A Campanato regularity theory for multi-valued functions with applications to minimal surface regularity theory

Abstract

The regularity theory of the Campanato space $L_k^{(q,\lambda )}\left(\Omega \right)$ 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 ([4]). 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 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 [16], [11], and [3], 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 [9], the results presented here establish a regularity theorem for stable codimension one stationary integral varifolds near classical cones of density $\frac{5}{2}$.