BV Functions In Heisenberg Groups And Free Discontinuity Problems

In chapter 1, we introduce all the basic ingredients of C-C spaces and their special cases: sub-Riemannian groups and Heisenberg groups. Some basic notions are introduced that are needed throughout the dissertation.In chapter 2 there are four goals: the first is to investigate some geometric properties of H-Caccioppoli sets, the second is to characterize the discontinuous set Su and jump set Ju of u ∈ BVH(Ω), the third is to study pointwise behavior of u ∈ BVH(Ω) and our effort is concentrated on showing approximate differentiability of u in the sense of Pansu's, while the last and the most important is to show that DHU with u ∈ BVH(Ω) as a Radon measure can be split into three parts (absolutely continuous part, jump part and Cantor part, respectively) just like the derivative of a BV function in the setting of Euclidean space.In chapter 3 we first show some important distributional characteristics of weak derivatives of BVH and SBVH functions as Radon measures and we also give some sufficient and necessary conditions that a BVH function becomes a SBVH function. Motivated by an idea of [5] , we secondly consider in this chapter the behaviour of u ∈E BVH(Ω) composed with a Lipschitz function to characterize SBVH fuctions, hence, to make preparation for proving the compactness theorem in the next section. We investigate the composed function v = f o u where u ∈ BVH(Ω) and f : R→ R is a Lipschitz function and we will see that the diffuse part, the jump part (see Definition 3.1.2) of the derivative DHV behave in a quite different way. In analogy with the classical chain rule formula, Hv= f'(u)Hu, while Hv = . When a class of problems of calculus variation are considered by means of the direct method, the compactness theorem of SBVH(Ω) will play an important role. Creating a criteria SBVH fuctions, we finally prove the compactness theorem ofSBVH(Ω).In chapter 4 we investigate the problem of variation of more general functionals of integral functions I(u) = fΩ f(x,u,Lu)dx for / being a Caratheodory function defined on Hn x R x R2n and for u ∈ SBVH(Ω), where Lu denotes the approximate derivative of the absolutely continuous part aHu with respect to DHu.Free discontinuity problems in Euclidean spaces have been attacked by many authors in the framework of BV and SBV spaces, one can find an intensive survey of this field in the monography [5] and the references therein. In chapter 5 we adress a class of free discontinuity problems in the Heisenberg group Hn. More precisely, we investigate the existence of minimising pairs (K, u) for the functional J(K, u) in (0.0.1), where α,β > 0,q > 1,g ∈ L9(Ω) n L∞(Ω), K varies in the closed subsets of Hn, u varies in CH1(Ω \ K), and / : R2n → R is fixed and is a convex function such that conditions (A1)(A2)(A3) in Section 5.1 are fulfilled.Differing from the Euclidean case, we adopt in ( 5.1.1) for the spherical Hausdorff measure SdQ-1 instead of Hausdorff measure Hd2n+1 because the intrinsic spherical Hausdorff measure SdQ-1 arises in the decomposition ( 2.6.34) of the derivative of a BVH function and it is an open problem whether it can be replaced by a geometric constant times the intrinsic Hausdorff measure Hd2n+1. As it is pointed out in [5] that the usual direct methods of calculus of variations do not apply easily to the functionals (0.0.1) . One of reasons is that there is no topology on closed sets which guarantees the compactness of the minimising sequences and the lower semiconti-nuity of the SDQ-1. To avoid this difficulty we introduce the relaxed functional F(u) in (0.0.2). For (0.0.2), using the SBVH compactness Theorem 3.4.2, one can get the existence of an absolute minimiser of ( 5.1.2). Based on the existence of minimisers of ( 5.1.2), we can deal with minimising pairs (K, u) for (0.0.1). At the same time some regularity results on minimisers of such free discontinuity problems have been obtained.