Improved energy decay estimate for Dir-stationary $Q$-valued functions and its applications

In this paper, we establish an improved decay estimate for the Dirichlet energy of Dir-stationary $Q$-valued functions. As a direct application of this estimate, we derive a Liouville-type theorem for bounded Dir-stationary $Q$-valued functions defined on ${\mathbb{R}}^m$. Additionally, in an attempt to establish the continuity of Dir-stationary $Q$-valued functions, we confirm that such functions exhibit the Lebesgue property at every point within their domain. Finally, we observe that the improved decay estimate implicates that Dir-stationary $Q$-valued functions reside within a generalized Campanato-Morrey space.