Higher integrability for solutions to a system of critical elliptic PDE
The paper is concerned with critical elliptic systems in the tradition of \textit{T. Rivière} [Invent. Math. 168, No. 1, 1--22 (2007; Zbl 1128.58010)] and \textit{T. Rivière} and \textit{M. Struwe} [Commun. Pure Appl. Math. 61, No. 4, 451--463 (2008; Zbl 1144.58011)] which include, most prominently, the harmonic map system. In the current paper, a right-hand side \(f\) is allowed to appear in the system, which formally reads \(-\Delta u-\Omega\cdot\nabla u=f\), with \(\Omega\) usually depending on \(u\) and \(\nabla u\). The domain dimension is allowed to be critical (\(n=2\)) or super-critical (\(n\geq3\)).NEWLINENEWLINEAssuming \(f\in L^p\) for \(p\in(n/2,n)\) and that both \(\nabla u\) and \(\Omega\) are in the Morrey space \(M^{2,n-2}\) natural for the problem, and that \(\Omega\) is sufficiently small in that space, the author proves higher regularity, namely local control of \(\nabla^2u\) in \(M^{2p/n,n-2}\) and of \(\nabla u\) in \(M^{2p/(n-p),n-2}\).NEWLINENEWLINEThe proof uses Rivière's technique of gauge fixing and requires subtle improvements of Adams-type potential estimates.
scientific article; zbMATH DE number 6366889