Regularity for solutions of H-systems and n-harmonic maps with n/2 square integrable derivatives

We study the regularity of weak solutions for two elliptic systems involving the $n$-Laplacian and a critical nonlinearity in the right hand side: $H$-systems and $n$-harmonic maps into compact Riemannian manifolds. Under the assumptions that the solutions belong to $W^{n\mathrm{/}2,2}$ in an even dimension $n$, we prove their continuty. The tools used in the proof involve Hardy spaces and BMO, and the Rivière–Uhlenbeck decomposition (with estimates in Morrey spaces). A prominent role is played by the Coifman–Rochberg–Weiss commutator theorem.