We consider critical points $u:\Omega \to N$ of the bi-energy ${\int }_{\Omega }{\left|\Delta u\right|}^2\mathrm{d}x,$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.

我们考虑关键点 $ü：\Omega \to ñ$ 的双能 ${\int }_{\Omega }{\left|\Delta ü\right|}^2\mathrm{d}X，$哪里 $\Omega \subset {\mathbb{[R}}^{米}$ 是维的有界光滑域 $米\ge 5$ 和 $ñ\subset {\mathbb{[R}}^{\mathrm{大号}}$无边界的紧凑子流形。更准确地说，我们考虑变分双调和图$ü\in {\mathrm{w\; ^}}^{2，2}（\Omega ，ñ）$，定义为满足一定平稳性条件直至边界的双能临界点。对于变分双调和图的弱收敛序列，我们证明了可以阻止强紧实直至边界的唯一障碍是目标流形中存在某些非恒定双调和4球或4半球。作为一种应用，我们推论了变分双调和图的完整边界正则性，前提是此类球不存在。