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

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

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

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

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

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数学物理学报(英文版) ›› 2005, Vol. 25 ›› Issue (3): 455-469.

宋迎清,杨孝平,秦姣华

Department of Mathematics &|Calculation, Hunan City University, Yiyang 413000, China
2School of Science, Nanjing University of Science &|Technology, Nanjing 210094, China

出版日期:2005-07-20 发布日期:2005-07-20
基金资助:
This work is supported by NNSF(10471063), Hunan NSF(03JJY4002) &
Hunan Education Administration Item(03A011)

SONG Ying-Qing, YANG Xiao-Beng, QIN Jiao-Hua

Online:2005-07-20 Published:2005-07-20
Supported by:
This work is supported by NNSF(10471063), Hunan NSF(03JJY4002) &
Hunan Education Administration Item(03A011)

摘要：

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

J (K, u) := \int_{Ω ∖ K} f (L_{u}) d x + α \int_{Ω ∖ K} | u - g |^{q} d x + β \sdqe (K \cap Ω),

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

Ω

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

\hn

$\hn$ ,

K

$K$ runs over all closed sets of

\hn

$\hn$ ,

u

$u$ varies in

C_{H}^{1} (Ω ∖ K)

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

α

$\alpha$ ,

β > 0, q \geq 1, g \in L^{q} (Ω) \cap L^{\infty} (Ω)

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

f : R^{2 n} \to R

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

_{1}

$_1$ )(H

_{2})

$_2)$ (H

_{3})

$_3)$ (see below).

Abstract:

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

J (K, u) := \int_{Ω ∖ K} f (L_{u}) d x + α \int_{Ω ∖ K} | u - g |^{q} d x + β \sdqe (K \cap Ω),

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

Ω

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

\hn

$\hn$ ,

K

$K$ runs over all closed sets of

\hn

$\hn$ ,

u

$u$ varies in

C_{H}^{1} (Ω ∖ K)

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

α

$\alpha$ ,

β > 0, q \geq 1, g \in L^{q} (Ω) \cap L^{\infty} (Ω)

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

f : R^{2 n} \to R

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

_{1}

$_1$ )(H

_{2})

$_2)$ (H

_{3})

$_3)$ (see below).

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

中图分类号:

26A45

宋迎清, 杨孝平, 秦姣华. EXISTENCE OF MINIMISERS FOR A CLASS OF FREE DISCONTINUITY PROBLEMS IN THE HEISENBERG GROUP Hn 1[J]. 数学物理学报(英文版), 2005, 25(3): 455-469.

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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