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

The regularity theory of the Campanato space L(q,λ)ₖ(Ω) has been successfully used to understand the regularity of solutions to certain geometric variational problems. Here we extend this theory to multi-valued functions, adapting for the most part Campanato's original ideas in {campanato1964proprieta}. We then give an application within the regularity theory of stationary integral varifolds. More precisely, we prove a regularity theorem for certain blow-up classes of 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 {wickramasekera2014general} and {BKW2021}, we are able to deduce a boundary regularity theory for multi-valued harmonic functions, which is the first of its kind. In conjunction with {minter2021}, the results here establish a regularity theorem for stable codimension one integral varifolds near classical cones of density 5/2.