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

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 $n\ge 2$, 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 $0<\alpha <1\mathrm{/}Q$. In dimensions $n\ge 3$, 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 ${\dim }_{\mathcal{H}}{\mathcal{B}}_u\le n-2$. This bound is sharp already in codimension one.