Boundary regularity of stationary biharmonic maps

TL;DR

This paper proves that stationary biharmonic maps with smooth boundary data are smooth outside a small, measure-zero singular set, under a boundary monotonicity condition, extending regularity results in higher dimensions.

Contribution

It establishes boundary regularity for stationary biharmonic maps with a boundary monotonicity inequality, identifying a small singular set where regularity fails.

Findings

Existence of a small singular set with zero (n-4)-dimensional measure.

Smoothness of solutions outside the singular set.

Regularity result holds for dimensions n ≥ 5.

Abstract

We consider the Dirichlet problem for stationary 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, in addition, $u$ satisfies a certain boundary monotonicity inequality, then there exists a closed subset $\Sigma\subset\bar{\Omega}$, with $H^{n-4}(\Sigma)=0$, such that $u\in C^\infty(\bar\Omega\setminus\Sigma, N)$.