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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 $u\inW^{2,2}(\Omega,N)$, which are defined as critical points of the bi-energy thatsatisfy a certain stationarity condition up to the boundary. For weaklyconvergent sequences of variationally biharmonic maps, we demonstrate that theonly obstruction that can prevent the strong compactness up to the boundary isthe presence of certain non-constant biharmonic $4$-spheres or $4$-halfspheresin the target manifold. As an application, we deduce full boundary regularityof variationally biharmonic maps provided such spheres do not exist.

Blow-up analysis and boundary regularity for variationally biharmonic maps