EXISTENCE OF MINIMISERS FOR A CLASS OF FREE DISCONTINUITY PROBLEMS IN THE HEISENBERG GROUP Hn 1

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Abstract:

The purpose of this paper is to prove existence of minimisers of
the functional

$J(K,u):=\int_{\Omega\setminus K}f(L_u){\rm d}x+\alpha \int_{\Omega\setminus K}|u-g|^q {\rm d}x+\beta \sdqe(K\cap\Omega),$
where

$\Omega$ is an open set of the Heisenberg group

$\hn$ ,

$K$ runs over all closed sets of

$\hn$ ,

$u$ varies in

$C_H^1(\Omega\setminus K)$ ,

$\alpha$ ,

$\beta >0,q\ge 1,g\in L^q(\Omega)\cap L^\infty(\Omega)$ and

$f:R^{2n} \rightarrow R$ is a
convex function satisfying some
structure conditions (H

$_1$ )(H

$_2)$ (H

$_3)$ (see below).

Key words: SBVH function, Heisenberg group, minimiser;energy;deviation, free dis-
continuity problem

CLC Number:

26A45

SONG Ying-Qing, YANG Xiao-Beng, QIN Jiao-Hua. EXISTENCE OF MINIMISERS FOR A CLASS OF FREE DISCONTINUITY PROBLEMS IN THE HEISENBERG GROUP Hn 1[J].Acta mathematica scientia,Series B, 2005, 25(3): 455-469.

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