Existence of solutions and regularity near the characteristic boundary for sub-Laplacian equations on Carnot groups

Dimiter Vassilev

Vol. 227 (2006), No. 2, 361–397

DOI: 10.2140/pjm.2006.227.361

Abstract

We prove that the best constant in the Folland–Stein embedding theorem on Carnot groups is achieved. This implies the existence of a positive solution of the Yamabe-type equation on Carnot groups. The second goal of the paper is to show a certain regularity of the Green’s function and solutions of the Yamabe equation involving the sub-Laplacian near the characteristic boundary of a domain in the considered groups.

Keywords

subelliptic regularity, Sobolev embedding, Carnot groups

Mathematical Subject Classification 2000

Primary: 35J70

Milestones

Received: 14 September 2004

Revised: 31 August 2005

Accepted: 10 October 2005

Published: 1 October 2006