Perturbative Schauder Estimates for $Q$-Valued Quasilinear Elliptic Systems and a Sharp Dimension Bound for Branch Sets of Stationary Graphs

Abstract

We establish a priori interior $C^{1,\alpha}$ and $C^{2,\alpha}$ 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 $C^{1,\alpha}$ 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 $C^{2,\alpha}$ estimate concerns strong solutions. In dimension $n=2$, both estimates hold for every $0<\alpha<1/Q$. In dimensions $n\geq3$, there exists $\delta=\delta(n,k,Q)>0$ such that both estimates hold for every $0<\alpha<\delta$. 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 $\gamma>0$, the branch set $\mathcal B_u$ of any $C^{1,\gamma}$ $Q$-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.