### Universit`

### a degli Studi di Pisa

DIPARTIMENTO DI MATEMATICACorso di Laurea in Matematica

Tesi di laurea triennale

### A discrete variational approximation of the

### Mumford-Shah functional in dimension one

Candidata

Clara Antonucci

Relatore

**Contents**

**1** **Introduction** **1**

1.1 The Mumford-Shah functional and its applications . . . 1

1.2 Lack of convexity . . . 2

1.3 The discrete setting . . . 3

1.4 The approximation proposed by Chambolle . . . 6

1.5 General results . . . 7

**2** **Gamma Convergence** **9**
2.1 Limsup inequality . . . 9

2.2 Liminf inequality . . . 10

**3** **Equicoerciveness** **15**
3.1 Proof of a property of equicoerciveness . . . 15

3.2 Existence of minimum for the Mumford-Shah functional . . . 16

**4** **Descending Metric Slope** **19**
4.1 Definition and main properties . . . 19

4.2 Γ-limsup of slopes . . . 22

**5** **Metric Slope of the Mumford-Shah functional** **27**
5.1 Slope of the Dirichlet functional . . . 27

5.2 Slope of the Mumford-Shah functional . . . 29

**6** **Convergence of Metric Slopes** **39**
6.1 Slope of the approximating functionals . . . 39

6.2 Main statement . . . 41

6.3 Limsup inequality . . . 41

6.3.1 First proof . . . 42

6.3.2 Second proof . . . 43

## 1

**Introduction**

**1.1**

**The Mumford-Shah functional and its applications**

In order to process an image in computer vision, it is often useful to simplify the image
by solving the so called “segmentation problem”, i.e. associating to the original picture,
*represented by g : R → R (where R denotes a domain of R*2), a set Γ of Hausdorff
*dimension one and a function f such that f is smooth in each connected component Ri*

*of R \ Γ.*

*Intuitively, Γ represents the edges of the image, while f represents the color in levels*
of gray of each point, which is supposed to vary smoothly within the edges of the picture.
This idea was first introduced in 1989 in a very important article by David Mumford
and Jayant Shah ([12]). As pointed out by Mumford and Shah themselves, this model
cannot be applied to every image; for example pictures containing textured objects or
partially transparent objects are too complex to fit into this model. Nevertheless there
are a lot of cases for which this model turns out to be good enough.

The first functional that they defined in their article is the following

*E(f, Γ) = µ*2
¨
*R*
*f (x, y) − g (x, y)*2 *dx dy +*
¨
*R\Γ*
*k∇f (x, y)k*2 *dx dy + ν |Γ| , (1.1.1)*
*where µ and ν are two real parameters that can be adjusted depending on the concrete*
problem one wants to solve.

A quick look to the functional gives us immediately an intuitive idea of the reason
*why minimizing E(f, Γ) is related to finding the best approximation of the original*
*picture g*

*the first term forces the reconstructed image f to be similar to the original image g;*
*the second term forces f to vary smoothly in each connected component of R \ Γ;*
the third term forces the edges to be as short as possible.

This thesis will deal with this problem only in dimension one, therefore the image

*g : R → R is replaced by a function h : [0, 1] → R, that can be interpreted as a signal. In*

*the sequel we will assume that µ = ν = 1, however all the statements presented in this*
*thesis remain valid, with minor changes in the proof, also for arbitrary µ, ν. For any*
*function u in the space of Special functions of Bounded Variation on (0, 1) the functional*
defined in (1.1.1) becomes
*E(u) =*
ˆ 1
0
*u (x) − h (x)*2 *dx +*
ˆ
*[0,1]\Su*
˙
*u (x)*2 *dx + #S _{u},* (1.1.2)

*where Su* *denotes the set of essential discontinuity points of u.*

We recall that in dimension one the space of Special functions of Bounded Variation
*can be described as the set of functions that are piecewise in H*1. A useful remark is
*that every function u in SBV (0, 1) can be decomposed in u = v + w where v is piecewise*
*constant (i.e. it has only jumps) and w belongs to H*1*(0, 1).*

In order to simplify the statements that follow, it is useful to define
*MS(u) =*

ˆ

*[0,1]\Su*

˙

*u (x)*2 *dx + #Su,* (1.1.3)

*then in Chapter 3 we will return to the original Mumford-Shah energy E(u), that includes*
the fidelity term ´*(u − h)*2.

**1.2**

**Lack of convexity**

An interesting remark is that it is not possible to approximate the Mumford-Shah functional by the Γ-convergence of local integral functional depending only on the first derivative (see [4]).

First, we recall the definition of Γ-convergence.

**Definition 1.2.1 (Γ-convergence).**

*If (X, d) is a metric space and Fn* : X → R is a sequence of functions, we say that the

*function F is the Γ-limit of the sequence Fn* if both the following inequalities hold.

*Liminf inequality: for every x*0 *in X, for every sequence xnsuch that xnconverges to x*0,

then

lim inf

*n→+∞Fn(xn) ≥ F (x*0*).*

*Limsup inequality: for every x*_{0} _{in X there exists a sequence (called recovery sequence)}

*xn* *such that xn* *converges to x*0 and

lim sup

*n→+∞*

*Fn(xn) ≤ F (x*0*).*

This notion of convergence was defined by E. De Giorgi (see [9]) and it is the most appropriate in the context of calculus of variation.

Therefore, given a family of local integral functionals depending only on the first derivative

*Fε(u) =*

ˆ 1 0

*fε*( ˙*u),*

*we know that, under appropriate hypotheses of growth on fε*, the relaxation is

*Fε(u) =*

ˆ 1 0

*f _{ε}*∗∗( ˙

*u),*

*where f _{ε}*∗∗

*denotes the convex envelope of fε*.

1.3. The discrete setting

• the Γ-limit of a family of functionals coincides with the Γ-limit of the corresponding relaxations,

• the functional ˆ 1

0

*f _{ε}*∗∗( ˙

*u) is convex for every ε,*

• the Γ-limit of a family of convex functionals is a convex functional.

Therefore such functionals can only approximate a convex functional, which is not the case of the Mumford-Shah functional, as we show in this example.

**Example 1.2.2. Let u : [0, 1] → R be defined by**

*u(x) :=*
*0 if x ∈*h*0,*1_{2}i*,*
*1 if x ∈*1_{2}*, 1*i*,*

*and let v : [0, 1] → R be identically zero.*
Then it holds that

*MS (u) = 1,*
*MS (v) = 0,*
MS
* _{u + v}*
2

*= 1,*and therefore 1 = MS

*2*

_{u + v}*>*

*MS (u) + MS (v)*2 = 1 2

*,*which shows that the Mumford-Shah functional is not convex.

For this reason a lot of different strategies have been developed since 1990, and some interesting approximations of the Mumford-Shah functional have been presented. Among the main ideas, the introduction of an auxiliary variable (see [2]), a second order perturbation (see [1]) and the Γ-convergence of a family of non-local functionals (see [5] and [11]).

In this thesis we present a discrete variational approximation of the Mumford-Shah functional introduced by A. Chambolle (in 1992 in dimension one, [6], and in 1995 in dimension two, [7]) and a Γ-convergence result for the respective metric slopes (a similar problem was considered by M. Gobbino in [10] for another approximating sequence).

**1.3**

**The discrete setting**

We now give some definitions and statements that are very useful in a discrete context.

**Definition 1.3.1 (P C**_{n}**(0, 1)).**

*We define P Cn(0, 1) as the set of all functions*

*u : [0, 1] → R such that u is constant in*

_{i}*n,*
*i + 1*
*n*
*for every i = 0, . . . , n − 1*
*.*

*Now we introduce a correspondence between P Cn(0, 1) and P An(0, 1) (i.e. the set*

of functions that are piecewise affine inh* _{n}i,i+1_{n}* i

*for i = 0, . . . , n − 1).*

**Definition 1.3.2 (Piecewise affine function associated to a piecewise constant function).**

*Given a function u in P Cn(0, 1), we associate to it the piecewise affine function v that*

joins the points_{n}i, u*i+1 _{n}*

*, as shown in Figure 1.1 (v is dashed). By definition, v is*constant inh

*n−1*

_{n}*, 1*i.

*u*

*v*

Figure 1.1

**Definition 1.3.3 (Discrete derivative).**

*Let [a, b] be an interval of R. For every function u : [a, b] → R we define the discrete*
derivative of step * _{n}*1

*of u as*

*D*±

*n*1

*u (x) =*˜

*u*

*x ±*1− ˜

_{n}*u (x)*±1

*n*

*,*

where ˜_{u : R → R is a function that coincides with u on [a, b] and such that}

(
˜
*u (x) = u (a) if x < a,*
˜
*u (x) = u (b) if x > b.*
*If u is a function of P Cn(0, 1), then D*
1

*nu| _{[0,1]}*

*is the function in P C*such that

_{n}
*D*1*nu*
*i*=
*ui+1−ui*
1
*n*
*= nhi* *if i = 1, . . . , n − 1,*
*D*1*nu*
*n= 0.*

1.3. The discrete setting

We notice that the discrete derivative is exactly the derivative of the piecewise affine
*function associated to u by the correspondence in Definition 1.3.2.*

**Theorem 1.3.4 (A discrete version of Arzel`a-Ascoli theorem).**

*Let nk* *be a sequence of natural numbers approaching +∞ and unk* be a sequence of

functions such that
*(1) unk* *∈ P Cnk(0, 1),*

*(2) u _{n}_{k}* are uniformly bounded,

(3)
*D*
1
*nk _{u}_{n}*

*k*

*2*

_{L}are uniformly bounded.

*Then, up to subsequences, there exists a function u*∞ *such that unk* *converges to u*∞

uniformly.

*Proof. Let us consider the sequence znk, where znk* is the piecewise affine function that

*corresponds to unk* by Definition 1.3.2.

*Hypothesis (2) implies that z _{n}_{k}* are uniformly bounded and hypothesis (3) ensures that

*znk* are uniformly

1

2-H¨older, and therefore equicontinuous. By the classical version of

Arzel`*a-Ascoli theorem, there exists a function u*∞ *such that znk* *→ u*∞uniformly, up to

subsequences.

We notice that, by definition,

*kunk− znk*k∞*= Mk*·

1

*nk*

*,*

*where Mk* *is the maximum of D*

1

*nk _{u}_{n}*

*k*.

*Moreover, by hypothesis (3), we know that there exists a constant H such that for every*

*k in N*
*H ≥*
*D*
1
*nk _{u}_{n}*

*k*

*2 = v u u t*

_{L}*n*X

*i=1*1

*nk*

*D*1

*nk*

_{u}_{n}*k*2

*,*and therefore

*H ≥*√1

*nk*max

*D*1

*nk*

_{u}_{n}*k*

*,*

*H*√

*nk*≥ max

*D*1

*nk*

_{u}_{n}*k*

*.*Hence

*ku*k

_{n}_{k}− z_{n}_{k}_{∞}

*≤ H ·*√

*nk*· 1

*nk*

*→ 0,*

*and this implies that u*

_{n}_{k}*→ u*∞ uniformly, as desired.

**1.4**

**The approximation proposed by Chambolle**

We now want to study the discrete variational approximation of the Mumford-Shah
*functional proposed by Antonin Chambolle. For every function u in P Cn(0, 1) he defined*

Ψ*n(u) :=*
*n−1*
X
*i=1*
1
*nψn(nhi*) =
ˆ 1
0
*ψn*
*Dn*1*u*
*,*
where
*ψn(x) := min*
n
*x*2*, n*o

is the truncated quadratic potential.

*n*

−√*n* √*n*

*Figure 1.2: ψn(x)*

*A possible interpretation of this definition is that a function u : [0, 1] → R can be*
conceived as a string subject to an elastic force. The energy of a configuration is therefore
proportional to the square of the elongation that the string undergoes. However, beyond
a certain threshold, which represents the maximum mechanical resistance of the system,
the string breaks, thus the potential stabilizes at a certain constant.

This is related to the weak membrane energy model (by A.Blake and A.Zissermann, [3]), also known as softening effect.

*Now we slightly modify the functions ψn* *and therefore the functionals Fn* in order

to gain more regularity.

**Definition 1.4.1 (Regularization of ψ**n**).**

*Let us choose any sequence rn*= O

1

*n*2

, and let us define

*ϕn(x) :=*
*ψn(x)* *if x ∈ [−*
√
*n + rn,*
√
*n − rn*]
*ψn(x)* *if |x| ≥*
√
*n*
*γn(x)* *if x ∈ [*
√
*n − rn,*
√
*n]*
*γn(−x)* *if x ∈ [−*
√
*n, −*√*n + rn*]

1.5. General results

*where γn* *is chosen properly so that for every n there exists an appropriate pn* in

(√*n − rn,*

√

*n) such that the following requirements hold*

*• x*2 _{≤ ϕ}*n(x) ≤ n in [*
√
*n − rn,*
√
*n],*
*• ϕn(x) ∈ C*1(R),

*• ϕn(x) is increasing in [0, +∞) (and therefore decreasing in (−∞, 0]),*

*• ϕ*0* _{n}(x) is increasing in [0, pn) and decreasing in (pn, +∞) (i.e. ϕn*is convex-concave

*in [0, +∞)),*
*• ϕ*0

*n(x) ≤ 2*

√

*n in R.*

These hypotheses also ensure that
*• ϕn(x) ≤ x*2*+ λn* on [
√
*n − rn,*
√
*n], where λn*→ 0,
*• ϕ*0* _{n}(pn) − ϕ*0

*n*( √

*n − rn) ≤ 2rn*.

It is easy to verify that the set of functions that satisfy these hypotheses is not empty (for example rewriting all the conditions in terms of the derivative, that has to be only continuous).

**Definition 1.4.2. Let u be a function in P C**n(0, 1). We define

*Fn(u) :=*
*n−1*
X
*i=1*
1
*nϕn(nhi*) =
ˆ 1
0
*ϕn*
*Dn*1*u*
*.*
*If u ∈ L*2*(0, 1) \ P C _{n}(0, 1), we define F_{n}(u) = +∞.*

These functionals are slightly different from the functionals introduced by Chambolle, however the difference is only important for the purposes of the computation of the metric slope (see Chapter 6), while the proofs of the Γ-convergence that we provide in Chapters 2 and 3 remain valid also in the context of Chambolle’s definition.

**1.5**

**General results**

In this section we recall a short list of well-known theorems about Γ-convergence that is useful to state at the beginning of this thesis.

**Theorem 1.5.1 (Convergence of minima and minimizers).**

*Let us consider a metric space (X, d) and a sequence of functions Gn* : X → R that

*Γ-converges to the function G. Let us assume that there exists a subset K of X that*
satisfies this properties

• inf

*u∈K{Gn*} = inf*u∈X{Gn} for every n,*

*• for every sequence of points (u _{n}) ⊆ K such that*

lim
*n→+∞*
*Gn(un*) − inf
*u∈XGn(u)*
*= 0,*

Then the following conclusions hold
*(1) G admits minimum in X,*
(2) lim
*n→+∞*
inf
*u∈XGn(u)*
= min
*u∈XG(u),*

*(3) if (un) ⊆ K is any sequence of points such that*

lim
*n→+∞*
*Gn(un*) − inf
*u∈XGn(u)*
*= 0,*

and if*u is any limit point of un(i.e. some subsequence of un* *converges to u), then*

*u is also a minimum point for G.*

**Theorem 1.5.2 (Stability under continuous perturbations).**

*Let (X, d) be a metric space, let Gn*: X → R be a sequence of functions Γ-converging to

*some function G. Let H be a continuous function on X.*
*Then Gn+ H* *→ G + H.*Γ

* Theorem 1.5.3. Let (X, d) be a metric space, let Gn*: X → R be a sequence of functions.

*The Γ-liminf and Γ-limsup of the sequence Gn* are defined as follows

Γ − lim inf
*n→+∞Gn(x) = inf*
lim inf
*n→+∞Gn(xn) s.t. xn→ x*
*,*
Γ − lim sup
*n→+∞*
*Gn(x) = inf*
lim sup
*n→+∞*
*Gn(xn) s.t. xn→ x*
*.*

*Then for every x in X there exist two sequences yn, zn* *converging to x such that*

lim inf
*n→+∞Gn(yn*) = Γ − lim inf*n→+∞Gn(x) ,*
lim sup
*n→+∞*
*Gn(zn*) = Γ − lim sup
*n→+∞*
*Gn(x) .*

## 2

**Gamma Convergence**

The aim of this chapter is a Γ-convergence result, that can be stated as follows.

* Theorem 2.0.1. The functionals F_{n}* defined in Definition 1.4.2 Γ-converge to MS with

*respect to the L*2 convergence, that is to say that both the following inequalities hold

Limsup inequality: *∀u ∈ L*2_{(0, 1), ∃u}

*n*
*L*2

*→ u,* lim sup

*n→+∞*

*Fn(un) ≤ MS(u),*

Liminf inequality: *∀u ∈ L*2_{(0, 1), ∀u}

*n*
*L*2

*→ u,* lim inf

*n→+∞Fn(un) ≥ MS(u).*

In the next sections we give a proof of this theorem that, as far as we know, is different from the original proof by Chambolle.

**2.1**

**Limsup inequality**

*Proof of the limsup inequality.*

*Let us consider any u*_{0} *in SBV (0, 1) (if u*_{0} *is not in SBV (0, 1), then MS (u*_{0}) = +∞
*and the conclusion is trivial). Let x*1_{, . . . , x}k_{be the jump points of u}

0. Let us consider

*the representative of u*0 *that is continuous in every (xi, xi+1*).

*Let us define unas the function belonging to P Cn* *such that un*(* _{n}i) = u*0(

*) for every*

_{n}i*i = 1, . . . , n. Since u*0 *is uniformly continuous on the compact subsets of [0, 1] that do*

*not contain any xi, then u _{n}→ uL*2

_{0}. We prove now that

lim sup
*n→+∞*
*Fn(un) ≤ MS(u*0*).*
*Let us define Ii* :=
h_{i−1}*n* *,*
*i*
*n*

*. If Ii* *contains one or more discontinuity points of u*0, then

1

*nϕn(nhi) ≤ MS(u*0|*Ii),*

because the right hand side is greater than or equal to 1, while the left hand side is less than or equal to 1 (by Definition 1.4.1).

*Otherwise u*_{0} *is continuous in I _{i}*, hence, using again the requirements in Definition 1.4.1,

1
*nϕn(nhi*) ≤
1
*n(nhi*)
2
+ 1
*nλn*=
ˆ *i+1*
*n*
*i*
*n*
˙
*vn*2+
1
*nλn,*

*where vn* *is the piecewise affine function corresponding to un* by Definition 1.3.2. Using

now classical convexity arguments it holds that
1
*nϕn(nhi*) ≤
ˆ *i+1*
*n*
*i*
*n*
˙
*u*02+
1
*nλn= MS(u*0|*Ii*) +
1
*nλn.*

In both cases we have shown that 1

*nϕn(nhi) ≤ MS(u*0|*Ii*) +

1

*nλn,*

*for i = 1, . . . , n, and this implies that for every n*

*Fn(un) ≤ MS(u*0*) + λn.*

Then the conclusion follows taking the limsup.

**2.2**

**Liminf inequality**

*It is useful to divide this proof in two cases, depending on whether or not u*0 is bounded

in k·k_{∞}.

*Proof of the liminf inequality for u*0 *bounded.*

**Step 1: Let us choose the representative of u**_{0} *such that Im(u*_{0}*) ⊆ [−D, D] for some*

*D > 0. Let us define un* *as the truncation of the functions un* *between [−D, D]. It’s*

immediate to see that

*un*
*L*2

*→ u*_{0}*,*

*Fn(un) ≤ Fn(un).*

*Then it is sufficient to prove the liminf inequality for un, i.e for uniformly bounded un*.

**Step 2: The conclusion is trivial when lim inf*** _{n→+∞}Fn(un*) = +∞, so in the sequel

we can assume lim inf*n→+∞Fn(un) = M ∈ R. Let us consider a subsequence (not*

*relabelled) un* *such that un* approaches the liminf. Since lim*n→+∞Fn(un) = M , it*

*follows that F _{n}(u_{n}) ≤ M + 1 eventually.*
Let us define

*an*= #*hi* *such that n|hi*| ≥

√

*n*

*.*

Then, by Definition 1.4.2, it follows that

*Fn(un) ≥ an*·
1
*n· ϕn*(
√
*n),*
*Fn(un) ≥ an,*

and this implies that eventually

*M + 1 ≥ an.*

*Let us now define a decomposition u _{n}= v_{n}+ w_{n}* as follows

*ji* :=
*hi* *if |hi*| ≥ √1* _{n},*
0

*if |hi| <*√1

*(2.2.1)*

_{n},2.2. Liminf inequality

*where ji* *denotes the value v(i+1 _{n}*

*) − v(*.

_{n}i) as in Definition 1.3.1. wnis defined as un− vn*In the sequel we will call vn* *the “jump part” of un* *and wn* the “absolutely continuous

*part” of u _{n}*.

*Since M + 1 ≥ an*, it is possible to extract a subsequence (not relabelled) such that

*an≡ a. Then each function vn* may be identified as a vector of R*2a*, for example by the

following bijection

*vn*∼

*x*1* _{n}, . . . , xa_{n}, j_{n}*1

*, . . . , j*

_{n}a*,*

*where xi _{n}*

*are the points of jump of vn*

*and jni*are the heights of each jump.

*Since xi _{n}*

*∈ [0, 1] and ji*

*n* *∈ [−2D, 2D], by the compactness of balls in R2a*, there exists a

*subsequence vn*such that

*x*1* _{n}→ x*1

*, . . . , j*1

_{n}*→ j*1

*, . . . .*

*Therefore v _{n}→ vL*2

_{0}

*, where v*

_{0}

*is the piecewise constant function that in xi*

_{has a jump of}

*height ji* *(if xi* *= xi+1* *the height is ji+ ji+1, and so on). Consequently v*0 has at most

*a jumps (possibly less). Hence*

*un= vn+ wn,*
*un*
*L*2
*→ u*_{0}*,*
*vnL*
2
*→ v*0*.*
Then
*wn*
*L*2
*→ w*_{0}*:= u*_{0}*− v*_{0}*.*

**Step 3: We can now conclude the proof as follows.**

*M + 1 ≥ Fn(un*)
= X
*|hi*|≥√1* _{n}*
1

*nϕn(nhi*) + X

*|hi|<*√1

*1*

_{n}*nϕn(nhi*)

*= a + Fn(wn*)

*≥ a +*ˆ 1 0 ˙

*wn*2

*,*(2.2.2)

*where (2.2.2) follows by the fact that ϕ _{n}(x) ≥ x*2

*if |x| ∈ [0,*√

*n] as we said in Definition*

1.4.1.

*Now, by the weak compactness of balls in L*2, we have that

*M + 1 ≥*

ˆ 1 0

˙

*wn*2 ⇒ ˙*wn* w*∞ *up to subsequences.*

*Since w _{n}L→ w*2

_{0}

*, then w*∞= ˙

*w*0

*, and in particular this implies that w*0

*is in H*1

*(0, 1).*

*Moreover, by the lower semicontinuity of the L*2 _{norm, it holds that}

lim inf
*n→+∞*
ˆ 1
0
˙
*wn*2 ≥
ˆ 1
0
˙
*w*02*.*

Therefore, taking the liminf in (2.2.2), it follows that
*M = lim inf*
*n→+∞Fn(un*)
≥ lim inf
*n→+∞*
(
*a +*
ˆ 1
0
˙
*wn*2
)
*≥ a +*
ˆ 1
0
˙
*w*02
*≥ #S _{v}*

_{0}+ ˆ 1 0 ˙

*w*02

*,*(2.2.3)

*where Sv*0 *denotes the number of jumps of v*0. Finally, we observe that the right hand

*side of (2.2.3) is exactly MS (u*0*), because u*0 *= v*0*+ w*0 *where w*0 *∈ H*1*(0, 1) and v*0 is

piecewise constant. Therefore the proof is concluded.

*Before giving the proof of the liminf inequality if u*_{0}is unbounded, we state and prove
a useful lemma.

**Lemma 2.2.1. Let us consider u any function in SBV (0, 1) (we always consider the**

*piecewise continuous representative). Given a, b in [0, 1] such that a < b, we define*

*ya:= u (a) ,*
*yb* *:= u (b) .*
Then
MS*u|[a,b]*
≥ min
(
*1,(yb− ya*)
2
*b − a*
)
*.*
*Proof. If u has one or more points of jump in [a, b], then*

MS*u| _{[a,b]}*

*≥ 1,*otherwise MS

*u|[a,b]*= ˆ

*b*

*a*˙

*u*2

*,*

*and since u is continuous in [a, b], by classical convexity arguments it follows that*
ˆ *b*
*a*
˙
*u*2 ≥
ˆ *b*
*a*
*˙v*2 =
ˆ *b*
*a*
_{y}*b− ya*
*b − a*
2
*,*

*where v is the affine function joining the points (a, ya) and (b, yb*). Therefore we have

MS*u| _{[a,b]}*≥

*(yb− ya*)

2

*b − a* *,*

2.2. Liminf inequality

*Proof of the liminf inequality for u*0 *unbounded.*

*We prove that for every unL*

2

*→ u*0 it holds that

lim inf

*n→+∞Fn(un) = +∞.*

*In fact, let us suppose by contradiction that there exists a sequence u _{n}→ uL*2

_{0}such that

lim inf

*n→+∞Fn(un) = M < +∞.*

Let us now consider this kind of sets

*An:= [n, n + 1) for n ∈ Z.*

*Since u*_{0} *is unbounded, it is possible to find k := 2 dM e + 3 sets A _{n}*

_{1}

*, . . . , Ank*such that

L

*u*−1_{0} *(Ani*)

*> 0, where*L denotes the Lebesgue measure. Without loss of generality,

*at least dM e + 2 sets are obtained for an odd n _{i}*.

*Let us fix D > max {|ni*|} and let us define

*unD* *:= the truncation of un* *between [−D, D] ,*

*u0D:= the truncation of u*0 *between [−D, D] .*

*Then unD* *L*

2

*→ u0D*.

*What we have proved assuming that u*_{0} was bounded can be applied to the sequence

*unD* *converging to u0D* and therefore it holds that

*M = lim inf*

*n→+∞Fn(un*)

≥ lim inf

*n→+∞Fn*(*unD*)

*≥ MS(u _{0D}).*

*Therefore u _{0D}*

*is in SBV (0, 1) (we consider from now on the continuous*

*represen-tative of u0D). Let us now choose an increasing sequence of points x*1

*, . . . , xdM e+2*such

*that u _{0D}*

*is continuous in every xi*

*and every u*

_{0D}*xi*

*belongs to a different Ani* *(all ni*

*are chosen odd). Since n _{i}*

*are different and odd, the difference between u*

_{0D}*xi*

and

*u0D* *xj**with i 6= j is at least 1.*

Now, by Lemma 2.2.1, it holds that

MS*u0D*|*(xi _{,x}i+1*

_{)}≥ min

*1,*

*u0D*

*xi+1*

*− u*

_{0D}*xi*2

*xi+1* ≥ min

_{− x}i*1,*

*u0D*

*xi+1*

*− u*

_{0D}*xi*2

*≥ 1.*

*Since xi* *is an increasing sequence, then xi, xi+1* are disjoint connected subsets of
*[0, 1] and therefore*
*MS (u _{0D}*) ≥

*dM e+1*X

*i=1*MS

*u0D*|

*(xi*

_{,x}i+1_{)}

*,*

*≥ dM e + 1.*and this gives an absurd.

## 3

**Equicoerciveness**

**3.1**

**Proof of a property of equicoerciveness**

To be able to apply Theorem 1.5.1 (convergence of minima and minimizers) to the approximation of the Mumford-Shah functional that we have introduced, we still have to prove an equicoerciveness property, that can be stated as follows.

**Theorem 3.1.1. Let u**nbe a sequence of functions un: [0, 1] → R. Let us assume that

*there exists a constant M such that*

*kun*k∞*+ Fn(un) ≤ M*

*for every n in N.*

*Then there exists a subsequence (not relabelled) u _{n}*

*and a function u such that u*2

_{n}→ u.L*Proof. In this proof we use the same decomposition of u introduced in (2.2.1). So let*
*vn+ wn* *be the decomposition of un*in jump part and absolutely continuous part. Since

*unare uniformly bounded by hypothesis, both vn* *and wn* are uniformly bounded.

Using the same argument of the proof of Theorem 2.0.1, we can deduce that there exists
*a subsequence (not relabelled) v _{n}*

*and a function v such that*

*vn*
*L*2

*→ v.*

*Now we observe that the subsequence wn* satisfies the hypotheses of Theorem 1.3.4

(discrete version of Arzel`*a-Ascoli theorem). In fact, we have already noticed that wn*are

uniformly bounded, and we know that

*M ≥ Fn(un) ≥ Fn(wn*) ≥
ˆ 1
0
˙
*wn*2*,*
that implies
√
*M ≥ k ˙wn*k* _{L}*2

*.*

**3.2**

**Existence of minimum for the Mumford-Shah functional**

Now we return to our original problem as stated in (1.1.2), that is minimizing the energy

*E(u) =*
ˆ 1
0
*u(x) − h(x)*2 *dx +*
ˆ
*[0,1]\Su*
˙
*u (x)*2 *dx + #Su,*

*where h is a function in L*∞*(0, 1) (our signal that we would like to reconstruct).*
Let us define these approximating functionals

*Gn(u) := Fn(u) +*

ˆ 1 0

*u(x) − h(x)*2 *dx,* (3.2.1)
*where Fn*are the discrete functionals introduced in Definition 1.4.2.

*We can observe that there exists a constant c such that inf G _{n}_{(u) ≤ c for every n in N.}*

*This can be checked for example defining c :=*´

_{0}1

*h*2

*and then noticing that for u*0 ≡ 0

*Gn(u*0*) = c for every n in N.*

*In Chapter 2 we have shown that the functionals F _{n}* Γ-converge to MS. Since
Γ-convergence is stable under continuous perturbations (see Theorem 1.5.2), it follows
that

*Gn(u)*

Γ

*→ E(u).* (3.2.2)

We can finally obtain an existence result for the minimum of the Mumford-Shah
*energy in dimension one. Our aim is to apply Theorem 1.5.1 to the functionals Gn*

*defined in (3.2.1). We have already shown that G _{n}(u)→ E(u). As for the compactness*Γ
property, it is possible to prove it as follows.

**Proposition 3.2.1. Let us consider a constant M greater than 0 and let us define**

*KM* :=

n

*u ∈ L*2*(0, 1) s.t. kuk*_{∞}*≤ M*o*.*

*Then, for every sequence of points u _{n}*

*in K*such that

_{M}lim

*n→+∞* *Gn(un*) −*u∈L*inf2* _{(0,1)}Gn(u)*
!

*= 0,*

*there exists a subsequence u _{n}_{k}*

*which converges in L*2 to some point

*u of KM*.

*Proof. Let us define*

*cn*:= inf

*u∈L*2_{(0,1)}Gn(u).

*As we noticed at the beginning of this section, there exists a constant c such that c _{n}≤ c.*

*Let us consider any sequence un*

*in KM*such that

lim
*n→+∞*
*Gn(un) − cn*
= 0
(if there are none, then the conclusion is trivial).

Then eventually it holds that

3.2. Existence of minimum for the Mumford-Shah functional

*Therefore, since Gn(u) = Fn(u) +*

´1

0*(u − h)*2, we deduce that

*Fn(un) ≤ c + 1.*

*Moreover, by definition of K _{M}, it holds that ku_{n}*k

_{∞}

*≤ M . Therefore we can apply*

*Theorem 3.1.1 and deduce that there exists a subsequence (not relabelled) un*and a

*function u such that un* *L*

2

*→ u. Finally, we observe that u ∈ KM*, as desired.

*Moreover, since h ∈ L*∞*(0, 1), there exists a constant M such that*

inf

*u∈KM*

*{G _{n}*} = inf

*u∈L*2* _{(0,1)}{Gn*}

*for every n. Therefore we have shown that both the hypotheses of Theorem 1.5.1 hold,*
and we gain the following conclusions

*(1) E(u) admits minimum in L*2*(0, 1) for every h in L*∞*(0, 1) ,*

(2) lim

*n→+∞* *u∈L*inf2* _{(0,1)}Gn(u)*
!

= min

*u∈L*2_{(0,1)}E(u),

*(3) If (un) ⊆ KM* is any sequence of points such that

lim

*n→+∞* *Gn(un*) −*u∈L*inf2* _{(0,1)}Gn(u)*
!

= 0

*and if u is any limit point of un(i.e. there exists a subsequence of un*that converges

## 4

**Descending Metric Slope**

**4.1**

**Definition and main properties**

First of all we recall the definition of descending metric slope.

**Definition 4.1.1 (Descending metric slope).**

*Let (X, d) be a metric space, let f : X → R be a function and let x*0 ∈ X be any point

*such that f (x*0*) is a real number. The metric slope of f at x*0 is defined as

*|∇f |(x*0) := lim sup

*r→0*+

*f (x*0*) − inf {f (x) : d(x, x*0*) ≤ r}*

*r* *.*

It is often useful, although not necessary, to define
*|∇f |(x*_{0}) = +∞
*if f (x*0) = +∞.

*We observe that |∇f |(x*_{0}) is always greater than or equal to 0 (including +∞),
*because x*0 is a competitor for the infimum.

*Through this chapter we denote for simplicity the closed balls of X with the symbol B.*

**Lemma 4.1.2. Let us assume that x**_{0} _{is not an isolated point of X. Then it holds that}

*|∇f |(x*0) = lim sup

*x→x*0

*max {f (x*0*) − f (x), 0}*

*d(x, x*0)

*.* (4.1.1)

*Proof. Let M denote the right hand side of (4.1.1).*

**Step 1: We show that |∇f |(x**_{0}*) ≥ M .*

*The conclusion is trivial when M = 0, so in the sequel we assume that M > 0. Since*
*the limsup is a maxlim, there exists a sequence xn→ x*0 such that

lim

*xn→x*0

*max {f (x*_{0}*) − f (x _{n}), 0}*

*d(xn, x*0)

*= M .*

*Since M > 0, the numerator must be eventually positive, hence*
lim

*xn→x*0

*f (x*0*) − f (xn*)

*d(xn, x*0)

*Now let us set rn:= d(xn, x*0*) and in:= inf {f (x) : d(x, x*0*) ≤ rn*}. It’s immediate to

*see that in≤ f (xn*), and therefore

*|∇f |(x*_{0}) = lim sup
*r→0*+
*f (x*0*) − inf {f (x) : d(x, x*0*) ≤ r}*
*r*
≥ lim sup
*n→+∞*
*f (x*0*) − in*
*rn*
≥ lim sup
*n→+∞*
*f (x*0*) − f (xn*)
*rn*
*= M.*

**Step 2: We show that |∇f |(x**_{0}*) ≤ M .*
*Let rn*→ 0+ be a sequence such that

*|∇f |(x*0) = lim

*n→+∞*

*f (x*0*) − in*

*rn*

*.*

*By definition of infimum, for every natural number n there exists yn* such that

(
*yn∈ B(x*0*, rn),*
*f (yn) < in+ rn*2*.*
It follows that
*M ≥* lim
*n→+∞*
*f (x*0*) − f (yn*)
*d(x*0*, yn*)
≥ lim
*n→+∞*
*f (x*0*) − in− rn*2
*d(x*0*, yn*)
≥ lim
*n→+∞*
*f (x*0*) − in− rn*2
*rn*
≥ lim
*n→+∞*
*f (x*0*) − in*
*rn*
*− r _{n}*

*= |∇f |(x*

_{0}

*),*as desired.

*We notice that asking that x*0 *is not an isolated point is necessary. In fact, if x*0 is

an isolated point, then the quantity lim sup

*x→x*0

*max {f (x*_{0}*) − f (x), 0}*

*d(x, x*0)

*is not well defined. However, if |∇f |(x*0*) 6= 0, then x*0 cannot be an isolated point, and

therefore Lemma 4.1.2 holds.

**Proposition 4.1.3. If X = R***n* *and f : X → R is a C*1 function, then the metric slope
is the norm of the gradient, i.e.

4.1. Definition and main properties

*Proof. If x*0 is a local minimum point, then

*|∇f | (x*0*) = k∇f (x*0*)k = 0.*

If not, we argue as follows.

**Step 1: We show that |∇f |(x**_{0}*) ≤ k∇f (x*_{0}*)k .*
*Let us consider the sequence r _{n}*→ 0 for which

lim

*n→∞*

*f (x*0*) − inf {f (x) : d(x, x*0*) ≤ rn*}

*rn*

*= |∇f | (x*_{0}*).*

*Since f is C*1 _{on R}*n*, the infima on closed balls are actually minima and therefore there
*exists a sequence xn* *such that xn→ x*0 and

lim

*n→∞*

*f (x*0*) − f (xn*)

*|x*0*− xn*|

*= |∇f |(x*0*).*

By definition of differentiability and by Cauchy-Schwarz inequality, it holds that

lim
*n→∞*
*f (x*0*) − f (xn*)
*|x*_{0}*− x _{n}*| = lim

*n→∞*

*h∇f (x*

_{0}

*) , x*

_{0}

*− x*i

_{n}*|x*

_{0}

*− x*|

_{n}*+ o(1) ≤ k∇f (x*0

*)k ,*

therefore we have shown that

*|∇f |(x*0*) ≤ k∇f (x*0*)k .*

* Step 2: We show that |∇f |(x*0

*) ≥ k∇f (x*0

*)k .*

*If k∇f (x*_{0}*)k = 0 this step is immediate, therefore we assume that ∇f (x*_{0}) 6= 0. For
*every v 6= 0 it holds that*

*|∇f |(x*_{0}) ≥ lim
*t→0*+
*f (x*0*) − f (x*0*− tv)*
*t kvk*
= lim
*t→0*+
*t h∇f (x*0*− ξtv) , vi*
*t kvk* *,*

*where ξt∈ [0, t] is given by the mean value theorem. Hence*

*|∇f |(x*_{0}) ≥ *h∇f (x*0*) , vi*

*kvk* *.*

*If we take v = ∇f (x*_{0}), it follows that

*|∇f |(x*_{0}*) ≥ k∇f (x*_{0}*)k ,*
and this completes the proof.

**4.2**

**Γ-limsup of slopes**

An interesting question about Γ-convergence is

*fn*

Γ

*→ f*∞ ⇒? *|∇fn*|

Γ

*→ ∇ |f*∞*| .*

Unfortunately the answer is negative and the question is also ill-posed, since the Γ-limit of slopes might not exist.

However, under appropriate hypotheses, we can prove at least one inequality between the Γ-limsup of slopes and the slope of the Γ-limit. This section is devoted to finding the optimal hypotheses of this statement.

**Theorem 4.2.1. Let (X, d) be a metric space and let f : X → R be a function. For**

*every x*_{0} _{∈ X we define}

*gx*0*(r) := inf {f (x) : d(x, x*0*) ≤ r} .*

Let us assume that

*(1) |∇f |(x) ≥ M for every x ∈ X,*

*(2) inf {f (x) : d(x, x*0*) ≤ r} is actually a minimum ∀r ∈ R, ∀x*0 ∈ X.

*Then g _{x}*0

0*(r) exists for almost every r ∈ R and g*

0

*x*0*(r) ≤ −M where it is defined.*

*Proof. Let us fix x*0 *∈ X. Since gx*0*(r) is a decreasing function, by Lebesgue’s theorem*

*for the differentiability of monotone functions, g _{x}*0

0*(r) exists almost everywhere. Let us*

suppose by contradiction that

*g*0_{x}_{0}*(r) > −M + 2δ*

*for some r, δ ∈ R*+*. Then there exists some h*_{0} *such that for every h ≤ h*_{0}

*gx*0*(r + h) − gx*0*(r)*

*h* *> −M + δ.* (4.2.1)

*Let us now take a point y ∈ B(x*_{0}*, r) such that f (y) = gx*0*(r) (it exists by hypothesis*

*(2)). |∇f (y)| ≥ M by hypothesis (1), hence*
lim sup

*r→0*+

*f (y) − inf {f (x) : d(x, y) ≤ r}*
*r* *≥ M.*

*Therefore there exists a radius r*0 *< h*0 such that

*f (y) − inf {f (x) : d(x, y) ≤ r*0}
*r*0
*> M −* *δ*
4*,*
*−f (y) + inf {f (x) : d(x, y) ≤ r*_{0}}
*r*0
*< −M +δ*
4*.*
But
*gx*0*(r + r*0*) − gx*0*(r)*
*r*0
*<* *−f (y) + inf {f (x) : d(x, y) ≤ r*0}
*r*0

4.2. Γ-limsup of slopes

(the inf is obtained on a smaller set). By (4.2.1) it follows that
*−M + δ < −M +* *δ*

4*,*
which is absurd, and therefore the theorem is proved.

* Lemma 4.2.2. Let X, f , x*0

*, gx*0 be as in Lemma 4.2.1 and let us assume that there

*exists a radius r*0 *such that g*0*x*0*(r) ≤ −M for every r ≤ r*0 *for which gx*0 is differentiable

*in r. Then*

*∀ε > 0, ∀r < r*_{0}*, ∃y ∈ B(x*0*, r) such that f (y) ≤ f (x*0*) − (M − ε)r.*

*Proof. By a general theorem, fixing r < r*0 we have

*gx*0*(r) − gx*0*(0) ≤ g*

0

*x*0*(c)r ≤ −M r,*

*for some c ∈ [0, r]. Therefore*

*inf {f (x) : d(x, x*0*) ≤ r} − f (x*0*) ≤ −M r.*

*It follows that for every ε > 0 there exists y in B(x*0*, r) such that*

*f (y) ≤ f (x*0*) − (M − ε)r,*

as desired.

Now we can prove the following theorem.

* Theorem 4.2.3. Let (X, d) be a metric space and fn*: X → R a sequence of functions

*Γ-converging to f*∞: X → R. Let us assume that

*• the function f _{n}*

_{is lower semicontinuous for every n in N (LSC),}*• for every r > 0, x*0 *∈ X, A ∈ R there exists a compact set K ⊆ X such that*

*{x ∈ X : d(x, x*0*) ≤ r, fn(x) ≤ A} ⊆ K for every n ∈ N (Local equicoerciveness).*

*Then ∀x ∈ X*

Γ − lim sup

*n→+∞*

*|∇fn|(x) ≤ |∇f*∞*|(x).*

*Proof. Let us fix x*0 *∈ X and ε > 0, and let us set*

*M := Γ − lim sup*

*n→+∞*

*|∇f _{n}|(x*

_{0}

*).*

*By definition of Γ-limsup, there exists a radius rε> 0 such that for infinitely many n*

*|∇fn|(x) ≥ M − ε* (4.2.2)

*for every x in B(x*_{0}*, 2rε). From now on, let fn* be a subsequence (not relabelled) for

*which the previous inequality holds. Let xn* *be a recovery sequence for x*0. Without

*loss of generality, we can assume that d(xn, x*0*) ≤ rε, so that B(xn, rε) ⊆ B(x*0*, 2rε*) for

local equicoerciveness, we can apply Theorem 4.2.1 and Lemma 4.2.2, that ensure that
*for every r < rε, for every n ∈ N there exists some yn∈ B(xn, r) such that*

*fn(yn) ≤ fn(xn) − (M − ε)r.* (4.2.3)

*By the equicoerciveness property, up to subsequences yn→ y*0 *for some y*0 ∈ X.

By taking the liminf in (4.2.3) and using the liminf inequality we obtain

*f*∞*(y*0) ≤ lim inf
*n→+∞fn(yn*) ≤ lim inf*n→+∞fn(xn) − (M − ε)r = f*∞*(x*0*) − (M − ε)r.*
Moreover
*d(x*0*, y*0) = lim
*n→+∞d(x*0*, yn) ≤ r.*
Therefore
*inf {f*∞*(x) : d(x, x*0*) ≤ r} ≤ f*∞*(x*0*) − (M − ε)r,*

and this implies

*|∇f*∞*|(x) ≥ M − ε,*

that concludes the proof.

We now show that this result is sharp.

**Example 4.2.4 (f**n**are locally equicoercive, but not lower semicontinuous).**

*Let us consider this sequence of functions fn: [0, 1] → R*

*fn(x) =*
1
*n− x for x ∈*
*0,* 1
*n*
*,*
*fn*
*x +* *k*
*n*
*= f _{n}(x) for k = 1, . . . , n − 1.*

*Since [0, 1] is compact the functions are locally equicoercive. The Γ-limit of the functions*

*fn* *is the function f*∞*≡ 0, and therefore |∇f*∞*|(x) = 0. However*

*|∇fn| (x) =*

(

1 *if x 6=* _{n}i*for i = 1, . . . , n,*
+∞ *if x =* _{n}i*for i = 1, . . . , n.*
Hence Γ − lim sup

*n→+∞*

*|∇f _{n}|(x) ≡ 1, and therefore Γ − lim sup*

*n→+∞*

*|∇f _{n}|(x) > |∇f*∞

*|(x).*

**Example 4.2.5 (f**n**are lower semicontinuous but not locally equicoercive).**

*Let us consider a real Hilbert space H with a countable base E = {ei* *: i ∈ N}. Let us*

define

*fn(x) :=*

(

*− |λ|* *if x = λen,*

+∞ *otherwise,*
*so that fn* *are lower semicontinuous. The slope of fn* is

*|∇f _{n}| (x) =*

(

1 *if x = λen,*

4.2. Γ-limsup of slopes

However it holds that

Γ − lim
*n→+∞fn(x) =*
(
0 *if x = 0,*
+∞ *otherwise,*
while
∇
Γ − lim
*n→+∞fn*
*(x) =*
(
0 *if x = 0,*
+∞ *otherwise,*
that implies
Γ − lim sup
*n→+∞*
*|∇f _{n}| (x) =*
(
1

*if x = 0,*+∞

*otherwise.*Therefore we have shown that Γ − lim sup

*n→+∞*

*|∇fn|(1) > |∇f*∞|(1), and this contradicts

Γ − lim sup

*n→+∞*

*|∇fn|(x) ≤ |∇f*∞*|(x).*

We now show that Theorem 4.2.3 cannot imply the existence of recovery sequences with bounded energy.

**Example 4.2.6. Let us consider this sequence of functions f**_{n}_{: [0, 1] → R}

*fn(x) =*
*x* *if x ≤* 1_{2} −1
*n,*
*n* *if x ∈*1_{2} −1
*n,*
1
2 +
1
*n*
*,*
*−x + 1 if x ≥* 1_{2} +1_{n}.

*These functions are lower semicontinuous. Since [0, 1] is compact, the property of local*
*equicoerciveness holds as well. This family of functions Γ-converges to f*∞, given by

*f*∞*(x) =*
(

*x* *if x ≤* 1_{2}*,*

*−x + 1 if x >* 1
2*.*

*For a fixed value of n it holds that*

*|∇fn| (x) =*
1 *if x ≤* 1_{2}− 1
*n,*
0 *if x ∈*1_{2}− 1* _{n},*1

_{2}+

*1*

_{n}*,*1

*if x ≥*1

_{2}+

*1*

_{n}*.*

*As n approaches infinity, this gives*

Γ − lim sup
*n→+∞*
*|∇fn| (x) =*
(
1 *if x 6=* 1_{2}*,*
0 *if x =* 1_{2}*.*

*Let us consider x*_{0} := 1_{2}*. Any recovery sequence xn* for Γ − lim sup*n→+∞∇ |fn*| in the

*point x*_{0} must satisfy

lim sup

*n→+∞*

*and this implies that xn*∈
_{1}
2 −
1
*n,*
1
2 +
1
*n*

*for infinitely many n. Therefore*
lim sup

*n→+∞*

*fn(xn) = +∞,*

*and so we have shown that there are no recovery sequences for the point x*_{0} having
bounded energy.

## 5

**Metric Slope of the**

**Mumford-Shah functional**

**5.1**

**Slope of the Dirichlet functional**

*Let us recall the definition of the Dirichlet functional in (a, b)*

*D(u) :=*
ˆ *b*
*a*
˙
*u*2 *if u ∈ H*1*(a, b),*
+∞ *if u ∈ L*2*(a, b) \ H*1*(a, b).*

**Theorem 5.1.1. Let us suppose that |∇D|(u) = M < +∞. Then it follows that**

*• (Regularity) u ∈ H*2* _{(a, b),}*
• (NBC)
(
˙

*u(a) = 0,*˙

*u(b) = 0.*

*We note that if u belongs to H*2*(a, b), then ˙u admits a continuous representative and*

therefore the Neumann Boundary Conditions make sense.

*Proof. Let us fix v ∈ H*1*(a, b) such that v 6= 0. By definition of metric slope*
*|∇D|(u) ≥ lim*
*t→0*
*D(u) − D(u + tv)*
*ktvk*
= lim
*t→0*
*D(u) − D(u + tv)*
*|t| kvk*
= lim
*t→0*
1
*|t| kvk* *−2t*
ˆ *b*
*a*
˙
*u ˙v − t*2
ˆ *b*
*a*
*˙v*2
!
= *−2 sign t*
*kvk*
ˆ *b*
*a*
˙
*u ˙v.* (5.1.1)
*Choosing opportunely either t → 0*+ *or t → 0*− in (5.1.1), it follows that

*M kvk ≥ 2*
ˆ *b*
*a*
˙
*u ˙v*
*.*

This implies that the linear functional
*H : L*2*(a, b) → R*
*v →*
ˆ *b*
*a*
˙
*u ˙v*

*is also continuous. Riesz’s theorem implies that there exists a function f ∈ L*2*(a, b) such*
that

*H (v) =*

ˆ *b*
*a*

*f v.*

*Therefore for every v ∈ H*1*(a, b)*
ˆ *b*
*a*
˙
*u ˙v =*
ˆ *b*
*a*
*f v,* (5.1.2)

*in particular this holds for every v ∈ C*_{0}∞*(a, b).* *This implies that −f is the weak*
derivative of ˙*u, therefore ˙u ∈ H*1* _{(a, b) and u ∈ H}*2

_{(a, b).}*However, since (5.1.2) holds also for every v ∈ H*1*(a, b), it follows that*
ˆ *b*
*a*
*f v =*
ˆ *b*
*a*
˙
*u ˙v =*
ˆ *b*
*a*
*f v + [ ˙uv]b _{a}*

and this implies that ˙*u(a) = 0 and ˙u(b) = 0, i.e. u satisfies the Neumann Boundary*

Conditions.

In particular in (5.1.1) we have proved that
*|∇D|(u) ≥* 2
*kvk*
ˆ *b*
*a*
¨
*uv*
(5.1.3)
*for every u in H*2*(a, b) that satisfies the Neumann Boundary Conditions and for every*

*v in H*1*(a, b).*

Now we can prove the following statement.

**Theorem 5.1.2 (Slope of the Dirichlet functional).**

*If u ∈ H*2*(a, b) and u satisfies conditions in Theorem 5.1.1, then*

*|∇D|(u) = 2 k¨uk .*
*Proof.*

**Step 1: |∇D|(u) ≥ 2 k¨**uk .

*Let w denote ¨u. We know that w is in L*2*(a, b), but w might not be in H*1*(a, b). However,*
*taking a sequence wnsuch that wn∈ H*1*(a, b) and wn* *L*

2

*→ w, and using (5.1.3), we obtain*
*|∇D|(u) ≥* 2
*kwn*k
ˆ *b*
*a*
¨
*uwn*
= 2
*kw _{n}*k
ˆ

*b*

*a*

*wnwn*+ ˆ

*b*

*a*(¨

*u − wn)wn*!

*C.S.*

*≥ 2 kwn*k − 2

*kwn*k k¨

*u − wnk kwn*k

*= 2 kw*k − 2 k¨

_{n}*u − wnk .*(5.1.4)

5.2. Slope of the Mumford-Shah functional

Taking the limit in (5.1.4) it follows that

*|∇D|(u) ≥ 2 k¨uk ,*

as desired.

**Step 2: |∇D|(u) ≤ 2 k¨**uk.

*Let us consider any w ∈ H*1*(a, b). Then*

*D (u) − D (w) = D (u) − D**(w − u) + u*
=
ˆ *b*
*a*
˙
*u*2−
ˆ *b*
*a*
( ˙*w − ˙u)*2− 2
ˆ *b*
*a*
( ˙*w − ˙u) ˙u −*
ˆ *b*
*a*
˙
*u*2
= −
ˆ *b*
*a*
( ˙*w − ˙u)*2− 2
ˆ *b*
*a*
( ˙*w − ˙u) ˙u*
≤ −2
ˆ *b*
*a*
( ˙*w − ˙u) ˙u*
= 2
ˆ *b*
*a*
*(w − u) ¨u − 2 [(w − u) ˙u]b _{a}*
= 2
ˆ

*b*

*a*

*(w − u) ¨u*

*≤ 2 kw − uk k¨uk .*Therefore

*D(u) − D(w)*

*kw − uk*≤ 2 k¨

*uk*

*for every w ∈ H*1

_{(a, b). This implies}*|∇D|(u) ≤ 2 k¨uk ,*

as desired.

**5.2**

**Slope of the Mumford-Shah functional**

This section is devoted to the computation of the metric slope of the Mumford-Shah functional. Before getting to the final theorem we state and prove some useful lemmas and we fix some notation.

* Definition 5.2.1. Let u*0

*be a function in SBV (0, 1) and let S*0:= n

*x*1*, . . . , xk*odenote
*the set of jump points of u*0. By definition of jump, we have

lim
*x→xi−*
*u*0*(x) = yi,*
lim
*x→xi+u*0
*(x) = zi,*
*with yi* *6= zi*. We define
*h := min {|zi− yi|} .*

*Let us consider any ε > 0. By definition of limit, for every i = 1, . . . , k there exists some*
*δ _{ε}i* such that

*u*0 h

*xi− δ*

_{ε}i, xi*⊆ [y*

_{i}− ε, y_{i}+ ε] ,*u*0

*xi, xi+ δ*i

_{ε}i*⊆ [zi− ε, zi+ ε] .*

*Let us define δ _{ε}*:= min

*δi _{ε}*

.

**Definition 5.2.2 (Little rectangles).**

*Let ε be any real positive number (it will be useful in the sequel to use ε <* *h*_{4}). We define
some little rectangles in R2 *nearby the points of jump (Li* is the “left-jump-rectangle”,
*while Ri* is the “right-jump-rectangle”)

*Li* :=
h
*xi− δε, xi*
i
*× [yi− 2ε, yi+ 2ε] ,*
*Ri* :=
h
*xi, xi+ δ _{ε}*i

*× [z*as shown in Figure 5.1.

_{i}− 2ε, z_{i}+ 2ε] ,*L*1

*R*1

*L*3

*R*2

*L*2

*R*3

*x*1

*x*2

*x*3 0 1 Figure 5.1

* Lemma 5.2.3. Let us consider x*0

*∈ [0, 1], y 6= z, h = |z − y|, ε <*

*|y−z|*

_{4}

*and δ <*

*(h−4ε)*

2

2 ,

*and let us define L _{ε}*

*and R*similarly to Definition 5.2.2

_{ε}*Lε:= [x*0*− δ, x*0*] × [y − 2ε, y + 2ε] ,*

*Rε* *:= [x*0*, x*0*+ δ] × [z − 2ε, z + 2ε] .*

*Let us consider any u ∈ SBV (0, 1) and let us assume that*

(

*graph(u) ∩ L _{ε}6= ∅,*

*graph(u) ∩ Rε*

*6= ∅.*

Then

5.2. Slope of the Mumford-Shah functional

*Proof. If u| _{[x}*

_{0}

*−δ,x*0

*+δ]*

*has at least one point of discontinuity, then MS(u|[x*0

*−δ,x*0

*+δ]*) ≥ 1.

*If not, we suppose without loss of generality that z > y.*
By classical convexity arguments

*MS(u| _{[x}*

_{0}

*−δ,x*0

*+δ]) ≥ MS(v),*(5.2.1)

*where v is the affine function joining the points (x*0*− δ, y + 2ε), (x*0*+ δ, z − 2ε), as shown*

in Figure 5.2. If we compute the right hand side of (5.2.1) we get

*xi*
Figure 5.2
*MS (v) =*
ˆ *x*0*+δ*
*x*0*−δ*
_{h − 4ε}*2δ*
2
= *(h − 4ε)*
2
*2δ* *> 1,*
therefore the proof is complete.

**Corollary 5.2.4. For every function that intersects both L**ε*and Rε* and is continuous

*in [x*_{0}*− δ, x*_{0}*+ δ], then*

*MS(u| _{[x}*

_{0}

*−δ,x*0

*+δ]*) ≥

*(h − 4ε)*2

*2δ* *.*

*Proof. The computation has already be done in the proof of Lemma 5.2.3.*

Now we need a last proposition, whose proof is a bit technical.

* Proposition 5.2.5. Let us consider u*0

*in SBV (0, 1) such that MS(u*0

*) = M . Let*

*S*0 :=
n

*x*1*, . . . , xk*o *be the set of points in which u*_{0} is discontinuous. We define

*X :=*n*u ∈ L*2*(0, 1) s.t u has exactly k points of jump*o*.*

*Then there exists a radius r*_{0} *(sufficiently small) such that for every r < r*_{0}*, for every u*
*in B (u*0*, r) ∩ X there exists ˜u ∈ X such that*

*• d (˜u, u*0*) ≤ d (u, u*0),

• MS (˜*u) ≤ MS (u),*

*Proof. If u /∈ SBV (0, 1), then MS (u) = +∞ and ˜u = u*0 verifies all the requirements.

*Therefore in the sequel we restrict ourselves only to u ∈ SBV (0, 1), for which we always*
consider the piecewise continuous representative.

*Let us choose ε such that 24ε < h. Let δε* *be as in Definition 5.2.1, and let us fix δ such*

that
*δ < min*
(
*δε,*
*(h − 4ε)*2
*2(M + 1),*
*h*2
*18 (M + 1)*
)
*.*

*Finally, let us put r*_{0}= *ε*

√

*δ*

2 .

* Step 1: We can observe that every u ∈ L*2

*(0, 1) such that d(u, u*0

*) ≤ r*0 must intersect

*all the little rectangles of u*_{0} introduced in Definition 5.2.2. In fact, if (without loss of
generality)
*graph(u) ∩ L*_{1} = ∅
then
*d(u, u*0) ≥
v
u
u
t
ˆ *x*1
*x*1_{−δ}*|u − u*0|2
≥
v
u
u
t
ˆ *x*1
*x*1_{−δ}*ε*2_{=}√* _{δε > r}*
0

*.*

*Therefore any u such that d(u, u*0*) < r*0 must intersect every little rectangle introduced

*in Definition 5.2.2. Now we recall that if u has no jump points in xi− δ, xi _{+ δ}*

for
*some i, then by Corollary 5.2.4 we know that*

*MS(u) ≥* *(h − 4ε)*

2

*2δ* *> M + 1,*
therefore ˜*u = u*0 verifies all the requirements.

*Otherwise the function u has at least one jump point in every xi− δ, xi _{+ δ}*

.

*Since u has exactly k points of jump, then u has exactly one point of jump pi* in every

*xi− δ, xi _{+ δ}*

.

*If pi= xi* *for every i, then ˜u = u verifies all the requirements. If pi* *6= xi* _{for some i, we}

*show that we can slightly modifify u and gain a ˜u that verifies all the requirements. Let*

*us suppose without loss of generality i = 1, y*1*< z*1 *and x*1*< p*1.

**Step 2: We now show that it is not restrictive to suppose that**

*u**x*1*− δ, p*1 ⊆
*−∞, y*_{1}*+ 2ε +h*
3
*,*
*u**p*1*, x*1*+ δ* ⊆
*z*1*− 2ε −*
*h*
3*, +∞*
*.*

*In fact, we know that L*1*∩ Im(u) 6= ∅, R*1_{∩ Im(u) 6= ∅. Hence, if the first inclusion is}

false, we can infer by classical convexity argument that
*MS(u| _{[x}*1

*1*

_{−δ,x}

_{+δ]}) ≥ MS(v),5.2. Slope of the Mumford-Shah functional

*where v is the segment joining the points (x*1*− δ, y*1*+ 2ε), (x*1*+ δ, y*1*+ 2ε +h*_{3}). But

*MS (v) =*
*h*
3
2
*2δ* *> M + 1,*

therefore ˜*u = u*0 verifies all the requirements, and we can use the same argument to

show that it is not restrictive to suppose that also the second inclusion holds.

**Step 3: Having assumed that both the inclusions of Step 2 hold, we slightly modify**

*u, as shown in the next figure.*

˜
*u(x) :=*
*u(x) if x < x*1*,*
*u(p*1_{)}+ _{if x ∈}*x*1* _{, p}*1

*,*

*u(x) if x > p*1

*.*

*u*0

*u*

*x*1

*1*

_{p}*u*0 ˜

*u*

*x*1

It is clear that MS(˜*u) ≤ MS(u), because the number of jumps stays constant and*

*the functions u and ˜u differ only in the interval x*1* _{, p}*1

, on which ˜*u is constant.*

*It is also clear that d(u*0*, ˜u) ≤ d(u*0*, u), because we have only modified u in**x*1*, p*1, and

in this interval we can compute the following estimates
ˆ *p*1
*x*1
*(u*0− ˜*u)*2 ≤
ˆ *p*1
*x*1
*4ε +* *h*
3
2
*,*
ˆ *p*1
*x*1
*(u*0*− u)*2 ≥
ˆ *p*1
*x*1
*z*1*− 2ε −*
*y*1*+ 2ε +*
*h*
3
2
*,*
and *z*1*− 4ε − y*1−*h*_{3}

*≥ 4ε +* *h*_{3} *because z*1*− y*1 *≥ h and h ≥ 24ε. This completes the*

proof.

Proposition 5.2.5 is the final step we need to be able to compute the metric slope of the Mumford-Shah functional using our previous computation of the slope of the Dirichlet functional.

* Theorem 5.2.6. Let us consider u*0

*in SBV (0, 1) such that MS(u*0) is a real number.

*Let S*0 :=
n

*x*1*, . . . , xk*o *be the set of points in which u*0 is discontinuous. We define

*x*0 *:= 0, xk+1* *:= 1. Let us assume that |∇ MS |(u*0*) = M < +∞. Then it follows that*

*• (Regularity) u*_{0} *∈ H*2_{(x}i_{, x}i+1_{) for every i = 0, . . . , k,}

• (NBC) *u*˙0*(xi) = 0 for every i = 0, . . . , k + 1.*

*Proof. |∇ MS |(u*_{0}*) < +∞ implies that |∇ MS |(u*_{0}|_{I}) < +∞ for every interval I ⊆ [0, 1].*Let us fix i, let us consider I = xi _{, x}i+1*

*and let us call u*_{0} *:= u*_{0}|* _{I}*.

*Let us consider any function w in H*1

*xi, xi+1*

and let us define

*w(x) :=*
(
*w(x)* *if x ∈ xi, xi+1*
*,*
*u*0*(x)* elsewhere.
Therefore
(
*d(u*0*, w) = d(u*0*, w),*
*MS(u*0*) − MS(w) ≥ D(u*0*) − D(w),*

*because the number of jumps of w is between #S*_{0}*− 2 and #S*_{0} *(D denotes as usual the*
*Dirichlet functional). Now we observe that |∇D|(u*0*) ≤ |∇ MS |(u*0). In fact, for every

*w such that*

(

*d(u*0*, w) = C*

*D(u*0*) − D(w) = C*0*,*

*there exists a function (for example w) such that*

(

*d(u*0*, w) = C*

*MS(u*_{0}*) − MS(w) ≥ C*0*,*

*and this implies exactly |∇D|(u*0*) ≤ |∇ MS |(u*0*). Therefore |∇D|(u*0*) < +∞, and we*

have proved in Theorem 5.1.1 that this implies

*u*0 *∈ H*2*(xi, xi+1*)
˙
*u*0*(xi*)+= 0
˙
*u*0*(xi+1*)−*= 0.*

*Since i is arbitrary, we obtain the conclusion.*

Now we can prove the following statement.

**Theorem 5.2.7 (Descending metric slope for the Mumford-Shah functional).**

*If u*_{0} *is a function in SBV (0, 1) that satisfies the conditions in Theorem 5.2.6, then*

*|∇ MS |(u*_{0}) = 2 k ¨*u*0*k ,*

where with a slight abuse of notation we define

k ¨*u*0k :=
v
u
u
t
*k*
X
*i=0*
ˆ *xi+1*
*xi*
¨
*u*0*(x)*2 *dx.*

5.2. Slope of the Mumford-Shah functional

* Proof. Part 1: We show that 2 k¨uk ≥ |∇ MS |(u*0).

**Step 1: Let us consider a sequence w**n*such that wn* *L*

2
*→ u*0 and
lim
*n→+∞*
*MS(u*_{0}*) − MS(w _{n}*)

*ku*

_{0}

*− w*k

_{n}*= |∇ MS| (u*0

*) ≥ 0,*

*(in particular MS (wn) ≤ MS (u*0)). We use an argument by Dal Maso and Toader ([8])

*to show that wn* *has the same number of jumps than u*0. We have already shown in

*Proposition 5.2.5 that every sequence w _{n}*

*→ uL*2

_{0}

*such that MS(w*

_{n}) ≤ M has at least k*points of jump, where k = #S*

_{0}

*. Therefore eventually it holds that #S*

_{0}

*− #S*≤ 0. Hence this estimate

_{w}lim
*n→+∞*
1
*ku*_{0}*− w _{n}*k
ˆ

*[0,1]\S*0 ˙

*u*02

*+ #S*0− ˆ

*[0,1]\Swn*˙

*wn*2

*− #Swn*! ≥ 0 implies that eventually

_{ˆ}

*[0,1]\S*0
˙
*u*02 ≥
ˆ
*[0,1]\Swn*
˙
*wn*2*.*

Let us define *un* *as the absolutely continuous part of u*0 *and wn* as the absolutely

*con-tinuous part of wn*.

*It is easy to see that the jump part of w _{n}converges to the jump part of u*

_{0}(if not, some

*jumps of u*

_{0}are being approximated with a continuous function, but we have shown in

*Corollary 5.2.4 that this makes the functionals tend to infinity). Therefore wn*

*L*

2
*→ u*0.
We know that ˆ
*[0,1]*
˙
*u*02 ≥
ˆ
*[0,1]*
˙
*wn*2*,*

therefore´* _{[0,1]}_{w}*˙

*2are uniformly bounded, hence ˙*

_{n}*wn* w*∞

*, and w*∞must be ˙

*u*0. Thanks

*to the lower semicontinuity of the L*2 norm with respect to the weak convergence

lim inf
*n→+∞*
*w*˙* _{n}*
≥

*u*˙

_{0}

*.*If we recall lim

*n→+∞*´

*[0,1]\S*˙0 2

_{u0}u*0− ´*

_{+ #S}*[0,1]\Swnw*˙

*n*2

_{− #S}*wn*

*ku*

_{0}

*− w*k ≥ 0

_{n}and use that

lim inf
*n→+∞*
*w*˙* _{n}*
≥

*u*˙

_{0}

*,*

*this gives that eventually #S _{w}_{n}*

*= #S*

_{0}. Therefore, since we are computing an upper

*bound for |∇ MS| (u*0

*), it is not restrictive to consider only functions w such that #Sw*=

*#Su*0. We also know, by Proposition 5.2.5, that it is not restrictive to consider only

*functions w such that S _{w}= S_{u}*

_{0}.

**Step 2: Let us consider any w such that w is in SBV (0, 1) and w has the same jump**

*points as u*0. Then
*MS(u*0*) − MS(w) =*
*k*
X
*i=0*
ˆ *xi+1*
*xi*
˙
*u*02− ˙*w*2
*.*

*Let us consider u*0*i* *and wi* *the restrictions of the previous functions to xi, xi+1*. The

previous equality becomes

*MS(u*0*) − MS(w) =*

*k*

X

*i=0*

*D(u*0*i) − D(wi).*

*We have already proved in Theorem 5.1.2 that, if u satisfies the Neumann Boundary*
Conditions, then
*D(u) − D(w) ≤ 2 k(u − w)k · k¨uk .*
Therefore
*MS(u*0*) − MS(w) ≤*
*k*
X
*i=0*
2
*(u*0
*i _{− w}i*

_{)}·

*u*¨0

*i*

*,*

*C.S.*≤ 2 v u u t

*k*X

*i=0*

*k(u*0

*i− wi*)k2 v u u t

*k*X

*i=0*

*u*¨0

*i*2

*,*

*= 2 ku*

_{0}

*− wk · k¨uk .*

This gives that

*MS(u*0*) − MS(w)*

*ku*_{0}*− wk* ≤ 2 k¨*uk ,*
that implies

*|∇ MS |(u*0) ≤ 2 k¨*uk ,*

as desired.

**Part 2: We show that |∇ MS |(u**_{0}) ≥ 2 k¨*uk .*

*If we consider any v in SBV (0, 1) such that v has x*1*, . . . , xk* as jump points, we can
apply the same argument used in the proof of Theorem 5.1.2. Therefore by definition of
metric slope
*|∇ MS |(u*_{0}) ≥ lim
*t→0*
*MS(u*_{0}*) − MS(u*_{0}*+ tv)*
*ktvk*
= lim
*t→0*
*k*
X
*i=0*
*MS ui*_{0}
*− MS ui*
0*+ tvi*
*ktvk*
= lim
*t→0*
*k*
X
*i=0*
*D ui*_{0}
*− D ui*
0*+ tvi*
*ktvk*
= lim
*t→0*
1
*ktvk*
*k*
X
*i=0*
*−2t*
ˆ *xi+1*
*xi*
˙
*ui*_{0}*˙vi− t*2
ˆ *xi+1*
*xi*
*˙vi*2
!
= _{q} *sign(t)*
P*k*
*i=0kvi*k
2
*k*
X
*i=0*
−2
ˆ *xi+1*
*xi*
˙
*ui*_{0}*˙vi.* (5.2.2)

5.2. Slope of the Mumford-Shah functional

*Choosing opportunely either t → 0*+ *or t → 0*− in (5.2.2), and integrating by parts, it
follows that
*|∇ MS |(u*_{0}) ≥ _{q} 2
P*k*
*i=0kvi*k
2
*k*
X
*i=0*
ˆ *xi+1*
*xi*
¨
*ui*_{0}*vi*

(we do not have the boundary term because of the Neumann Boundary Conditions as explained in Theorem (5.2.6)).

*Since this holds for every v in SBV (0, 1) such that v has x*1*, . . . , xk* as jump points,
*we can find a particular sequence of functions v _{n}* such that

*(1) vn* *has x*1*, . . . , xk* as jump points,

*(2) v _{n}i*

*∈ H*1

_{(x}i_{, x}i+1_{) for every i = 0, . . . , k,}*(3) v _{n}i* → ¨

*L*2

*ui*

_{0},

(we are using the hypothesis of regularity explained in Theorem (5.2.6)). Therefore

*|∇ MS |(u*0) ≥
2
q
P*k*
*i=0kvni*k2
*k*
X
*i=0*
ˆ *xi+1*
*xi*
¨
*ui*_{0}*˙v _{n}i*
!
=

_{q}2 P

*k*

*i=0kvni*k2

*k*X

*i=0*ˆ

*xi+1*

*xi*

*vnivni*+ ˆ

*xi+1*

*xi*¨

*u*0

*− vni*

*vni*! ≥ 2 v u u t

*k*X

*i=0*

*kv*

_{n}i_{k}2

_{−}2 q P

*k*

*i=0kvni*k2

*k*X

*i=0*

*u*¨

*i*0

*− vni*

*vn*

*i*

*,*(5.2.3)

where (5.2.3) follows by Cauchy-Schwarz inequality. Taking the limit in (5.2.3) and using
*that vi _{n}L*→ ¨2

*ui*

_{0}, it follows that

*|∇ MS |(u) ≥ 2 k¨uk ,*