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):=∫Ω\K f(Lu)dx α∫Ω\K |u - g|qdx βSQ-1d(K∩Ω),where Ω is an open set of the Heisenberg group Hn, K runs over all closed sets of Hn, u varies in C1H(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).