EXISTENCE OF MINIMISERS FOR A CLASS OF FREE DISCONTINUITY PROBLEMS IN THE HEISENBERG GROUP Hn

The purpose of this paper is to prove existence of minimisers of the functional J(K,u):=∫f(Lu)dx+α∫‖u−g‖qdx+βDdQ−1(K∩Ω),where Ω is an open set of the Heisenberg group Hn, K runs over all closed sets of Hn, u varies in CH1(Ω/K),α,β>0,q≥1,g∈Lq(Ω)∩L∞(Ω) and f : R2n → R is a convex function satisfying some structure conditions (H1)(H2)(H3) (see below).