A Campanato Regularity Theory for Multi-Valued Functions with Applications to Minimal Surface Regularity Theory

Abstract: The regularity theory of the Campanato space $\mathcal{L}^{(q,\lambda)}_k(\Omega)$ has found many applications within the regularity theory of solutions to various geometric variational problems. Here we extend this theory from single-valued functions to multi-valued functions, adapting for the most part Campanato's original ideas (\cite{campanato}). We also give an application of this theory within the regularity theory of stationary integral varifolds. More precisely, we prove a regularity theorem for certain \textit{blow-up classes} of multi-valued functions, which typically arise when studying blow-ups of sequences of stationary integral varifolds converging to higher multiplicity planes or unions of half-planes. In such a setting, based in part on ideas in \cite{wickstable}, \cite{minterwick}, and \cite{beckerwick}, we are able to deduce a boundary regularity theory for multi-valued harmonic functions; such a boundary regularity result would appear to be the first of its kind for the multi-valued setting. In conjunction with \cite{minter}, the results presented here establish a regularity theorem for stable codimension one stationary integral varifolds near classical cones of density $\frac{5}{2}$.