Higher integrability for solutions to a system of critical elliptic PDE

Abstract

We give new estimates for a critical elliptic system introduced by Rivi\`ere-Struwe in \cite{riviere_struwe} (see also the work of Rupflin \cite{rupflin} and Schikorra \cite{schikorra_frames}), which generalises PDE solved by harmonic (and almost harmonic) maps from a Euclidean ball $B_1 \In \R^n$ into Riemannian manifolds. Solutions take the form $$-\Dl u = \Om.\D u $$ where $\Om$ is an anti-symmetric potential with $\Om$ and $\D u$ belonging to the Morrey space $\M^{2,n-2}$ making the PDE critical from a regularity perspective (classical theory gives one estimates on $\D u$ in the weak-Morrey space $\M^{(2,\infty),n-2}$, see Sections \ref{adams_decay} and \ref{Morrey} for definitions if necessary). We use the Coulomb frame method employed in \cite{riviere_struwe} along with the H\"older regularity already acquired in \cite{rupflin}, coupled with an extension of a Riesz potential estimate of Adams \cite{adams_riesz} in order to attain estimates on $\D^2 u \in \M^{s, n-2}$ for any $s<2$. These methods apply when $n=2$ thereby re-proving the full regularity in this case (see \cite{Sh_To}) using Coulomb gauge methods. Moreover they lead to a self contained proof of the local regularity of stationary harmonic maps in high dimension (see Corollary \ref{highint}).