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 R2), 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) = µ2 ¨ R f (x, y) − g (x, y)2 dx dy + ¨ R\Γ k∇f (x, y)k2 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 + #Su, (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 H1. 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 H1(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 x0 in X, for every sequence xnsuch that xnconverges to x0,
then
lim inf
n→+∞Fn(xn) ≥ F (x0).
Limsup inequality: for every x0 in X there exists a sequence (called recovery sequence)
xn such that xn converges to x0 and
lim sup
n→+∞
Fn(xn) ≤ F (x0).
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
fε∗∗( ˙u),
where fε∗∗ 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
fε∗∗( ˙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 ∈h0,12i, 1 if x ∈12, 1i,
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 u + v 2 > 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 Cn(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 inhni,i+1n 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 pointsni, ui+1n , as shown in Figure 1.1 (v is dashed). By definition, v is constant inhn−1n , 1i.
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 n1 of u as D±n1u (x) = ˜ ux ±n1− ˜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 Cn such that
D1nu i= ui+1−ui 1 n = nhi if i = 1, . . . , n − 1, D1nu 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) unk are uniformly bounded,
(3) D 1 nkun k L2
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 znk 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− znkk∞= Mk·
1
nk
,
where Mk is the maximum of D
1
nkun k.
Moreover, by hypothesis (3), we know that there exists a constant H such that for every
k in N H ≥ D 1 nkun k L2 = v u u t n X i=1 1 nk D 1 nkun k 2 , and therefore H ≥ √1 nk max D 1 nkun k , H√nk≥ max D 1 nkun k . Hence kunk− znkk∞≤ H ·√nk· 1 nk → 0, and this implies that unk → 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
Ψn(u) := n−1 X i=1 1 nψn(nhi) = ˆ 1 0 ψn Dn1u , where ψn(x) := min n x2, no
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 ψn).
Let us choose any sequence rn= O
1
n2
, 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
• x2 ≤ ϕ n(x) ≤ n in [ √ n − rn, √ n], • ϕn(x) ∈ C1(R),
• ϕn(x) is increasing in [0, +∞) (and therefore decreasing in (−∞, 0]),
• ϕ0n(x) is increasing in [0, pn) and decreasing in (pn, +∞) (i.e. ϕnis convex-concave
in [0, +∞)), • ϕ0
n(x) ≤ 2
√
n in R.
These hypotheses also ensure that • ϕn(x) ≤ x2+ λn on [ √ n − rn, √ n], where λn→ 0, • ϕ0n(pn) − ϕ0n( √ 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 Cn(0, 1). We define
Fn(u) := n−1 X i=1 1 nϕn(nhi) = ˆ 1 0 ϕn Dn1u . If u ∈ L2(0, 1) \ P Cn(0, 1), we define Fn(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} = infu∈X{Gn} for every n,
• for every sequence of points (un) ⊆ 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 ifu 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 infn→+∞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 Fn defined in Definition 1.4.2 Γ-converge to MS with respect to the L2 convergence, that is to say that both the following inequalities hold
Limsup inequality: ∀u ∈ L2(0, 1), ∃u
n L2
→ u, lim sup
n→+∞
Fn(un) ≤ MS(u),
Liminf inequality: ∀u ∈ L2(0, 1), ∀u
n L2
→ 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 u0 in SBV (0, 1) (if u0 is not in SBV (0, 1), then MS (u0) = +∞ and the conclusion is trivial). Let x1, . . . , xk be the jump points of u
0. Let us consider
the representative of u0 that is continuous in every (xi, xi+1).
Let us define unas the function belonging to P Cn such that un(ni) = u0(ni) for every
i = 1, . . . , n. Since u0 is uniformly continuous on the compact subsets of [0, 1] that do
not contain any xi, then un→ uL2 0. We prove now that
lim sup n→+∞ Fn(un) ≤ MS(u0). Let us define Ii := hi−1 n , i n
. If Ii contains one or more discontinuity points of u0, then
1
nϕn(nhi) ≤ MS(u0|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 u0 is continuous in Ii, 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 ˙ vn2+ 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 ˙ u02+ 1 nλn= MS(u0|Ii) + 1 nλn.
In both cases we have shown that 1
nϕn(nhi) ≤ MS(u0|Ii) +
1
nλn,
for i = 1, . . . , n, and this implies that for every n
Fn(un) ≤ MS(u0) + 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 u0 is bounded
in k·k∞.
Proof of the liminf inequality for u0 bounded.
Step 1: Let us choose the representative of u0 such that Im(u0) ⊆ [−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 L2
→ u0,
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 infn→+∞Fn(un) = +∞, so in the sequel
we can assume lim infn→+∞Fn(un) = M ∈ R. Let us consider a subsequence (not
relabelled) un such that un approaches the liminf. Since limn→+∞Fn(un) = M , it
follows that Fn(un) ≤ 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 un= vn+ wn as follows
ji := hi if |hi| ≥ √1n, 0 if |hi| < √1n, (2.2.1)
2.2. Liminf inequality
where ji denotes the value v(i+1n ) − v(ni) 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 un.
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 R2a, for example by the
following bijection
vn∼
x1n, . . . , xan, jn1, . . . , jna,
where xin are the points of jump of vn and jni are the heights of each jump.
Since xin ∈ [0, 1] and ji
n ∈ [−2D, 2D], by the compactness of balls in R2a, there exists a
subsequence vnsuch that
x1n→ x1, . . . , jn1 → j1, . . . .
Therefore vn→ vL2 0, where v0 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 v0 has at most
a jumps (possibly less). Hence
un= vn+ wn, un L2 → u0, vnL 2 → v0. Then wn L2 → w0:= u0− v0.
Step 3: We can now conclude the proof as follows.
M + 1 ≥ Fn(un) = X |hi|≥√1n 1 nϕn(nhi) + X |hi|<√1n 1 nϕn(nhi) = a + Fn(wn) ≥ a + ˆ 1 0 ˙ wn2, (2.2.2)
where (2.2.2) follows by the fact that ϕn(x) ≥ x2 if |x| ∈ [0,√n] as we said in Definition
1.4.1.
Now, by the weak compactness of balls in L2, we have that
M + 1 ≥
ˆ 1 0
˙
wn2 ⇒ ˙wn* w∞ up to subsequences.
Since wnL→ w2 0, then w∞= ˙w0, and in particular this implies that w0 is in H1(0, 1).
Moreover, by the lower semicontinuity of the L2 norm, it holds that
lim inf n→+∞ ˆ 1 0 ˙ wn2 ≥ ˆ 1 0 ˙ w02.
Therefore, taking the liminf in (2.2.2), it follows that M = lim inf n→+∞Fn(un) ≥ lim inf n→+∞ ( a + ˆ 1 0 ˙ wn2 ) ≥ a + ˆ 1 0 ˙ w02 ≥ #Sv0+ ˆ 1 0 ˙ w02, (2.2.3)
where Sv0 denotes the number of jumps of v0. Finally, we observe that the right hand
side of (2.2.3) is exactly MS (u0), because u0 = v0+ w0 where w0 ∈ H1(0, 1) and v0 is
piecewise constant. Therefore the proof is concluded.
Before giving the proof of the liminf inequality if u0is 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 MSu|[a,b] ≥ min ( 1,(yb− ya) 2 b − a ) . Proof. If u has one or more points of jump in [a, b], then
MSu|[a,b]≥ 1, otherwise MSu|[a,b] = ˆ b a ˙ u2,
and since u is continuous in [a, b], by classical convexity arguments it follows that ˆ b a ˙ u2 ≥ ˆ b a ˙v2 = 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
MSu|[a,b]≥ (yb− ya)
2
b − a ,
2.2. Liminf inequality
Proof of the liminf inequality for u0 unbounded.
We prove that for every unL
2
→ u0 it holds that
lim inf
n→+∞Fn(un) = +∞.
In fact, let us suppose by contradiction that there exists a sequence un→ uL2 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 u0 is unbounded, it is possible to find k := 2 dM e + 3 sets An1, . . . , Ank such that
L
u−10 (Ani)
> 0, whereL denotes the Lebesgue measure. Without loss of generality,
at least dM e + 2 sets are obtained for an odd ni. Let us fix D > max {|ni|} and let us define
unD := the truncation of un between [−D, D] ,
u0D:= the truncation of u0 between [−D, D] .
Then unD L
2
→ u0D.
What we have proved assuming that u0 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(u0D).
Therefore u0D 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 x1, . . . , xdM e+2 such
that u0D is continuous in every xi and every u0D xi
belongs to a different Ani (all ni
are chosen odd). Since ni are different and odd, the difference between u0D xi
and
u0D xjwith i 6= j is at least 1.
Now, by Lemma 2.2.1, it holds that
MSu0D|(xi,xi+1) ≥ min 1, u0D xi+1 − u0D xi2 xi+1− xi ≥ min 1,u0D xi+1− u0Dxi2 ≥ 1.
Since xi is an increasing sequence, then xi, xi+1 are disjoint connected subsets of [0, 1] and therefore MS (u0D) ≥ dM e+1 X i=1 MSu0D|(xi,xi+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 unbe a sequence of functions un: [0, 1] → R. Let us assume that
there exists a constant M such that
kunk∞+ Fn(un) ≤ M
for every n in N.
Then there exists a subsequence (not relabelled) un and a function u such that un→ u.L2
Proof. In this proof we use the same decomposition of u introduced in (2.2.1). So let vn+ wn be the decomposition of unin 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) vn and a function v such that
vn L2
→ 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 wnare
uniformly bounded, and we know that
M ≥ Fn(un) ≥ Fn(wn) ≥ ˆ 1 0 ˙ wn2, that implies √ M ≥ k ˙wnkL2.
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 Fnare the discrete functionals introduced in Definition 1.4.2.
We can observe that there exists a constant c such that inf Gn(u) ≤ c for every n in N. This can be checked for example defining c :=´01h2 and then noticing that for u0 ≡ 0
Gn(u0) = c for every n in N.
In Chapter 2 we have shown that the functionals Fn 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 Gn(u)→ E(u). As for the compactnessΓ 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 ∈ L2(0, 1) s.t. kuk∞≤ Mo.
Then, for every sequence of points un in KM such that
lim
n→+∞ Gn(un) −u∈Linf2(0,1)Gn(u) !
= 0,
there exists a subsequence unk which converges in L2 to some point u of KM.
Proof. Let us define
cn:= inf
u∈L2(0,1)Gn(u).
As we noticed at the beginning of this section, there exists a constant c such that cn≤ 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 KM, it holds that kunk∞ ≤ 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
{Gn} = inf
u∈L2(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 L2(0, 1) for every h in L∞(0, 1) ,
(2) lim
n→+∞ u∈Linf2(0,1)Gn(u) !
= min
u∈L2(0,1)E(u),
(3) If (un) ⊆ KM is any sequence of points such that
lim
n→+∞ Gn(un) −u∈Linf2(0,1)Gn(u) !
= 0
and if u is any limit point of un(i.e. there exists a subsequence of unthat 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 x0 ∈ X be any point
such that f (x0) is a real number. The metric slope of f at x0 is defined as
|∇f |(x0) := lim sup
r→0+
f (x0) − inf {f (x) : d(x, x0) ≤ r}
r .
It is often useful, although not necessary, to define |∇f |(x0) = +∞ if f (x0) = +∞.
We observe that |∇f |(x0) is always greater than or equal to 0 (including +∞), because x0 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 x0 is not an isolated point of X. Then it holds that
|∇f |(x0) = lim sup
x→x0
max {f (x0) − f (x), 0}
d(x, x0)
. (4.1.1)
Proof. Let M denote the right hand side of (4.1.1).
Step 1: We show that |∇f |(x0) ≥ 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→ x0 such that
lim
xn→x0
max {f (x0) − f (xn), 0}
d(xn, x0)
= M .
Since M > 0, the numerator must be eventually positive, hence lim
xn→x0
f (x0) − f (xn)
d(xn, x0)
Now let us set rn:= d(xn, x0) and in:= inf {f (x) : d(x, x0) ≤ rn}. It’s immediate to
see that in≤ f (xn), and therefore
|∇f |(x0) = lim sup r→0+ f (x0) − inf {f (x) : d(x, x0) ≤ r} r ≥ lim sup n→+∞ f (x0) − in rn ≥ lim sup n→+∞ f (x0) − f (xn) rn = M.
Step 2: We show that |∇f |(x0) ≤ M . Let rn→ 0+ be a sequence such that
|∇f |(x0) = lim
n→+∞
f (x0) − in
rn
.
By definition of infimum, for every natural number n there exists yn such that
( yn∈ B(x0, rn), f (yn) < in+ rn2. It follows that M ≥ lim n→+∞ f (x0) − f (yn) d(x0, yn) ≥ lim n→+∞ f (x0) − in− rn2 d(x0, yn) ≥ lim n→+∞ f (x0) − in− rn2 rn ≥ lim n→+∞ f (x0) − in rn − rn = |∇f |(x0), as desired.
We notice that asking that x0 is not an isolated point is necessary. In fact, if x0 is
an isolated point, then the quantity lim sup
x→x0
max {f (x0) − f (x), 0}
d(x, x0)
is not well defined. However, if |∇f |(x0) 6= 0, then x0 cannot be an isolated point, and
therefore Lemma 4.1.2 holds.
Proposition 4.1.3. If X = Rn and f : X → R is a C1 function, then the metric slope is the norm of the gradient, i.e.
4.1. Definition and main properties
Proof. If x0 is a local minimum point, then
|∇f | (x0) = k∇f (x0)k = 0.
If not, we argue as follows.
Step 1: We show that |∇f |(x0) ≤ k∇f (x0)k . Let us consider the sequence rn→ 0 for which
lim
n→∞
f (x0) − inf {f (x) : d(x, x0) ≤ rn}
rn
= |∇f | (x0).
Since f is C1 on Rn, the infima on closed balls are actually minima and therefore there exists a sequence xn such that xn→ x0 and
lim
n→∞
f (x0) − f (xn)
|x0− xn|
= |∇f |(x0).
By definition of differentiability and by Cauchy-Schwarz inequality, it holds that
lim n→∞ f (x0) − f (xn) |x0− xn| = limn→∞ h∇f (x0) , x0− xni |x0− xn| + o(1) ≤ k∇f (x0)k ,
therefore we have shown that
|∇f |(x0) ≤ k∇f (x0)k .
Step 2: We show that |∇f |(x0) ≥ k∇f (x0)k .
If k∇f (x0)k = 0 this step is immediate, therefore we assume that ∇f (x0) 6= 0. For every v 6= 0 it holds that
|∇f |(x0) ≥ lim t→0+ f (x0) − f (x0− tv) t kvk = lim t→0+ t h∇f (x0− ξtv) , vi t kvk ,
where ξt∈ [0, t] is given by the mean value theorem. Hence
|∇f |(x0) ≥ h∇f (x0) , vi
kvk .
If we take v = ∇f (x0), it follows that
|∇f |(x0) ≥ k∇f (x0)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 x0 ∈ X we define
gx0(r) := inf {f (x) : d(x, x0) ≤ r} .
Let us assume that
(1) |∇f |(x) ≥ M for every x ∈ X,
(2) inf {f (x) : d(x, x0) ≤ r} is actually a minimum ∀r ∈ R, ∀x0 ∈ X.
Then gx0
0(r) exists for almost every r ∈ R and g
0
x0(r) ≤ −M where it is defined.
Proof. Let us fix x0 ∈ X. Since gx0(r) is a decreasing function, by Lebesgue’s theorem
for the differentiability of monotone functions, gx0
0(r) exists almost everywhere. Let us
suppose by contradiction that
g0x0(r) > −M + 2δ
for some r, δ ∈ R+. Then there exists some h0 such that for every h ≤ h0
gx0(r + h) − gx0(r)
h > −M + δ. (4.2.1)
Let us now take a point y ∈ B(x0, r) such that f (y) = gx0(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 r0 < h0 such that
f (y) − inf {f (x) : d(x, y) ≤ r0} r0 > M − δ 4, −f (y) + inf {f (x) : d(x, y) ≤ r0} r0 < −M +δ 4. But gx0(r + r0) − gx0(r) r0 < −f (y) + inf {f (x) : d(x, y) ≤ r0} r0
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 , x0, gx0 be as in Lemma 4.2.1 and let us assume that there
exists a radius r0 such that g0x0(r) ≤ −M for every r ≤ r0 for which gx0 is differentiable
in r. Then
∀ε > 0, ∀r < r0, ∃y ∈ B(x0, r) such that f (y) ≤ f (x0) − (M − ε)r.
Proof. By a general theorem, fixing r < r0 we have
gx0(r) − gx0(0) ≤ g
0
x0(c)r ≤ −M r,
for some c ∈ [0, r]. Therefore
inf {f (x) : d(x, x0) ≤ r} − f (x0) ≤ −M r.
It follows that for every ε > 0 there exists y in B(x0, r) such that
f (y) ≤ f (x0) − (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 fn is lower semicontinuous for every n in N (LSC),
• for every r > 0, x0 ∈ X, A ∈ R there exists a compact set K ⊆ X such that
{x ∈ X : d(x, x0) ≤ 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 x0 ∈ X and ε > 0, and let us set
M := Γ − lim sup
n→+∞
|∇fn|(x0).
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(x0, 2rε). From now on, let fn be a subsequence (not relabelled) for
which the previous inequality holds. Let xn be a recovery sequence for x0. Without
loss of generality, we can assume that d(xn, x0) ≤ rε, so that B(xn, rε) ⊆ B(x0, 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→ y0 for some y0 ∈ X.
By taking the liminf in (4.2.3) and using the liminf inequality we obtain
f∞(y0) ≤ lim inf n→+∞fn(yn) ≤ lim infn→+∞fn(xn) − (M − ε)r = f∞(x0) − (M − ε)r. Moreover d(x0, y0) = lim n→+∞d(x0, yn) ≤ r. Therefore inf {f∞(x) : d(x, x0) ≤ r} ≤ f∞(x0) − (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 (fn 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 = fn(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= ni for i = 1, . . . , n, +∞ if x = ni for i = 1, . . . , n. Hence Γ − lim sup
n→+∞
|∇fn|(x) ≡ 1, and therefore Γ − lim sup
n→+∞
|∇fn|(x) > |∇f∞|(x).
Example 4.2.5 (fn 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
|∇fn| (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→+∞ |∇fn| (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 fn: [0, 1] → R
fn(x) = x if x ≤ 12 −1 n, n if x ∈12 −1 n, 1 2 + 1 n , −x + 1 if x ≥ 12 +1n.
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 ≤ 12,
−x + 1 if x > 1 2.
For a fixed value of n it holds that
|∇fn| (x) = 1 if x ≤ 12− 1 n, 0 if x ∈12− 1n,12 +n1, 1 if x ≥ 12+n1.
As n approaches infinity, this gives
Γ − lim sup n→+∞ |∇fn| (x) = ( 1 if x 6= 12, 0 if x = 12.
Let us consider x0 := 12. Any recovery sequence xn for Γ − lim supn→+∞∇ |fn| in the
point x0 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 x0 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 ˙ u2 if u ∈ H1(a, b), +∞ if u ∈ L2(a, b) \ H1(a, b).
Theorem 5.1.1. Let us suppose that |∇D|(u) = M < +∞. Then it follows that
• (Regularity) u ∈ H2(a, b), • (NBC) ( ˙ u(a) = 0, ˙ u(b) = 0.
We note that if u belongs to H2(a, b), then ˙u admits a continuous representative and
therefore the Neumann Boundary Conditions make sense.
Proof. Let us fix v ∈ H1(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 − t2 ˆ b a ˙v2 ! = −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 : L2(a, b) → R v → ˆ b a ˙ u ˙v
is also continuous. Riesz’s theorem implies that there exists a function f ∈ L2(a, b) such that
H (v) =
ˆ b a
f v.
Therefore for every v ∈ H1(a, b) ˆ b a ˙ u ˙v = ˆ b a f v, (5.1.2)
in particular this holds for every v ∈ C0∞(a, b). This implies that −f is the weak derivative of ˙u, therefore ˙u ∈ H1(a, b) and u ∈ H2(a, b).
However, since (5.1.2) holds also for every v ∈ H1(a, b), it follows that ˆ b a f v = ˆ b a ˙ u ˙v = ˆ b a f v + [ ˙uv]ba
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 H2(a, b) that satisfies the Neumann Boundary Conditions and for every
v in H1(a, b).
Now we can prove the following statement.
Theorem 5.1.2 (Slope of the Dirichlet functional).
If u ∈ H2(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 L2(a, b), but w might not be in H1(a, b). However, taking a sequence wnsuch that wn∈ H1(a, b) and wn L
2
→ w, and using (5.1.3), we obtain |∇D|(u) ≥ 2 kwnk ˆ b a ¨ uwn = 2 kwnk ˆ b a wnwn+ ˆ b a (¨u − wn)wn ! C.S. ≥ 2 kwnk − 2 kwnk k¨u − wnk kwnk = 2 kwnk − 2 k¨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 ∈ H1(a, b). Then
D (u) − D (w) = D (u) − D(w − u) + u = ˆ b a ˙ u2− ˆ b a ( ˙w − ˙u)2− 2 ˆ b a ( ˙w − ˙u) ˙u − ˆ b a ˙ u2 = − 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]ba = 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 ∈ H1(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 u0 be a function in SBV (0, 1) and let S0:= n
x1, . . . , xkodenote the set of jump points of u0. By definition of jump, we have
lim x→xi− u0(x) = yi, lim x→xi+u0 (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 δεi such that u0 h xi− δεi, xi⊆ [yi− ε, yi+ ε] , u0 xi, xi+ δεii⊆ [zi− ε, zi+ ε] .
Let us define δε:= min
δiε
.
Definition 5.2.2 (Little rectangles).
Let ε be any real positive number (it will be useful in the sequel to use ε < h4). 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+ δεi× [zi− 2ε, zi+ 2ε] , as shown in Figure 5.1. L1 R1 L3 R2 L2 R3 x1 x2 x3 0 1 Figure 5.1
Lemma 5.2.3. Let us consider x0 ∈ [0, 1], y 6= z, h = |z − y|, ε < |y−z|4 and δ < (h−4ε)
2
2 ,
and let us define Lε and Rε similarly to Definition 5.2.2
Lε:= [x0− δ, x0] × [y − 2ε, y + 2ε] ,
Rε := [x0, x0+ δ] × [z − 2ε, z + 2ε] .
Let us consider any u ∈ SBV (0, 1) and let us assume that
(
graph(u) ∩ Lε6= ∅, graph(u) ∩ Rε 6= ∅.
Then
5.2. Slope of the Mumford-Shah functional
Proof. If u|[x0−δ,x0+δ] has at least one point of discontinuity, then MS(u|[x0−δ,x0+δ]) ≥ 1.
If not, we suppose without loss of generality that z > y. By classical convexity arguments
MS(u|[x0−δ,x0+δ]) ≥ MS(v), (5.2.1)
where v is the affine function joining the points (x0− δ, y + 2ε), (x0+ 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 x0+δ x0−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 Lε and Rε and is continuous
in [x0− δ, x0+ δ], then
MS(u|[x0−δ,x0+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 u0 in SBV (0, 1) such that MS(u0) = M . Let
S0 := n
x1, . . . , xko be the set of points in which u0 is discontinuous. We define
X :=nu ∈ L2(0, 1) s.t u has exactly k points of jumpo.
Then there exists a radius r0 (sufficiently small) such that for every r < r0, for every u in B (u0, r) ∩ X there exists ˜u ∈ X such that
• d (˜u, u0) ≤ d (u, u0),
• MS (˜u) ≤ MS (u),
Proof. If u /∈ SBV (0, 1), then MS (u) = +∞ and ˜u = u0 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), h2 18 (M + 1) ) .
Finally, let us put r0= ε
√
δ
2 .
Step 1: We can observe that every u ∈ L2(0, 1) such that d(u, u0) ≤ r0 must intersect
all the little rectangles of u0 introduced in Definition 5.2.2. In fact, if (without loss of generality) graph(u) ∩ L1 = ∅ then d(u, u0) ≥ v u u t ˆ x1 x1−δ |u − u0|2 ≥ v u u t ˆ x1 x1−δ ε2=√δε > r 0.
Therefore any u such that d(u, u0) < r0 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 = u0 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, y1< z1 and x1< p1.
Step 2: We now show that it is not restrictive to suppose that
ux1− δ, p1 ⊆ −∞, y1+ 2ε +h 3 , up1, x1+ δ ⊆ z1− 2ε − h 3, +∞ .
In fact, we know that L1∩ Im(u) 6= ∅, R1∩ Im(u) 6= ∅. Hence, if the first inclusion is
false, we can infer by classical convexity argument that MS(u|[x1−δ,x1+δ]) ≥ MS(v),
5.2. Slope of the Mumford-Shah functional
where v is the segment joining the points (x1− δ, y1+ 2ε), (x1+ δ, y1+ 2ε +h3). But
MS (v) = h 3 2 2δ > M + 1,
therefore ˜u = u0 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 < x1, u(p1)+ if x ∈ x1, p1 , u(x) if x > p1. u0 u x1 p1 u0 ˜ u x1
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 x1, p1
, on which ˜u is constant.
It is also clear that d(u0, ˜u) ≤ d(u0, u), because we have only modified u inx1, p1, and
in this interval we can compute the following estimates ˆ p1 x1 (u0− ˜u)2 ≤ ˆ p1 x1 4ε + h 3 2 , ˆ p1 x1 (u0− u)2 ≥ ˆ p1 x1 z1− 2ε − y1+ 2ε + h 3 2 , and z1− 4ε − y1−h3
≥ 4ε + h3 because z1− y1 ≥ 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 u0 in SBV (0, 1) such that MS(u0) is a real number.
Let S0 := n
x1, . . . , xko be the set of points in which u0 is discontinuous. We define
x0 := 0, xk+1 := 1. Let us assume that |∇ MS |(u0) = M < +∞. Then it follows that
• (Regularity) u0 ∈ H2(xi, xi+1) for every i = 0, . . . , k,
• (NBC) u˙0(xi) = 0 for every i = 0, . . . , k + 1.
Proof. |∇ MS |(u0) < +∞ implies that |∇ MS |(u0|I) < +∞ for every interval I ⊆ [0, 1]. Let us fix i, let us consider I = xi, xi+1
and let us call u0 := u0|I. Let us consider any function w in H1 xi, xi+1
and let us define
w(x) := ( w(x) if x ∈ xi, xi+1 , u0(x) elsewhere. Therefore ( d(u0, w) = d(u0, w), MS(u0) − MS(w) ≥ D(u0) − D(w),
because the number of jumps of w is between #S0− 2 and #S0 (D denotes as usual the Dirichlet functional). Now we observe that |∇D|(u0) ≤ |∇ MS |(u0). In fact, for every
w such that
(
d(u0, w) = C
D(u0) − D(w) = C0,
there exists a function (for example w) such that
(
d(u0, w) = C
MS(u0) − MS(w) ≥ C0,
and this implies exactly |∇D|(u0) ≤ |∇ MS |(u0). Therefore |∇D|(u0) < +∞, and we
have proved in Theorem 5.1.1 that this implies
u0 ∈ H2(xi, xi+1) ˙ u0(xi)+= 0 ˙ u0(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 u0 is a function in SBV (0, 1) that satisfies the conditions in Theorem 5.2.6, then
|∇ MS |(u0) = 2 k ¨u0k ,
where with a slight abuse of notation we define
k ¨u0k := v u u t k X i=0 ˆ xi+1 xi ¨ u0(x)2 dx.
5.2. Slope of the Mumford-Shah functional
Proof. Part 1: We show that 2 k¨uk ≥ |∇ MS |(u0).
Step 1: Let us consider a sequence wn such that wn L
2 → u0 and lim n→+∞ MS(u0) − MS(wn) ku0− wnk = |∇ MS| (u0) ≥ 0,
(in particular MS (wn) ≤ MS (u0)). We use an argument by Dal Maso and Toader ([8])
to show that wn has the same number of jumps than u0. We have already shown in
Proposition 5.2.5 that every sequence wn → uL2 0 such that MS(wn) ≤ M has at least k points of jump, where k = #S0. Therefore eventually it holds that #S0 − #Sw ≤ 0. Hence this estimate
lim n→+∞ 1 ku0− wnk ˆ [0,1]\S0 ˙ u02+ #S0− ˆ [0,1]\Swn ˙ wn2− #Swn ! ≥ 0 implies that eventually ˆ
[0,1]\S0 ˙ u02 ≥ ˆ [0,1]\Swn ˙ wn2.
Let us define un as the absolutely continuous part of u0 and wn as the absolutely
con-tinuous part of wn.
It is easy to see that the jump part of wnconverges to the jump part of u0 (if not, some jumps of u0 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 → u0. We know that ˆ [0,1] ˙ u02 ≥ ˆ [0,1] ˙ wn2,
therefore´[0,1]w˙n2are uniformly bounded, hence ˙wn* w∞, and w∞must be ˙u0. Thanks
to the lower semicontinuity of the L2 norm with respect to the weak convergence
lim inf n→+∞ w˙n ≥ u˙0 . If we recall lim n→+∞ ´ [0,1]\Su0u˙0 2+ #S 0− ´ [0,1]\Swnw˙n 2− #S wn ku0− wnk ≥ 0
and use that
lim inf n→+∞ w˙n ≥ u˙0 ,
this gives that eventually #Swn = #S0. Therefore, since we are computing an upper bound for |∇ MS| (u0), it is not restrictive to consider only functions w such that #Sw=
#Su0. We also know, by Proposition 5.2.5, that it is not restrictive to consider only
functions w such that Sw= Su0.
Step 2: Let us consider any w such that w is in SBV (0, 1) and w has the same jump
points as u0. Then MS(u0) − MS(w) = k X i=0 ˆ xi+1 xi ˙ u02− ˙w2 .
Let us consider u0i and wi the restrictions of the previous functions to xi, xi+1. The
previous equality becomes
MS(u0) − MS(w) =
k
X
i=0
D(u0i) − 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(u0) − MS(w) ≤ k X i=0 2 (u0 i− wi) · u¨0 i , C.S. ≤ 2 v u u t k X i=0 k(u0i− wi)k2 v u u t k X i=0 u¨0i 2 , = 2 ku0− wk · k¨uk .
This gives that
MS(u0) − MS(w)
ku0− wk ≤ 2 k¨uk , that implies
|∇ MS |(u0) ≤ 2 k¨uk ,
as desired.
Part 2: We show that |∇ MS |(u0) ≥ 2 k¨uk .
If we consider any v in SBV (0, 1) such that v has x1, . . . , 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 |(u0) ≥ lim t→0 MS(u0) − MS(u0+ tv) ktvk = lim t→0 k X i=0 MS ui0 − MS ui 0+ tvi ktvk = lim t→0 k X i=0 D ui0 − D ui 0+ tvi ktvk = lim t→0 1 ktvk k X i=0 −2t ˆ xi+1 xi ˙ ui0˙vi− t2 ˆ xi+1 xi ˙vi2 ! = q sign(t) Pk i=0kvik 2 k X i=0 −2 ˆ xi+1 xi ˙ ui0˙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 |(u0) ≥ q 2 Pk i=0kvik 2 k X i=0 ˆ xi+1 xi ¨ ui0vi
(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 x1, . . . , xk as jump points, we can find a particular sequence of functions vn such that
(1) vn has x1, . . . , xk as jump points,
(2) vni ∈ H1(xi, xi+1) for every i = 0, . . . , k,
(3) vni → ¨L2 ui0,
(we are using the hypothesis of regularity explained in Theorem (5.2.6)). Therefore
|∇ MS |(u0) ≥ 2 q Pk i=0kvnik2 k X i=0 ˆ xi+1 xi ¨ ui0˙vni ! = q 2 Pk i=0kvnik2 k X i=0 ˆ xi+1 xi vnivni+ ˆ xi+1 xi ¨ u0− vni vni ! ≥ 2 v u u t k X i=0 kvnik2− 2 q Pk i=0kvnik2 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 vinL→ ¨2 ui0, it follows that
|∇ MS |(u) ≥ 2 k¨uk ,
