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： 卡普兰托空间的正则性理论$\mathcal{L}^{(q,\lambda)}_k(\Omega)$在各种几何变分问题解的正则性理论中找到了许多应用。 在这里，我们将这一理论从单值函数扩展到多值函数，大部分情况下借鉴了卡普兰托最初的思路（\cite{campanato}）。 我们还给出了该理论在静态积分变分元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}多值函数的正则性定理，这些函数通常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元序列的爆破时出现。 在这样的背景下，基于\cite{wickstable}、\cite{minterwick}和\cite{beckerwick}中的一些想法，我们能够推导出多值调和函数的边界正则性理论；这种边界正则性结果似乎是在多值情形下的首次此类结果。 与\cite{minter}相结合，这里的结果在密度$\frac{5}{2}$的经典锥体附近建立了稳定的一维余维静止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 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 $\frac{5}{2}$.