Higher integrability for solutions to a system of critical elliptic PDE

We give new estimates for a critical elliptic system introduced by Rivière-Struwe which generalises PDE solved by (almost) harmonic maps from a Euclidean ball $B_1 \subset \mathbb{R}^n$ into closed Riemannian manifolds $\mathcal{N} \hookrightarrow \mathbb{R}^m$. Solutions $u : B_1 \to \mathbb{R}^m$ take the form$$-\Delta u^i = \Omega^i_j . \nabla u^j$$where $\Omega$ maps into antisymmetric $m \times m$ matrices with entries in $\mathbb{R}^n$. Here $\Omega$ and $\nabla u$ belong to a Morrey space which makes the PDE critical from a regularity perspective. We use the Coulomb framemethod employed by Rivière-Struwe along with the Hölder regularity already acquired therein, coupled with an extension of a Riesz potential estimate, in order to improve the known regularity and estimates for solutions $u$. These methods apply when $n = 2$ thereby re-proving the full regularity in this case using Coulomb gauge methods. Moreover they lead to a self contained proof of the local regularity of stationary harmonic maps in high dimension.

Keywords

harmonic maps, regularity for critical elliptic systems, Riesz potential estimates for Morrey-Hardy spaces

2010 Mathematics Subject Classification

35B65, 42B37, 58E20

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Published 13 August 2014