Partial regularity for sub-elliptic systems with Dini continuous coefficients involving natural growth terms in the Heisenberg group (Q2019164)

In this paper, the authors study the partial regularity of weak solutions, satisfying a natural growth condition, of a nonlinear sub-elliptic system in divergence form on the Heisenberg group \(\mathbb{H}^n\) of the kind \[ -\sum_{j=1}^{2n}X_jA_j^\alpha(z,u(z),Xu(z))=B^\alpha(z,u(z),Xu(z)),\,\,\,\,z\in\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{H}^n\), \(\alpha=1,\ldots,N,\) \(X=(X_1,\ldots,X_{2n})\) (the \(X_j\) being the basic vector fields of the Lie algebra of \(\mathbb{H}^n\)), \(u:\Omega\rightarrow\mathbb{R}^N\). The coefficients \(A_j^\alpha\) of the principal part are assumed to be Dini-continuous. Hence the authors weaken the regularity hypothesis from Hölder-continuous to Dini-continuous. Here, they apply the technique of \(\mathcal{A}\)-harmonic approximation adapted to the setting of the Heisenberg group. Their result is optimal, in that when the coefficients are Hölder-continuous they immediately get the optimal Hölder exponent for the horizontal gradient \(Xu\), of a weak solution \(u\), on \(\Omega\setminus\mathrm{Sing}(u)\).