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 QQ (ℝn)-valued functions in the Sobolev space γ2(Ω, QQ (ℝn)), where the domain Ω is open in ℝm . In this article, we introduce the class of weakly stationary-harmonic Q Q (ℝn)-valued functions. These functions are the critical points of Dirichlet's integral under smooth domain-variations and range-variations. We prove that if Ω is a two-dimensional domain in ℝ2 and f∈γ2 (Ω,QQ(ℝ n)) is weakly stationary-harmonic, then f is continuous in the interior of the domain Ω.
ASJC Scopus subject areas
- Geometry and Topology