标题： 多值函数的卡姆panato正则性理论及其在极小曲面正则性理论中的应用 显示英文标题

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

摘要： 卡普拉托空间的正则性理论 $\mathcal{L}^{(q,\lambda)}_k(\Omega)$已被成功用于理解某些几何变分问题解的正则性。 在这里，我们将这一理论扩展到多值函数，大部分情况下适应了卡普拉托在\cite{campanato1964proprieta}中的原始思想。 然后，我们给出在静态积分变分法正则性理论中的一个应用。 更准确地说，我们证明了一个关于某些\textit{爆破类}的正则性定理，这些函数通常出现在研究收敛到更高重数平面或半平面并集的静态积分变分序列的爆破分析中。 在这种情况下，部分基于\cite{wickramasekera2014general}和\cite{BKW2021}中的思想，我们能够推导出多值调和函数的边界正则性理论，这是首创的。 结合\cite{minter2021}，这里的结果确立了在密度为$\frac{5}{2}$的经典锥体附近稳定的一维余维积分变分的正则性定理。 显示英文摘要

摘要： The regularity theory of the Campanato space $\mathcal{L}^{(q,\lambda)}_k(\Omega)$ 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 \cite{campanato1964proprieta}. We then give an application within the regularity theory of stationary integral varifolds. More precisely, we prove a regularity theorem for certain \textit{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 \cite{wickramasekera2014general} and \cite{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 \cite{minter2021}, the results here establish a regularity theorem for stable codimension one integral varifolds near classical cones of density $\frac{5}{2}$.