Boundary partial regularity for a class of biharmonic maps

Abstract

We consider the Dirichlet problem for biharmonic maps u from a bounded, smooth domain \({\Omega\subset\mathbb R^n (n\ge 5)}\) to a compact, smooth Riemannian manifold \({N\subset{\mathbb {R}}^l}\) without boundary. For any smooth boundary data, we show that if u is a stationary biharmonic map that satisfies a certain boundary monotonicity inequality, then there exists a closed subset \({\Sigma\subset\overline{\Omega}}\) , with \({H^{n-4}(\Sigma)=0}\) , such that \({\displaystyle u\in C^\infty(\overline\Omega\setminus\Sigma, N)}\).