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 C1,α and C2,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≥2, and target dimension k. The C1,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 C2,α estimate concerns strong solutions. In dimension n=2, both estimates hold for every 0<α<1/Q. In dimensions n≥3, 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 ℬu of any C1,γ Q-valued map u whose graph varifold is stationary satisfies dimℋℬu≤ n−2. This bound is sharp already in codimension one.
Related
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BibTeX
@misc{luchese2026perturbative,
title = {{Perturbative Schauder Estimates for $Q$-Valued Quasilinear Elliptic Systems and a Sharp Dimension Bound for Branch Sets of Stationary Graphs}},
author = {Mattia Luchese},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12950},
eprint = {2609.12950},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12950},
}
RIS
TY - UNPB
TI - Perturbative Schauder Estimates for Q-Valued Quasilinear Elliptic Systems and a Sharp Dimension Bound for Branch Sets of Stationary Graphs
AU - Mattia Luchese
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12950
UR - https://arxiv.org/abs/2609.12950
AB - 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≥2, 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≥3, 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 ℬ_u of any C^1,γ Q-valued map u whose graph varifold is stationary satisfies dim_ℋℬ_u≤ n−2. This bound is sharp already in codimension one.
ER -