C1,α-regularity for quasilinear degenerate elliptic equation with a drift term on the Heisenberg group

Abstract

The aim of the paper is to derive the C1,α-regularity of weak solutions to a quasilinear degenerate elliptic equation with a drift term on the Heisenberg group. Establishing a Caccioppoli inequality (i.e. energy estimate) for the horizontal gradient of weak solutions, we use it and the Sobolev embedding theorem on the Heisenberg group and the Moser iteration method to prove the local boundedness of the horizontal gradient of weak solutions. Then we prove an oscillation estimate for the horizontal gradient of weak solutions and combine the iteration Lemma to show that the horizontal gradient of weak solutions is Hölder continuous.