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

TL;DR

This paper proves continuity of weak solutions to certain elliptic systems involving the n-Laplacian and critical nonlinearities, using advanced harmonic analysis tools in even dimensions.

Contribution

It establishes regularity results for solutions of H-systems and n-harmonic maps with derivatives in L^{n/2}, employing Hardy spaces, BMO, and the Rivi e{}re--Uhlenbeck decomposition.

Findings

Solutions are continuous under specified conditions.

Uses harmonic analysis techniques like Hardy spaces and BMO.

Employs Rivi e{}re--Uhlenbeck decomposition with Morrey space estimates.

Abstract

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 $W^{n/2,2}$ in an even dimension $n$, we prove their continuty. The tools used in the proof involve Hardy spaces and BMO, and the Rivi\`{e}re--Uhlenbeck decomposition (with estimates in Morrey spaces). A prominent role is played by the Coifman--Rochberg--Weiss commutator theorem.