Heisenberg群上的有界变差函数及其自由不连续问题

摘要

第一章为以后的需要我们引入C-C空间和其特殊情形:次Riemann群和Heisenberg群.第二章有四个目标;第三章我们先讨论Heisenberg群H<'n>上有界变差函数和特殊有界变差函数的弱导数作为Radon测度的若干重要分布特征.第四章我们研究一类泛函I(u)=f<,Ω>f(x,u,L<,u>)dx的下半连续性.第五章我们提出了H<'n>上的一类与偏微分方程和极小曲面联系十分紧密的自由不连续问题.

目录

摘要

Abstract

preface

1C-CSpaces and Heisenberg Groups

1.1 C-C spaces

1.2 Sub-Riemannian groups

1.3 Heisenberg groups

2The Measure Decomposition

2.1 Introduction

2.2 Preliminary results

2.3 Geometric properties of H-Caccioppoli sets

2.4 Approximate continuity of u ∈ BVH(Ω)

2.5 Approximate differentiability

2.6 Decomposition of DHu for u ∈ BVH(Ω)

3Characterization of BV and SBV Functions

3.1 Properties of DHu

3.2 Chain rule of BVH functions

3.3 Criterion on SBVH functions

3.4 Compactness theorem

3.5 SBVH compactness theorems in full generality

4LowerSemicontinuityInSBVH

4.1 Lusin approximation of BVH functions

4.2 Lower semicontinuity

5Existence of Minimisers

5.1 Introduction

5.2 Poincare inequality in SBVH

5.3 Limit behaviour of sequences in SBVH

5.4 The density lower bound

5.5 The existence of minimisers

Acdnowledgement

致谢

Bibliography