Morrey regularity theory of Rivière's equation (Q6602155)

The authors deal with the system \N\[\N-\Delta u=\Omega\cdot\nabla u+f\N\]\N on the unit ball \(B\) of \(\mathbb R^2\), and extend previous results on the regularity of weak solution to the ambient of Morrey spaces. More precisely, if \(u\in W^{1,2}(B)\) is a weak solution, assuming that \(||\Omega||_{L^2(B)}\le \varepsilon\), then there exists a constant \(C\) such that \N\[\N||\nabla u||_{M^{2,2}\lambda(B_{1/2})} \le C \left( ||\nabla u||_{L^2(B)} + ||f||_{M^{1,\lambda}(B_{1/2})}\right),\N\]\Nanytime \(f\in M^{1,\lambda}(B_{1/2})\) for some \(0<\lambda<1\). Additionally, if \(f\in M^{p,p\lambda}(B_{1/2})\) for some \(1<p<2\) and \(0\le \lambda< 2/p-1\), then \N\[\N||\nabla^2 u||_{M^{p,p\lambda}(B_{1/2})}+||\nabla u||_{M^{p^*,p^*\lambda}(B_{1/2})} \le C \left( ||\nabla u||_{L^2(B)} + ||f||_{M^{p,p\lambda}(B_{1/2})}\right),\N\]\Nwhere \(p^*=2p/(2-p)\).