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## Reduced problem results in dimension n k

### The lim inf inequality for the dimension-reduced problem .

#### Origin of the model and preliminaries

**Table of contents :**

Introduction

**1 Ane cost function **

1.1 Introduction

1.2 Preliminaries for the chapter

1.3 Local Result

1.4 Equicoercivity and -liminf

1.5 -limsup inequality

1.6 Numerical Approximation

**2 Multidimensional case **

2.1 Introduction

2.2 Reduced problem results in dimension n k

2.3 Compactness

2.4 -liminf inequality

2.5 -limsup inequality

**3 The k-dimensional problem **

3.1 Introduction

3.2 Compactness and k-rectiability

3.3 -liminf inequality

3.4 -limsup inequality

3.5 Discussion about the results

**4 Piecewise ane cost functions **

4.1 Introduction

4.2 Remarks

4.3 The -limit of the phase eld functional

4.3.1 The lim inf inequality for the dimension-reduced problem .

4.3.2 The lim inf inequality

4.3.3 Equicoercivity

4.3.4 The lim sup inequality

4.4 Numerical experiments

4.4.1 Discretization

4.4.2 Optimization

4.4.3 Experimental results

**5 Generalized cost functions **

5.1 Introduction

5.2 Origin of the model and preliminaries

5.3 Proof of Theorem 5.1

Conclusion

**A Density result for vector measures in R2 **

**B Reduced problem in dimension n k **

B.1 Auxiliary problem

B.2 Study of the transition energy

B.3 Proof of Proposition 2.1

B.4 Proof of Proposition 2.2

B.5 Proof of Proposition 2.3

**C Slicing of measures **

**D Resume substantiel en langue francaise**