AI 中文总结
本文为$Q$-值拟线性椭圆系统建立微扰Schauder估计，并证明平稳图的分支集维数不超过$n-2$，该界在余维数一时是锐利的。
AI 中文摘要
我们为一类$Q$-值拟线性椭圆系统建立了内部先验$C^{1,\alpha}$和$C^{2,\alpha}$ Schauder估计，该系统是Laplace系统的微扰，适用于任意重数$Q$、域维数$n\geq2$和目标维数$k$。$C^{1,\alpha}$估计推广了Simon和Wickramasekera在文献\cite{SW16}中关于线性系统$2$-值解的工作，并涉及散度形式系统的弱解，而$C^{2,\alpha}$估计则涉及强解。在维数$n=2$时，两个估计对所有$0<\alpha<1/Q$成立。在维数$n\geq3$时，存在$\delta=\delta(n,k,Q)>0$使得两个估计对所有$0<\alpha<\delta$成立。作为应用，我们获得了$Q$-值映射的小斜率Schauder估计和小斜率Bernstein定理，这些映射的图varifold是平稳的。将Schauder估计与Krummel--Minter--Wickramasekera在文献\cite{KMW26}中最近的分支集分层理论相结合，我们进一步证明，对于每个$\gamma>0$，任何图varifold平稳的$C^{1,\gamma}$ $Q$-值映射$u$的分支集$\mathcal B_u$满足$\dim_{\mathcal H}\mathcal B_u\leq n-2$。这个界在余维数为一的情况下已经是锐利的。
英文摘要
We establish a priori interior $C^{1,α}$ and $C^{2,α}$ Schauder estimates for a class of $Q$-valued quasilinear elliptic systems which are perturbations of the Laplace system, for arbitrary multiplicity $Q$, domain dimension $n\geq2$, and target dimension $k$. The $C^{1,α}$ estimate generalises the work of Simon and Wickramasekera in \cite{SW16} on $2$-valued solutions of linear systems and concerns weak solutions of divergence-form systems, while the $C^{2,α}$ estimate concerns strong solutions. In dimension $n=2$, both estimates hold for every $0<α<1/Q$. In dimensions $n\geq3$, there exists $δ=δ(n,k,Q)>0$ such that both estimates hold for every $0<α<δ$. As applications, we obtain a small-slope Schauder estimate and a small-slope Bernstein theorem for $Q$-valued maps whose graph varifolds are stationary. Combining the Schauder estimate with the recent branch-set stratification theory of Krummel--Minter--Wickramasekera in \cite{KMW26}, we further prove that, for every $γ>0$, the branch set $\mathcal B_u$ of any $C^{1,γ}$ $Q$-valued map $u$ whose graph varifold is stationary satisfies $\dim_{\mathcal H}\mathcal B_u\leq n-2$. This bound is sharp already in codimension one.
