Hölder Continuity for Sub-Elliptic Systems Under the Sub-Quadratic Controllable Growth in Carnot Groups

Abstract

This paper is devoted to optimal partial regularity of weak solutions to nonlinear sub-elliptic systems for the case $1<m<2$ under the controllable growth condition in Carnot groups. We begin with establishing a Sobolev-Poincaré type inequality for the function $u\in HW^{1,m}(\Omega),\mathbb {R}^{N})$ with $m\in (1,2)$, and then partial regularity with optimal local Hölder exponent for horizontal gradients of weak solutions to the systems is established by using ${\mathcal A}$-harmonic approximation technique.