$C^{1, \alpha}$-Regularity for p-Harmonic Functions in SU(3)

Abstract

This artical concerns the $C_{\mathrm{loc}}^{1, \alpha}$-regularity of weak solutions u to the degenerate subelliptic p-Laplacian equation

$\Delta_{\mathcal{H}, p} u(x)=\sum_{i=1}^{6} X_{i}^{*}\left(\left|\nabla_{\mathcal{H}} u\right|^{p-2} X_{i} u\right)=0,$

where $\mathcal{H}$ is the orthogonal complement of a Cartan subalgebra in SU(3) with the orthonormal basis composed of the vector fields X₁,...,X₆. When $1<p<2$, we prove that $\nabla_{\mathcal{H}} u \in C_{\mathrm{loc}}^{\alpha}$.

Keywords

p-Laplacian equation / $C^{1, \alpha}$-regularity / SU(3) / Caccioppoli inequality / De Giorgi / p-harmonic function