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

Abstract

The regularity theory of the Campanato space 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 ([Cam64]). 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 [Wic14], [MW24], and [BKMW25], 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 [Min21], the results presented here establish a regularity theorem for stable codimension one stationary integral varifolds near classical cones of density .