Blow-up analysis and boundary regularity for variationally biharmonic maps

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.