Interior continuity of two-dimensional weakly stationary-harmonic multiple-valued functions
In his big regularity paper, Almgren has proven the regularity theorem for mass-minimizing integral currents. One key step in his paper is to derive the regularity of Dirichlet-minimizing $\mathbf{Q}_{Q}(\mathbb{R}^{n})$-valued functions in the Sobolev space $\mathcal{Y}_{2}(\Omega, \mathbf{Q}_{Q} (\mathbb{R}^{n}))$, where the domain $\Omega$ is open in $\mathbb{R}^{m}$. In this article, we introduce the class of weakly stationary-harmonic $\mathbf{Q}_{Q} (\mathbb{R}^n)$-valued functions. These functions are the critical points of Dirichlet integral under smooth domain-variations and range-variations. We prove that if $\Omega$ is a two-dimensional domain in $\mathbb{R}^{2}$ and $f\in\mathcal{Y}_{2}(\Omega,\mathbf{Q}_{Q}(\mathbb{R}^{n}))$ is weakly stationary-harmonic, then $f$ is continuous in the interior of the domain $\Omega$.