多值函数的Campanato正则性理论及其在极小曲面正则性理论中的应用
A Campanato Regularity Theory for Multi-Valued Functions with Applications to Minimal Surface Regularity Theory
AI总结：
本文将Campanato空间正则性理论从单值函数推广至多值函数，推导了多值调和函数的边界正则性结果，结合已有研究建立了密度为5/2的经典锥附近稳定余维一稳态积分变分的正则性定理。
摘要：
Campanato空间$\u2715^{(q,λ)}_k(Ω)$的正则性理论已在各类几何变分问题解的正则性理论中得到诸多应用。本文将该理论从单值函数推广至多值函数，主要沿用了Campanato的原始思路（\u300acampanato\u300b）。我们还给出了该理论在稳态积分变分（stationary integral varifold）正则性理论中的一项应用。更确切地说，我们证明了一类多值函数的“blow-up类（爆破类）”的正则性定理，这类函数通常出现在研究收敛到高重数平面或半平面并集的稳态积分变分序列的爆破过程中。在此框架下，我们部分基于\u300awickstable\u300b、\u300aminterwick\u300b和\u300abeckerwick\u300b中的思路，推导出多值调和函数的边界正则性理论；这类边界正则性结果在多值情形下尚属首次。结合\u300aminter\u300b的研究，本文给出的结果建立了稳定余维一稳态积分变分在密度为$\frac{5}{2}$的经典锥附近的正则性定理。
英文摘要：
The regularity theory of the Campanato space $\mathcal{L}^{(q,λ)}_k(Ω)$ 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 $\frac{5}{2}$.
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