linking potential singularities to the existence of biharmonic spheres in the target manifold.

## Key findings

- Weak convergence of maps is obstructed only by biharmonic spheres or half-spheres.
- Boundary regularity holds if such biharmonic spheres do not exist in the target.
- Identifies conditions under which boundary regularity is achieved.

## Abstract

We consider critical points $u:\Omega\to N$ of the bi-energy \[ \int_\Omega |\Delta u|^2\,d x, \] where $\Omega\subset\mathbb{R}^m$ is a bounded smooth domain of dimension $m\ge 5$ and $N\subset\mathbb{R}^L$ a compact submanifold without boundary. More precisely, we consider variationally biharmonic maps $u\in W^{2,2}(\Omega,N)$, which are defined as critical points of the bi-energy that satisfy a certain stationarity condition up to the boundary. For weakly convergent sequences of variationally biharmonic maps, we demonstrate that the only obstruction that can prevent the strong compactness up to the boundary is the presence of certain non-constant biharmonic $4$-spheres or $4$-halfspheres in the target manifold. As an application, we deduce full boundary regularity of variationally biharmonic maps provided such spheres do not exist.