Universit`
a degli Studi di Pisa
DIPARTIMENTO DI MATEMATICACorso di Laurea in Matematica
Tesi di laurea triennale
Nonlocal characterizations of Sobolev spaces and
functions of bounded variation in dimension one
Candidato
Nicola Picenni
Relatore
Contents
1 Introduction 1
1.1 Preliminaries . . . 1
2 “Horizontal” functionals 5 2.1 A characterization based on the difference quotient . . . 5
2.2 A more general characterization . . . 7
3 “Vertical” functionals 11 3.1 Definitions and main results . . . 11
3.2 Pointwise convergence . . . 12
3.3 Liminf inequality . . . 14
3.4 Limsup inequality . . . 26
1
Introduction
In this thesis we present some recent characterizations of Sobolev spaces and functions of bounded variation of one variable which are based on nonlocal functionals. In particular, we consider two kinds of functionals that we call “horizontal” and “vertical”.
The first kind of functionals is based on the norm of some horizontal increments that are similar to the classic difference quotient. These functionals have been studied by J. Bourgain, H. Brezis and P. Mironescu in [1].
The second kind of functionals is based on vertical differences and is much more com-plicated. The asymptotic behaviour of this family of functionals has been studied by H.-M. Nguyen and J. Bourgain in a series of papers ([2], [5], [6], [7], [8]). In particular, the pointwise convergence was proved in [5] and the Γ-convergence was proved in [7]. Moreover, in [3], H. Brezis and H.-M. Nguyen extended these results to a more general class of functionals. However, the proof of Γ-convergence presented in [7] involves an unknown constant, whose value was only conjectured.
In the sequel, we present a new proof of the Γ-convergence, based on a joint work with C. Antonucci, M. Gobbino and M. Migliorini, that provides the exact value of that constant thanks to a new estimate for the liminf inequality. This also leads to a simpler and more explicit proof of the limsup inequality that allows us to give an answer to Open problem 3 in [3], in a special case.
Both horizontal and vertical functionals have some interesting applications, since they depend only on the metric and the measure of R (or Rn), and not on the linear structure. Therefore, they could be used to give a notion of Sobolev spaces and functions of bounded variation for functions between metric measure spaces, under suitable assumptions on the relation between metric and measure, as in [4]. Moreover, the Γ-convergence result makes possible to give a formulation in metric spaces for variational problems involving the Lp-norm of the derivative or the total variation. However, we have not investigated this possibility thoroughly.
For the sake of simplicity, we present all these results only in dimension one, even if almost all of them hold in any dimension.
1.1
Preliminaries
First of all we recall the definition of Sobolev spaces and space of functions with bounded variation.
Definition 1.1.1 (Sobolev spaces W1,p(R)). For every p ≥ 1, we say that a function
Chapter 1. Introduction such that ˆ R u(x) ˙ϕ(x) dx = − ˆ R v(x)ϕ(x) dx ∀ϕ ∈ Cc∞(R). In this case, we say that v is the weak derivative of u and we denote it with ˙u.
Remark 1.1.2. In the sequel we denote with k ˙ukp the Lp-norm of the weak derivative of
u, namely k ˙ukp := ˆ R | ˙u(x)|pdx 1p . If u /∈ W1,p(R), we set k ˙uk p := +∞.
Definition 1.1.3. We say that µ : B(R) → R is a bounded signed measure on R if
• µ(∅) = 0, • µ is σ-additive,
• There exists M ∈ [0, +∞) such that |µ(A)| ≤ M for every Borel set A ⊆ R. We denote with M(R) the set of bounded signed measure on R. If µ ∈ M(R), we denote with kµk the total variation of µ, i. e.
kµk := sup{µ(A) − µ(B) : A, B ∈ B(R)}.
Definition 1.1.4 (Space of functions of bounded variation BV (R)). We say that a
function u ∈ L1(R) belongs to the space of functions of bounded variation BV (R) if there exists a bounded signed measure µ ∈ M(R) such that
ˆ R u(x) ˙ϕ(x) dx = − ˆ R ϕ(x) dµ(x) ∀ϕ ∈ Cc∞(R).
In this case we say that the measure µ is the weak derivative of u and we denote it with
Du.
Remark 1.1.5. In the sequel we denote with TV(u) the total variation of u, that is
TV(u) := kDuk. If u /∈ BV (R), we set TV(u) := +∞.
In dimension one, functions of bounded variation are characterized by an interesting property that makes more clear the notion of total variation.
Proposition 1.1.6. Let u ∈ L1(R). Then u ∈ BV (R) if and only if
sup
( n X
i=1
|u(xi) − u(xi−1)| : −∞ < x0 < · · · < xn< +∞ Lebesgue points
)
< +∞.
Moreover, the left hand side is equal to TV(u).
1.1. Preliminaries
Sketch of the proof. Let us call TV(u) the supremum at the left hand side. It is easy to
prove that, if TV(u) < +∞, u admits a representative that is continuous at every point except countably many and that is right continuous at every point. Then we can take this representative for u and we can define µ((a, b]) := u(b) − u(a) and extend it to B(R). It is possible to verify that Du = µ ∈ M(R). On the other hand, if u ∈ BV (R), it is easy to see that u(y) − u(x) = Du((x, y]) for every x, y ∈ R Lebesgue points for u. Then, if TV(u) is not finite, it is possible to find finite unions of intervals with arbitrarily large measure, and this contradicts the fact that Du ∈ M(R).
Remark 1.1.7. We recall that Lp functions are actually equivalence classes of functions that coincide almost everywhere. As a consequence, in the sequel we sometimes define functions without specifying their value at all points, but only for almost every point. However, this is enough to determinate a unique class of function. We also say that a function is continuous if it has a continuous representative and we treat it as its continu-ous representative. In particular, we identify Sobolev functions with their continucontinu-ous representative.
Now, we recall also the definition of Γ-convergence.
Definition 1.1.8 (Γ-convergence). Let (X, d) be a metric space and let Fδ : X → R
be a family of functions, indexed by δ ∈ (0, δ0) for some δ0 > 0. We say that {Fδ}
Γ-converges to a function F : X → R and we write Γ − lim
δ→0Fδ = F
if the following two properties hold:
(Liminf inequality) For every x ∈ X and for every family xδ→ x,
lim inf
δ→0 Fδ(xδ) ≥ F (x),
(Limsup inequality) For every x ∈ X there exists a family xδ→ x such that
lim sup
δ→0
Fδ(xδ) ≤ F (x).
Moreover, we call recovery family any family {xδ} such that xδ→ x and Fδ(xδ) → F (x). Actually, it is not necessary to prove the Limsup inequality for every x ∈ X, since there exists a useful fact that simplify the proof of Γ-convergence. Before stating it, we give the definition of dense in energy set.
Definition 1.1.9 (Dense in energy set). Let (X, d) be a metric space, and F : X → R.
A subset D ⊆ X is dense in energy with respect to F if
∀x ∈ X ∃{xδ} ⊆ D such that xδ→ x and F (xδ) → F (x).
Proposition 1.1.10. Let (X, d) be a metric space, and let D ⊆ X be a dense in energy
Chapter 1. Introduction
indexed by δ ∈ (0, δ0) for some δ0 > 0. Let us assume that for every x ∈ D there exists
a family xδ→ x such that
lim sup
δ→0
Fδ(xδ) ≤ F (x).
Then, for every x ∈ X there exists a family xδ→ x such that
lim sup
δ→0
Fδ(xδ) ≤ F (x).
2
“Horizontal” functionals
2.1
A characterization based on the difference quotient
In this section we prove a classical characterization of Sobolev spaces and functions of bounded variation in one variable that is based on the Lp-norm of the difference quotient. To be more precise, for every measurable function u : R → R and every ε > 0, we consider the difference quotient
Rεu(x) :=
u(x + ε) − u(x)
ε .
Then, for every ε > 0, we define the functional
Fε(u) := ˆ R u(x + ε) − u(x) ε p dx = kRεukpp.
Now, we can state and prove the following characterization of Sobolev spaces.
Theorem 2.1.1. Let u ∈ Lp(R), and p > 1. Then the following facts are equivalent:
(1) u ∈ W1,p(R), (2) lim inf ε→0 Fε(u) < +∞, (3) Γ − lim inf ε→0 Fε(u) < +∞. Moreover lim
ε→0Fε(u) = Γ − limε→0Fε(u) = k ˙uk p p.
Proof. We divide the proof in two steps.
Step 1 We show that for every u ∈ W1,p(R) and every ε > 0 we have
Fε(u) ≤ k ˙ukpp. (2.1.1)
Since Sobolev functions are the antiderivatives of their weak derivatives, we obtain that 1 ε(u(x + ε) − u(x)) = 1 ε ˆ x+ε x ˙ u(t) dt.
Therefore, by H¨older’s inequality, we deduce that
1 ε ˆ x+ε x ˙ u(t) dt ≤ ε−1ε1−1p ˆ x+ε x | ˙u(t)|pdt !1p = 1 ε ˆ x+ε x | ˙u(t)|pdt !p1 .
Chapter 2. “Horizontal” functionals
Finally, we conclude that
Fε(u) = ˆ R u(x + ε) − u(x) ε p dx ≤ ˆ R 1 ε ˆ x+ε x | ˙u(t)|pdt ! dx = ˆ R (| ˙u(t)|p) dt = k ˙ukpp.
Step 2 We show that (3) ⇒ (1) ⇒ Γ − lim inf Fε(u) ≥ k ˙ukpp.
From (3), it follows that there exists a family {uε} ⊆ Lp such that u
ε→ u with respect
to the Lp-norm and
lim inf
ε→0 Fε(uε) < +∞.
Then there exists a sequence εn→ 0 such that Fεn(uεn) is uniformly bounded and
lim inf
ε→0 Fε(uε) =n→+∞lim Fεn(uεn) =n→+∞lim kRεnuεnk p
p < +∞.
So, up to subsequences, Rεnuεn * v ∈ L
p. Now, for every ϕ ∈ C∞
c (R), we have ˆ R vϕ = lim n→+∞ ˆ R (Rεnuεn)ϕ = lim n→+∞− ˆ R uεn(R−εnϕ) = − ˆ R u ˙ϕ.
This implies that v ∈ Lp(R) is the weak derivative of u, so u ∈ W1,p(R). Moreover, by the lower semicontinuity of the Lp-norm with respect to the weak convergence of Lp, it follows that
lim inf
ε→0 Fε(uε) =n→+∞lim kRεnuεnk p
p ≥ kvkpp = k ˙ukpp.
Since (2) ⇒ (3) is trivial, we have proved that (1), (2) and (3) are equivalent. We have also proved that
k ˙ukpp ≥ lim
ε→0Fε(u) ≥ Γ − limε→0Fε(u) ≥ k ˙uk p p.
This completes the proof.
Remark 2.1.2. If p = 1, the proof of Step 2 does not work because a bounded sequence
in L1(R) might not have a subsequence that converges weakly to a function in L1(R) (this is because L1(R) is not a reflexive Banach space). However, L1(R) is isometrically included in the space M(R) that is the dual space of C00(R). So, a bounded sequence
in L1(R) has a subsequence that converges to a bounded signed measure with respect to the weak* topology of M(R) (namely, it converges weakly as sequence of measures). This remark leads to the following theorem, whose proof is the same of Theorem 2.1.1 (using some µ ∈ M(R) instead of v ∈ Lp(R) in Step 2).
Theorem 2.1.3. Let u ∈ L1(R). Then the following facts are equivalent:
(1) u ∈ BV (R), (2) lim inf ε→0 Fε(u) < +∞, (3) Γ − lim inf ε→0 Fε(u) < +∞. Moreover lim
ε→0Fε(u) = Γ − limε→0Fε(u) = TV(u).
2.2. A more general characterization
2.2
A more general characterization
The classical results about Sobolev and BV spaces that we have proved in the previ-ous section have been generalized by J. Bourgain, H. Brezis and P. Mironescu, who introduced another family of functionals in [1]. In this section we state their results (in dimension one) and we give a sketch of the proof. To this aim, we first give a definition.
Definition 2.2.1 (Family of mollifiers). Let us consider a family of measurable and non
negative functions ρε: R → [0, +∞). We say that {ρε} is a family of mollifiers if:
• ˆ R ρε(x) dx = 1 for every ε > 0, • lim ε→0 ˆ δ −δ ρε(x) dx = 1 for every δ > 0.
Given a family of mollifiers {ρε} and p ≥ 1, we can consider the following family of functionals, defined for every measurable function u : R → R as
Gε(u) = ¨ R2 u(y) − u(x) y − x p ρε(y − x) dx dy.
Remark 2.2.2. We have said that these functionals are more general than the previous
ones because Fεis obtained from Gεusing a family of Dirac deltas at the point y − x = ε instead of the family of mollifiers ρε(y − x).
These functionals give us this characterization of Sobolev spaces.
Theorem 2.2.3. Let u ∈ Lp(R), and p > 1. Then the following facts are equivalent:
(1) u ∈ W1,p(R), (2) lim inf ε→0 Gε(u) < +∞, (3) Γ − lim inf ε→0 Gε(u) < +∞. Moreover lim
ε→0Gε(u) = Γ − limε→0Gε(u) = k ˙uk p p.
Sketch of the proof. By a change of variable we can write
Gε(u) = ¨ R2 u(x + h) − u(x) h p ρε(h) dx dh = ˆ R Fh(u)ρε(h) dh.
Now, by (2.1.1), we deduce that for every u ∈ W1,p(R) we have
Gε(u) = ˆ R Fh(u)ρε(h) dh ≤ ˆ R k ˙ukppρε(h) dh = k ˙ukpp.
On the other hand, if we have a sequence uε → u in Lp(R) such that
lim inf
Chapter 2. “Horizontal” functionals
we can find two sequences εn→ 0 and hn→ 0 for which Fhn(uεn) ≤ L. This implies that
Γ − lim inf Fε(u) < +∞ and, by Theorem 2.1.1, we deduce that u ∈ W1,p(R). Moreover,
kRhnuεnk
p
p = Fhn(uεn) ≤ L. Therefore, up to subsequences, Rhnuεn* v ∈ L
p(R). Now,
for every ϕ ∈ Cc∞(R), we have ˆ R vϕ = lim n→+∞ ˆ R (Rhnuεn)ϕ = lim n→+∞− ˆ R uεn(R−hnϕ) = − ˆ R u ˙ϕ.
Then, the semicontinuity of the norm implies lim inf
ε→0 Gε(uε) = L ≥ lim infn→+∞Fhn(uεn) ≥ k ˙uk p p.
As before, we also have a characterization of BV space, whose proof is similar to the previous one, and has an interesting corollary that is useful in the sequel.
Theorem 2.2.4. Let u ∈ L1(R). Then the following facts are equivalent:
(1) u ∈ BV (R), (2) lim inf ε→0 Gε(u) < +∞, (3) Γ − lim inf ε→0 Gε(u) < +∞. Moreover lim
ε→0Gε(u) = Γ − limε→0Gε(u) = TV(u).
Corollary 2.2.5. Let Ω = (a, b) be a bounded interval and A a measurable subset of Ω.
Let us assume that ¨
A×(Ω\A)
1
|y − x|2dx dy < +∞.
Then either |A| = 0 or |Ω \ A| = 0.
Proof. We apply Theorem 2.2.4 with ρε(t) = (ε/2)|t|ε−11[−1,1](t). It is easy to see that
{ρε} is a family of mollifiers. Then we set u(x) = 1A(x). Therefore we have TV(1A) = lim ε→0Gε(1A) = limε→0 ¨ A×(R\A) ε |y − x|2−ε1[−1,1](y − x) dx dy.
Now, let us set E = (a − 1, a) ∪ (b, b + 1). Then, since |y − x| ≤ b − a for every (x, y) ∈ A × (Ω \ A), we have TV(1A) ≤ lim ε→0 ¨ A×(Ω\A) ε(b − a)ε |y − x|2 dx dy + ¨ Ω×E ε |y − x|2−εdx dy.
Now, because of the hypothesis, the first integral tends to 0, while the second one can be explicitly computed. Therefore we have
TV(1A) ≤ lim ε→0 2 1 − ε(1 + (b − a) ε− (b − a + 1)ε) = 2. 8
2.2. A more general characterization
This means that A is a finite perimeter set in R with perimeter less then or equal to 2. So, up to null sets, A can be either empty or an interval contained in Ω. If A is an interval, it is easy to see that it must be the whole Ω. In fact, if we assume by contradiction that A = (c, d) with c > a or d < b, we see that
ˆ d c dx ˆ c a dy 1 |y − x|2 + ˆ b d dy 1 |y − x|2 ! = +∞.
3
“Vertical” functionals
3.1
Definitions and main results
This chapter is entirely devoted to the study of the asymptotic behaviour of the family of functionals Iδ,Ω, defined for every measurable function u : R → R, for every p ∈ [1, +∞) and for every δ > 0 as
Iδ,Ω(u) := ¨ Ω2 |u(x)−u(y)|>δ δp |x − y|p+1dx dy.
From now on, if Ω = R, we write Iδ(u) instead of Iδ,R(u).
Remark 3.1.1. We can notice that, if v(x) = λu(αx + β) + µ, for some α ∈ (0, +∞), λ ∈ R, β ∈ R, µ ∈ R then Iδ,[a−β α , b−β α ] (v) = αp−1|λ|pI δ |λ|,[a,b] (u).
We can now state the main results related to this family of functionals, that provide a characterization of Sobolev and BV spaces.
Theorem 3.1.2. Let u ∈ Lp(R), and p > 1. Then the following are equivalent:
(1) u ∈ W1,p(R), (2) lim inf δ→0 Iδ(u) < +∞, (3) Γ − lim inf δ→0 Iδ(u) < +∞. Moreover • lim δ→0Iδ(u) = 2 pk ˙uk p p, • Γ − lim δ→0Iδ(u) = 2 pCpk ˙uk p p, where Cp = p−11 1 −2p−11 .
Remark 3.1.3. It is easy to see that Cp< 1 for every p > 1, so the Γ-limit of Iδ is strictly
Chapter 3. “Vertical” functionals
Theorem 3.1.4. Let u ∈ L1(R). Then the following are equivalent:
(1) u ∈ BV (R), (2) Γ − lim inf δ→0 Iδ(u) < +∞. Moreover Γ − lim δ→0Iδ(u) = 2C1TV(u), where C1= log 2.
Remark 3.1.5. We notice that we do not have any pointwise convergence results for p = 1. The reason is that Iδ is very sensitive to jumps. In fact, if we consider the
function j : R → R defined by j(x) := 0 if x ∈ (−∞, 0] ∪ [2, +∞), 1 if x ∈ (0, 1], 2 − x if x ∈ (1, 2),
we have j ∈ BV (R) but Iδ(j) = +∞ for every δ < 1. However, we can also find an
absolutely continuous function g ∈ W1,1(R) such that Iδ(g) → +∞ for δ → 0 (for
example this happens if we consider g(x) = | log(x)|−1/2 for x ∈ (0, 1/2) and we extend it to R so that g = 0 in (0, 1)c).
In the next sections we first prove the pointwise convergence of Iδ, then we prove the Γ-convergence of Iδ showing that the following propositions hold.
Proposition 3.1.6. For every family {uδ} ⊆ Lp(R) converging to some u ∈ Lp(R) with
respect to the Lp-norm, we have
lim inf δ→0 Iδ(uδ) ≥ 2 pCpk ˙uk p p,
with TV(u) instead of k ˙ukpp if p = 1.
Proposition 3.1.7. For every u ∈ Lp(R) there exists a family {uδ} ⊆ Lp(R) converging
to u with respect to the Lp-norm such that
lim sup δ→0 Iδ(uδ) ≤ 2 pCpk ˙uk p p, (3.1.1)
with TV(u) instead of k ˙ukpp if p = 1.
3.2
Pointwise convergence
In this section we prove the pointwise convergence of Iδ(u) for every p ≥ 1 and every
u ∈ P Cc1(R), where
P Cc1(R) =nu ∈ Cc0(R) : ∃x1, . . . , xns.t. u ∈ C1([xi, xi+1]) ∀i, supp(u) ⊆ [x1, xn]
o
.
Actually, as we have said before, if p > 1 the pointwise convergence holds also for every
u ∈ W1,p(R), but the proof is more complicated. So we prove it only for u ∈ P Cc1(R),
that is what we need in the sequel to prove the Γ-convergence. 12
3.2. Pointwise convergence
Proposition 3.2.1. Let u ∈ P Cc1(R). Then
lim δ→0Iδ(u) = 2 pk ˙uk p p.
Proof. By a change of variable we have
Iδ(u) = ¨ R2 δp |k|1+p1{|u(x+k)−u(x)|>δ}(x, k) dx dk = 2 ˆ R dx ˆ +∞ 0 dk δ p k1+p1(1,+∞) |u(x + k) − u(x)| δ .
Since u has compact support, there exists R > 1 such that supp(u) ⊆ [−R + 1, R − 1]. Then lim δ→02 ˆ |x|>R dx ˆ +∞ 0 dk δ p k1+p1(1,+∞) |u(x + k) − u(x)| δ ≤ lim δ→02δ p ˆ −R −∞ dx ˆ −x+R−1 −x−R+1 dk k1+p ≤ lim δ→02δ p ˆ −R −∞ dx ˆ −x+R−1 −x−R+1 dk k2 = lim δ→02δ plog(2R − 1) = 0. It follows that lim δ→0Iδ(u) = limδ→02 ˆ +R −R dx ˆ +∞ 0 dk δ p k1+p1(1,+∞) |u(x + k) − u(x)| δ .
By another change of variables, we have lim δ→0Iδ(u) = limδ→02 ˆ R −R dx ˆ +∞ 0 dh 1 h1+p1(1,+∞) |u(x + δh) − u(x)| δh h . Since 1 h1+p1(1,+∞) |u(x + δh) − u(x)| δh h ≤ 1 h1+p1(1,+∞)(k ˙uk∞h) , and 2 ˆ R −R dx ˆ +∞ 0 dh 1 h1+p1(1,+∞)(k ˙uk∞h) = 2 ˆ R −R dx ˆ +∞ k ˙uk−1 ∞ dh 1 h1+p = 4R p k ˙uk p ∞,
we can apply Lebesgue’s dominated convergence theorem and we obtain lim δ→0Iδ(u) = 2 ˆ R −R dx ˆ +∞ 0 dh lim δ→0 1 h1+p1(1,+∞) |u(x + δh) − u(x)| δh h .
Now, since 1(1,+∞)(t) is continuous at every point except t = 1, we can notice that lim δ→0 1 h1+p1(1,+∞) |u(x + δh) − u(x)| δh h = 1 h1+p1(1,+∞)(| ˙u(x)|h) ,
Chapter 3. “Vertical” functionals
for every (x, h) ∈ [−R, R] × (0, +∞) such that h 6= | ˙u(x)|−1.
But {(x, h) ∈ [−R, R] × (0, +∞) : h = | ˙u(x)|−1} is the graph of a piecewise continuous function defined on an open subset of R, because u ∈ P Cc1(R). Since any open set of R is
covered by countably many compact subsets, {(x, h) ∈ [−R, R]×(0, +∞) : h = | ˙u(x)|−1} is covered by countably many graphs of uniformly continuous functions. Then it has measure zero. It follows that
lim δ→0 1 h1+p1(1,+∞) |u(x + δh) − u(x)| δh h = 1 h1+p1(1,+∞)(| ˙u(x)|h) ,
for almost every (x, h) ∈ [−R, R] × (0, +∞). Therefore we have lim δ→0Iδ(u) = 2 ˆ R −R dx ˆ +∞ 0 dh 1 h1+p1(1,+∞)(| ˙u(x)|h) = 2 ˆ R −R dx ˆ +∞ | ˙u(x)|−1 dh h1+p = 2 pk ˙uk p p.
3.3
Liminf inequality
In this section we prove Proposition 3.1.6. The proof follows from an estimate of the asymptotic cost of oscillations as in [2], but with the exact constant Cp, that comes
from the computation of Iδ on a family of piecewise constant monotone functions. The main idea of the proof is reducing the problem to a finite combinatorial rearrangement inequality, that is proved in Lemma 3.3.5. To this aim, we first prove some lemmas.
Lemma 3.3.1. Let Ω be an interval and p ≥ 1. Let us fix δ > 0 and uδ ∈ Lp(Ω). We
consider ¯ uδ:= X k∈Z kδ1{u−1 δ ([kδ,(k+1)δ))}. Then Iδ,Ω(uδ) ≥ Iδ,Ω(¯uδ).
Proof. It is sufficient to observe that the following inclusion holds
{|¯uδ(x) − ¯uδ(y)| > δ} ⊆ {|uδ(x) − uδ(y)| > δ}.
Lemma 3.3.2. Let p ≥ 1 and δ > 0. Let us set N =
δ−1
. Let uδ: [0, 1] → δZ ∩ [0, 1] such that Iδ,[0,1](uδ) < +∞. Let us set, for every k ∈ {0, .., N },
Ak:= u−1δ (kδ), µk:= |Ak|,
k0 := min{k ∈ {0, . . . , N } : µk> 0}, k1:= max{k ∈ {0, . . . , N } : µk > 0}.
Then µk> 0 for every k ∈ {k0, . . . , k1}.
Proof. Let us assume, by contradiction, that µi = 0 for some i ∈ {k0, . . . , k1}. Let us
set A = A0∪ · · · ∪ Ai−1. Therefore
Iδ,[0,1](uδ) ≥ ¨ A×Ac δp |x − y|p+1dx dy ≥ ¨ A×Ac δp |x − y|2 dx dy.
Then, by Corollary 2.2.5, either A = ∅ or A = [0, 1]. This is absurd because of the definition of k0 and k1.
3.3. Liminf inequality
Lemma 3.3.3. Let A0, . . . , AN be a partition of [0, 1] in N + 1 measurable subsets and
let X be the set of finite union of intervals with rational endpoints. Then, for every ε > 0, there exists another partition of [0, 1] in N + 1 sets B0, . . . , BN ∈ X such that
|Ai4Bi| ≤ ε for every i ∈ {0, . . . , N }.
Proof. Since Lebesgue measure is regular, for every ε > 0 and every measurable set A,
there exists an open set B0 such that A ⊆ B0 and |A \ B0| < ε/4. Then, since open sets in R are countably (or finite) union of disjoint open intervals, it is easy to find a finite union of open intervals B00 such that |A4B00| < ε/2 and also a finite union of open intervals with rational endpoints B (i. e. B ∈ X) such that |A4B| < ε. Now, we apply this argument to find B0 ∈ X such that |A04B0| ≤ ε/2N +1. Then we apply the
same argument to A1\ B0 and we find B1 ∈ X such that |(A1 \ B0)4B1| ≤ ε/2N +1.
We can also assume that B0∩ B1 = ∅. So, |A14B1| ≤ ε/2N. Then we repeat the same
argument to find N + 1 disjoint sets B0, . . . , BN ∈ X such that |Ai4Bi| ≤ ε/2N +1−i for
every i ∈ {0, . . . , N }. We also have that
N X i=0 |Bi| ≥ N X i=0 |Ai| − ε 2N +1−i ≥ 1 − ε 1 − 1 2N +1 . Therefore, if we set C := N [ i=0 Bi !c ,
it follows that |C| ≤ ε−ε/2N +1. Now, since X is an algebra, we have that B0∪C ∈ X and |A04(B0∪ C)| ≤ ε. If we take B0∪ C instead of B0we have the requested partition.
Now we can state and prove the combinatorial inequality that is the key point of the proof of Proposition 3.1.6.
Definition 3.3.4 (Nondecreasing rearrangement). Let d be a positive integer and let
f : {1, . . . , d} → N. We say that the nondecreasing rearrangement of f is the function f∗(i) := min{n ∈ N : |{j ∈ {1, . . . , d} : f (j) ≤ n}| ≥ i}.
Lemma 3.3.5. Let d be a positive integer and f : {1, . . . , d} → N. Let c : N → R be a
nonincreasing function and E ⊆ N2 such that (i, j) ∈ E if and only if (j, i) ∈ E. We set J (E, f ) := X
i≤j
(f (i),f (j))∈E
c(j − i).
Then J (E2, f ) ≥ J (E2, f∗), where E2 = {(i, j) ∈ N2 : |i − j| ≥ 2}.
Proof. We first introduce some useful notation and then we divide the proof in some
steps. Let us set m := max{f (i) : i ∈ {1, . . . , d}}, and let k ∈ {1, . . . , d} be the largest integer such that f (k) = m. If d ≥ 2, we consider the function Rf : {1, . . . , d − 1} → N defined by
Rf (i) :=
(
f (i) if i < k,
Chapter 3. “Vertical” functionals
Now we can define
∆(E, f ) := J (E, f ) − J (E, Rf ). It follows that ∆(E, f ) = X (f (i),f (k))∈E c(|i − k|) − X i<k<j (f (i),f (j))∈E (c(j − i − 1) − c(j − i)), (3.3.1)
where the first sum is due to the fact that in Rf we “removed the value at k”, while the second one is due to the reduction of the distances between the “values before k” and the “values after k”. Finally, if L and R are subsets of N \ {0}, we define
G(L, R) := c(0) +X `∈L c(`) +X r∈R c(r) − X (`,r)∈L×R (c(` + r − 1) − c(` + r)).
Step 1 We prove that
G(L, R) ≤
|L|+|R|
X
i=0
c(i). (3.3.2)
We argue by induction on |R|. If R = ∅, since c is nonincreasing, we have
G(L, R) = c(0) +X `∈L c(`) ≤ |L| X i=0 c(i) = |L|+|R| X i=0 c(i).
Now, let us assume that the conclusion is true if |R| = n and let us consider L, R ⊆ N with |R| = n + 1. Let us set
a := max R, b := min{n ∈ N \ {0} : n 6∈ L}, L1:= L ∪ {b}, R1 := R \ {a}.
and let us observe that |R1| = n and |L1| + |R1| = |L| + |R|. Therefore, by the inductive
assumption, (3.3.2) is equivalent to
G(L1, R1) ≥ G(L, R).
By definition of G(L, R), this is equivalent to
c(b) +X
`∈L
(c(` + a − 1) − c(` + a)) ≥ c(a) + X
r∈R1
(c(b + r − 1) − c(b + r)). (3.3.3)
If b > 1, since L ⊇ {1, . . . , b − 1} and all terms are nonnegative, we have that
c(b) +X `∈L (c(` + a − 1) − c(` + a)) ≥ c(b) + b−1 X `=1 (c(` + a − 1) − c(` + a)) = c(b) + c(a) − c(a + b − 1). 16
3.3. Liminf inequality
Moreover, the inequality is obviously true also if b = 1. On the other hand, since
R1⊆ {1, . . . , a − 1}, we have c(a) + X r∈R1 (c(b + r − 1) − c(b + r)) ≤ c(a) + a−1 X r=1 (c(b + r − 1) − c(b + r)) = c(a) + c(b) − c(a + b − 1) ≤ c(b) +X `∈L (c(` + a − 1) − c(` + a)).
Therefore we have proved (3.3.3), which implies (3.3.2).
Step 2 We prove that
∆(E2, f ) ≥ ∆(E2, f∗). (3.3.4)
for any f : {1, . . . , d} → N. Let us set
Em := N2\ {m − 1, m}2.
We claim that
∆(E2, f ) ≥ ∆(Em, f ) ≥ ∆(Em, f∗) = ∆(E2, f∗).
The equality ∆(Em, f∗) = ∆(E2, f∗) is an easy consequence of (3.3.1), since k = d for
f∗ because it is nondecreasing and (f∗(i), m) ∈ E2 if and only if (f∗(i), m) ∈ Em. The inequality ∆(E2, f ) ≥ ∆(Em, f ) is again an easy consequence of (3.3.1), since we
have
{i : (f (i), m) ∈ E2} = {i : (f (i), m) ∈ Em}, and
{(i, j) : i < k < j ∧ (f (i), f (j)) ∈ E2} ⊆ {(i, j) : i < k < j ∧ (f (i), f (j)) ∈ Em}.
So we have to prove ∆(Em, f ) ≥ ∆(Em, f∗). Let us consider
Emc := N2\ Em = {m − 1, m}2.
By definition of J (E, f ) it is clear that
J (Em, f ) = J (N2, f ) − J (Emc, f ),
for every f , and therefore
∆(Em, f ) = ∆(N2, f ) − ∆(Ecm, f ).
So, ∆(Em, f ) ≥ ∆(Em, f∗) if and only if ∆(Ecm, f ) ≤ ∆(Emc, f∗). Now, let us set
R(f ) := {r ≥ 1 : f (k + r) ∈ {m − 1, m}} , L(f ) := {` ≥ 1 : f (k − `) ∈ {m − 1, m}} .
With this notation we have
Chapter 3. “Vertical” functionals Therefore X (f (i),f (k))∈E c(|i − k|) = c(0) + X `∈L(f ) c(`) + X r∈R(f ) c(r), and X i<k<j (f (i),f (j))∈E (c(j − i − 1) − c(j − i)) = X (`,r)∈L(f )×R(f ) (c(` + r − 1) − c(` + r)). Thus, by (3.3.1), we have ∆(Emc, f ) = c(0) + X `∈L(f ) c(`) + X r∈R(f ) c(r) + X (`,r)∈L(f )×R(f ) (c(` + r − 1) − c(` + r)) = G(L(f ), R(f )).
On the other hand, since
R(f∗) = ∅, L(f∗) = {1, . . . , |L(f )| + |R(f )|}, it follows that ∆(Emc, f∗) = |L(f )|+|R(f )| X i=0 c(i).
Therefore, by (3.3.2), we have ∆(Emc, f ) ≤ ∆(Emc, f∗). So, we have proved (3.3.4).
Step 3 We prove that J (E2, f ) ≥ J (E2, f∗) for every f : {1, . . . , d} → N arguing by
induction on d.
If d = 1, f = f∗ so there is nothing to prove. Let us assume that the conclusion holds for every f : {1, . . . , d} → N for some d ≥ 1 and let us consider a function
f : {1, . . . , d + 1} → N. Since Rf : {1, . . . , d} → N, by inductive assumption we have
J (E2, Rf ) ≥ J (E2, (Rf )∗). Since (Rf )∗= R(f∗), by (3.3.4) we have J (E2, f ) = J (E2, Rf ) + ∆(E2, f ) ≥ J (E2, (Rf )∗) + ∆(E2, f∗) = J (E2, R(f∗)) + ∆(E2, f∗) = J (E2, f∗).
So, we completed the proof.
Now we are ready to prove the main lemma for the proof of Proposition 3.1.6, that puts together all the previous ones. Given a family of functions {uδ} ⊆ Lp(R), this lemma
allows us to find another family of functions that are much simpler and for which the value of Iδ does not increase much. This provides an estimate from below for the value of Iδ(uδ) that leads directly to Proposition 3.1.6.
3.3. Liminf inequality
Lemma 3.3.6. Let us fix δ > 0 and uδ: [0, 1] → δZ ∩ [0, 1] such that
|u−1δ ([0, ε])| ∧ |u−1δ ([1 − ε, 1])| > 2η,
for some ε > 0, η > 0. Then, there exists a nondecreasing function u∗δ : [0, 1] → δZ∩[0, 1] of the form u∗δ = N X k=0 kδ1[tk,tk+1),
for some t0, . . . , tN +1∈ Q with 0 = t0 ≤ t1 ≤ · · · ≤ tN ≤ tN +1= 1 and N =δ−1, such
that Iδ,[0,1](uδ) ≥ Iδ,[0,1](u∗δ) − δp and
|u∗δ−1([0, ε])| ∧ |u∗δ−1([1 − ε, 1])| > η.
Proof. Let Ak, µk, k0, k1 be as in Lemma 3.3.2 and µ := min{µk: k ∈ {k0, . . . , k1}}. If
Iδ,[0,1](uδ) = +∞ the conclusion is trivial. So, we can assume Iδ,[0,1](uδ) < +∞. Then,
by Lemma 3.3.2, µ > 0. Let U := (x, y) ∈ [0, 1]2: |x − y| > µ 4 .
Let M be a real number large enough to ensure that 1
|x − y|1+p ≤ M, ∀(x, y) ∈ [0, 1]
2\ U. (3.3.5)
Now, we consider the functional JδM : Lp(R) → R defined by
JδM(v) := ¨ [0,1]2 |v(x)−v(y)|>δ δp 1 |x − y|p+1 ∧ M dx dy.
Clearly, Iδ,[0,1](uδ) ≥ JδM(uδ). Now we take Bk0, . . . , Bk1 given by Lemma 3.3.3 such
that |Bk4Ak| ≤ η N + 1∧ 1 2M (N + 1)2 ∧ µ 4, (3.3.6)
for every k ∈ {k0, . . . , k1}. So, we can define ˜uδ :=Pkδk1Bk.
Let us set Cij := (Bi × Bj) \ (Ai × Aj). Since Ai × Aj ⊇ (Bi × Bj) \ Cij for every
i, j ∈ {0, . . . , N }, it follows that JδM(uδ) = X |i−j|≥2 ¨ Ai×Aj δp 1 |x − y|p+1∧ M dx dy ≥ X |i−j|≥2 ¨ Bi×Bj δp 1 |x − y|p+1∧ M dx dy − ¨ Cij δp 1 |x − y|p+1 ∧ M dx dy ≥ JδM(˜uδ) − X |i−j|≥2 δpM |Cij|.
Now, since Cij ⊆ ((Bi\ Ai) × [0, 1]) ∪ ([0, 1] × (Bj\ Aj)), we have, by (3.3.6), that
|Cij| ≤ 1 2M (N + 1)2 + 1 2M (N + 1)2 = 1 M (N + 1)2.
Chapter 3. “Vertical” functionals
From the two previous inequalities, it follows that
JδM(uδ) ≥ JδM(˜uδ) − X |i−j|≥2 δpM 1 M (N + 1)2 ≥ J M δ (˜uδ) − δp.
Now, we notice that ˜uδis a finite sum of characteristic functions of intervals with rational
endpoints weighted with integer multiples of δ. So, we can write it as a finite sum of weighted characteristic functions of intervals with the same (rational) length 1/d, for some d ∈ N. Namely, it exists f : {1, . . . , d} → N such that
˜ uδ= d X i=1 f (i)δ1[i−1 d , i d).
So, we can apply Lemma 3.3.5 with the function c : N → R defined by
c(i) = ˆ 1 d 0 dx ˆ i+1 d i d dy δp 1 |x − y|p+1∧ M ,
so that JδM(˜uδ) = 2J (E2, f ). Then we deduce that JδM(˜uδ) ≥ JδM(u
∗ δ), where u ∗ δ is defined by u∗δ := d X i=1 f∗(i)δ1[i−1 d , i d),
where f∗(i) is the nondecreasing rearrangement of f . Therefore, we have obtained that
JδM(uδ) ≥ JδM(u
∗
δ) − δp.
Now, since |u∗δ−1(kδ)| = |Bk| and (3.3.6) implies that |Bk4Ak| ≤ µ/4, we notice that
|u∗δ−1(kδ)| ≥ 3µ/4 for every k ∈ {0, . . . , N } and therefore {|u∗δ(x) − u∗δ(y)| > δ} ∩ U = ∅. Combining this with (3.3.5), we notice that
JδM(u∗δ) = Iδ,[0,1](u∗δ). Thus, we proved all these inequalities
Iδ,[0,1](uδ) ≥ JδM(uδ) ≥ JδM(˜uδ) − δp ≥ JδM(u
∗
δ) − δp = Iδ,[0,1](u∗δ) − δp.
Finally, by (3.3.6) we also have that
|u∗δ−1([0, ε])| ≥ |u−1δ ([0, ε])| −
k1
X
k=k0
|Bk4Ak| ≥ 2η − η = η.
The same holds for |u∗δ−1([1 − ε, 1])|. So, u∗δ satisfies all the requirements. We can now prove Proposition 3.1.6.
3.3. Liminf inequality
Proof of Proposition 3.1.6 in the case p = 1. Let us fix ε > 0. Let x1, . . . , xn ∈ R be
Lebesgue points for u such that
n−1
X
j=1
|u(xj+1) − u(xj)| ≥ TV(u) − ε,
if u ∈ BV (R), or such that the left hand side is greater than 1/ε otherwise. Since lim inf δ→0 Iδ(uδ) ≥ n−1 X j=1 lim inf δ→0 Iδ,[xj,xji+1](uδ),
it is sufficient to prove that lim inf
δ→0 Iδ,[xj,xj+1](uδ) ≥ 2 log 2 |u(xj+1) − u(xj)|,
for every j ∈ {1, . . . , n − 1}.
Let us fix j ∈ {1, . . . , n − 1}. By a change of variable (as we have seen in Remark 3.1.1), we can assume without loss of generality that xj = 0, xj+1 = 1, u(0) = 0, u(1) = 1. We
can also assume that every uδ takes only values in [0, 1], otherwise we can consider their truncations. Since uδ→ u with respect to the L1-norm and 0, 1 are Lebesgue points for
u, there exist η > 0 and δ0 > 0 such that
|u−1δ ([0, ε])| ∧ |u−1δ ([1 − ε, 1])| > 2η,
for every δ < δ0. By Lemma 3.3.1 and Lemma 3.3.6, for every δ < δ0 we can find u∗δ of the form u∗δ = Nδ X k=0kδ1[ tδ k,t δ k+1), for some tδ0, . . . , tδN δ+1 ∈ Q with 0 = t δ 0 ≤ tδ1 ≤ · · · ≤ tδNδ ≤ t δ Nδ+1 = 1 and Nδ = δ−1 such that |u∗δ−1([0, ε])| ∧ |u∗δ−1([1 − ε, 1])| > η, and Iδ,[0,1](uδ) ≥ Iδ,[0,1](u∗δ) − δ.
In particular, since u∗δ is nondecreasing, u∗δ(η) ≤ ε and u∗δ(1 − η) ≥ 1 − ε. Now we can estimate the value of Iδ,[0,1](u∗δ) by an explicit computation
Iδ,[0,1](u∗δ) = ¨ [0,1]2 |u∗ δ(x)−u ∗ δ(y)|>δ δ |x − y|2 dx dy ≥ ¨ [η,1−η]×[0,1] |u∗ δ(x)−u ∗ δ(y)|>δ δ |x − y|2 dx dy ≥ ¨ [η,1−η]×R |u∗ δ(x)−u ∗ δ(y)|>δ δ |x − y|2 dx dy − ¨ [η,1−η]×[0,1]c δ |x − y|2 dx dy ≥ ¨ [η,1−η]×R |u∗ δ(x)−u ∗ δ(y)|>δ δ |x − y|2 dx dy − 2δ log 1 − η η .
Chapter 3. “Vertical” functionals Let us call a := min{i ∈ {0, . . . , Nδ+ 1} : tδi > η}, b := max{i ∈ {0, . . . , Nδ+ 1} : tδi < 1 − η}. Therefore we have ¨ [η,1−η]×R |u∗ δ(x)−u ∗ δ(y)|>δ δ |x − y|2 dx dy ≥ b−1 X i=a ˆ tδi+1 tδ i dx ˆ tδi−1 −∞ δ |x − y|2 dy + ˆ +∞ tδ i+2 δ |x − y|2 dy ! .
Computing the explicit value of the right hand side we find
b−1 X i=a ˆ tδ i+1 tδ i dx ˆ tδ i−1 −∞ δ |x − y|2dy + ˆ +∞ tδ i+2 δ |x − y|2dy ! =δ b−1 X i=a log t δ i+1− tδi−1 tδ i − tδi−1 ! + log t δ i+2− tδi tδ i+2− tδi+1 ! =δ b−1 X i=a log t δ i+1− tδi−1 tδi − tδ i−1 ! + δ b X i=a+1 log t δ i+1− tδi−1 tδi+1− tδ i ! ≥δ b−1 X i=a+1 log t δ i+1− tδi−1 tδi − tδ i−1 ! + log t δ i+1− tδi−1 tδi+1− tδ i ! . Let us set lδ i := tδi − tδi−1. So we obtain δ b−1 X i=a+1 log t δ i+1− tδi−1 tδ i − tδi−1 ! + log t δ i+1− tδi−1 tδ i+1− tδi ! =δ b−1 X i=a+1 log l δ i+1+ lδi lδi ! + log l δ i+1+ liδ lδi+1 ! =δ b−1 X i=a+1 log 1 +l δ i+1 lδi ! 1 + l δ i lδi+1 !! ≥δ b−1 X i=a+1 log 4 = δ(b − a − 1)2 log 2.
Now, since u∗δ(η) ≤ ε and u∗δ(1 − η) ≥ 1 − ε, we notice that u∗δ(tδa) ≤ ε + δ and u∗δ(tδb) ≥ 1 − ε − δ. From the definition of u∗δ, it follows that
aδ ≤ ε + δ, bδ ≥ 1 − ε − δ. Therefore b − a ≥ 1 − 2ε δ − 2. Thus, we have δ(b − a − 1)2 log 2 ≥ (1 − 2ε − 3δ)2 log 2. 22
3.3. Liminf inequality
Combining all the inequalities we get
Iδ,[0,1](uδ) ≥ Iδ,[0,1](u∗δ) − δ ≥ (1 − 2ε − 3δ)2 log 2 − 2δ log 1 − η η − δ. Then lim inf δ→0 Iδ,[0,1](uδ) ≥ lim infδ→0 Iδ,[0,1](u ∗ δ) − δ ≥ (1 − 2ε)2 log 2.
Since ε is arbitrary, the conclusion follows.
Now, we state and prove a lemma about Sobolev norms that is useful to extend the proof for p = 1 to the case of p > 1. It is similar to the classical characterization of BV functions in one variable that we have used in the previous proof.
Lemma 3.3.7. Let u ∈ Lp(R) for some p > 1. Then
k ˙ukpp = sup
(n−1 X
i=1
|u(xi+1) − u(xi)|p
(xi+1− xi)p−1 : x1< · · · < xn are Lebesgue points for u
)
.
Proof. We notice that the quantity
n−1
X
i=1
|u(xi+1) − u(xi)|p
(xi+1− xi)p−1
is equal to k ˙gkpp where g is a piecewise affine function defined on [x1, xn] that agrees
with u on {x1, x2, . . . , xn}. Since the function ψ : R → R defined by ψ(t) = tp is strictly
convex, the unique minimizers of the functional k ˙vkp
p defined for every v ∈ Lp(Ω) (where
Ω ⊆ R is an interval) with Dirichlet boundary conditions are affine functions. Therefore k ˙ukpp ≥ sup
(n−1 X
i=1
|u(xi+1) − u(xi)|p
(xi+1− xi)p−1 : x1 < · · · < xn are Lebesgue points for u
)
.
On the other hand, if u is not continuous the equality is trivial, since both sides are equal to +∞. If u is continuous, we can consider a sequence {gk} of piecewise affine function such that gk is null outside [−k, k], affine on every interval of the form [i/k, (i + 1)/k] and gk(i/k) = u(i/k) for every i ∈ {−k2+ 1, . . . , k2− 1}. It is easy to see that gk→ u
uniformly on compact subsets of R. Then, if u ∈ W1,p(R), we have that k ˙ukpp is finite,
so k ˙gkkpp is bounded. Therefore, up to subsequences, ˙gk * ˙u weakly in Lp(R) and we
deduce that
lim inf k ˙gkkpp≥ k ˙ukpp.
Otherwise, if u /∈ W1,p(R), the sum must diverge. In fact, if it were bounded, there
would be a subsequence of ˙gk weakly converging to some v ∈ Lp(R), and this would
imply that u ∈ W1,p(R).
Proof of Proposition 3.1.6 in the case p > 1. Let us take x1, . . . , xn Lebesgue points for
u. By Lemma 3.3.7, it is sufficient to prove that
lim inf δ→0 Iδ,[xj,xj+1](uδ) ≥ 2 pCp |u(xj+1) − u(xj)|p (xj+1− xj)p−1 ,
Chapter 3. “Vertical” functionals
for every j ∈ {1, . . . , n − 1}. Let us fix j. By a change of variable, we can assume without loss of generality that xj = 0, xj+1 = 1, u(0) = 0, u(1) = 1. Let us fix ε > 0. Since
uδ→ u with respect to the Lp-norm and 0, 1 are Lebesgue points, there exist η > 0 and
δ0 > 0 such that
|u−1δ ([0, ε])| ∧ |u−1δ ([1 − ε, 1])| > 2η,
for every δ < δ0. By Lemma 3.3.1 and Lemma 3.3.6, for every δ < δ0 we can find u∗δ of the form u∗δ= Nδ X k=0 kδ1[tδ k,tδk+1), for some tδ0, . . . , tδN δ+1 ∈ Q with 0 = t δ 0 ≤ tδ1 ≤ · · · ≤ tδNδ ≤ t δ Nδ+1 = 1 and Nδ = δ−1 , such that |u∗δ−1([0, ε])| ∧ |u∗δ−1([1 − ε, 1])| > η, and Iδ,[0,1](uδ) ≥ Iδ,[0,1](u∗δ) − δp.
In particular, since u∗δ is nondecreasing, u∗δ(η) ≤ ε and u∗δ(1 − η) ≥ 1 − ε. Now we can estimate the value of Iδ,[0,1](u∗δ) by an explicit computation
Iδ,[0,1](u∗δ) = ¨ [0,1]2 |u∗ δ(x)−u ∗ δ(y)|>δ δp |x − y|1+pdx dy ≥ ¨ [η,1−η]×[0,1] |u∗ δ(x)−u ∗ δ(y)|>δ δp |x − y|1+pdx dy ≥ ¨ [η,1−η]×R |u∗ δ(x)−u ∗ δ(y)|>δ δp |x − y|1+pdx dy − ¨ [η,1−η]×[0,1]c δp |x − y|1+pdx dy ≥ ¨ [η,1−η]×R |u∗ δ(x)−u ∗ δ(y)|>δ δp |x − y|1+pdx dy − 2δp p(p − 1) 1 ηp−1 − 1 (1 − η)p−1 . Let us call a := min{i ∈ {0, . . . , Nδ+ 1} : tδi > η}, b := max{i ∈ {0, . . . , Nδ+ 1} : tδi < 1 − η}. Therefore we have ¨ [η,1−η]×R |u∗ δ(x)−u ∗ δ(y)|>δ δp |x − y|1+pdx dy ≥ b−1 X i=a ˆ tδ i+1 tδ i dx ˆ tδ i−1 −∞ δp |x − y|1+pdy + ˆ +∞ tδ i+2 δp |x − y|1+pdy ! . 24
3.3. Liminf inequality
Computing the explicit value of the right hand side we find
b−1 X i=a ˆ tδ i+1 tδ i dx ˆ tδ i−1 −∞ δp |x − y|1+pdy + ˆ +∞ tδ i+2 δp |x − y|1+pdy ! = δ p p(p − 1) b−1 X i=a 1 (tδi − tδ i−1)p−1 − 1 (tδi+1− tδ i−1)p−1 + 1 (tδi+2− tδ i+1)p−1 − 1 (tδi+2− tδ i)p−1 ! = δ p p(p − 1) b−1 X i=a 1 (tδ i − tδi−1)p−1 − 1 (tδ i+1− tδi−1)p−1 ! + δ p p(p − 1) b X i=a+1 1 (tδi+1− tδ i)p−1 − 1 (tδi+1− tδ i−1)p−1 ! ≥ δ p p(p − 1) b−1 X i=a+1 1 (tδi − tδ i−1)p−1 + 1 (tδi+1− tδ i)p−1 − 2 (tδi+1− tδ i−1)p−1 ! . Let us set lδi := tδi − tδ i−1. So we obtain δp p(p − 1) b−1 X i=a+1 1 (tδi − tδ i−1)p−1 + 1 (tδi+1− tδ i)p−1 − 2 (tδi+1− tδ i−1)p−1 ! = δ p p(p − 1) b−1 X i=a+1 1 (liδ)p−1+ 1 (lδi+1)p−1− 2 (lδi+1+ lδi)p−1 ! .
Since the function t 7→ t1−p is convex on (0, +∞), by Jensen’s inequality we have
δp p(p − 1) b−1 X i=a+1 1 (lδi)p−1+ 1 (lδi+1)p−1− 2 (lδi+1+ lδi)p−1 ! ≥ δ p p(p − 1) b−1 X i=a+1 2p− 2 (lδi+1+ liδ)p−1 ! ≥(2 p− 2)δp p(p − 1) (b − a − 1)p Pb−1 a+1 (li+1δ + lδi) p−1 ≥(2 p− 2)δp p(p − 1) (b − a − 1)p 2p−1 =(b − a − 1)pδp2 pCp.
Now, since u∗δ(η) ≤ ε and u∗δ(1 − η) ≥ 1 − ε, we notice that u∗δ(tδa) ≤ ε + δ and u∗δ(tδb) ≥ 1 − ε − δ. It follows from the definition of u∗δ that
aδ ≤ ε + δ, bδ ≥ 1 − ε − δ.
Therefore
b − a ≥ 1 − 2ε δ − 2.
Chapter 3. “Vertical” functionals Thus, we have (b − a − 1)pδp2 pCp ≥ (1 − 2ε − 3δ) p2 pCp.
Combining all the inequalities we get
Iδ,[0,1](uδ) ≥ Iδ,[0,1](u∗δ) − δp ≥ (1 − 2ε − 3δ)p 2 pCp− 2δp p(p − 1) 1 ηp−1 − 1 (1 − η)p−1 − δp. Then lim inf δ→0 Iδ,[0,1](uδ) ≥ lim infδ→0 (1 − 2ε − 3δ) p2 pCp− 0 = (1 − 2ε) p2 pCp.
Since ε is arbitrary, the conclusion follows.
3.4
Limsup inequality
In this section we want to prove Proposition 3.1.7. By Proposition 1.1.10 it is sufficient to prove (3.1.1) only for every u that belongs to some subset D ⊆ Lp(R) which is dense in energy with respect to the Sobolev norm (or the total variation if p = 1). It is also known that both Cc∞(R) and P Ac(R) (that is the set of continuous piecewise affine functions
with compact support) have this property. However, we can prove Proposition 3.1.7 for the larger set P Cc1(R). This is useful since the recovery families for Cc∞(R) can be used to extend our proof to higher dimension, while, in the next section, we use recovery families for piecewise affine functions (that are easy to treat) to show that we can build recovery families that are C∞ (this provides a positive answer to Open problem 3 in [3] at least in dimension one and for ϕ = c1ϕ1). First of all, we prove the following lemma,
which uses an idea from [7], and that allows us to compute Iδ(uδ) as the sum of simpler terms.
Lemma 3.4.1. Let Ω ⊆ R be an interval and x1 < · · · < xn ∈ Ω a finite number of
points. Then, for every family {uδ} ⊆ Lp(Ω) such that supp(u
δ) ⊆ [−R, R] for some
R > 0 and for every δ, we have
lim sup δ→0 Iδ,Ω(uδ) ≤ lim sup δ→0 n X i=0 Iδ,[xi,xi+1](uδ) + n X i=1 Iδ,[x i− √ δ,xi+ √ δ](uδ),
where x0 = inf Ω and xn+1= sup Ω.
Proof. First of all, since supp(uδ) ⊆ [−R, R], we notice that the integration set for
Iδ,Ω(uδ) is included in R × [−R, R] ∪ [−R, R] × R. Therefore, since we are enlarging the
integration set, it is clear that
Iδ,Ω(uδ) ≤ n X i=0 Iδ,[xi,xi+1](uδ) + n X i=1 Iδ,[x i− √ δ,xi+ √ δ](uδ) + ¨ E δp |x − y|p+1dx dy. Where E = E1∪ E2∪ E3 with E1= (−∞, −R − √ δ] × [−R, R], E2= [−R − √ δ, R + √ δ] × R ∩ {|x − y| > √ δ}, E3= [R + √ δ, +∞) × [−R, R]. 26
3.4. Limsup inequality
So, we only have to show that
lim sup δ→0 ¨ E δp |x − y|p+1dx dy = 0.
This is an easy computation. If p > 1 we find ¨ E1 δp |x − y|p+1dx dy = δp p(p − 1) 1 δp−12 − 1 2R +√δp−1 → 0, ¨ E2 δp |x − y|p+1dx dy = 2R + 2√δ2 pδ p 2 → 0, ¨ E3 δp |x − y|p+1dx dy = δp p(p − 1) 1 δp−12 − 1 2R +√δp−1 → 0. If p = 1 we find ¨ E1 δ |x − y|2 dx dy = δ log 1 +2R√ δ → 0, ¨ E2 δ |x − y|2 dx dy = 2 2R + 2√δ√δ → 0, ¨ E3 δ |x − y|2 dx dy = δ log 1 +2R√ δ → 0.
Thus, the proof is complete.
Proof of Proposition 3.1.7. We define
uδ=
X
k∈Z
kδ1{u−1([kδ,(k+1)δ))}.
We notice that, since u is continuous and has compact support, u is bounded and this implies that the sum has only a finite number of terms that are not identically null. Now we proceed step by step.
Step 1 We prove (3.1.1) for u ∈ C1([a, b]) with | ˙u(x)| ≥ m > 0 for every x ∈ [a, b].
In particular, u is striclty monotone. Without loss of generality, we can assume a = 0,
b = 1, u(0) = 0 and u(1) = 1. Then
uδ= Nδ X k=0 kδ1[tδ k,t δ k+1). Where Nδ = bδ−1c, u(tδ
k) = kδ for every k ∈ {0, . . . , Nδ} and tδNδ+1 = 1. Therefore, we
Chapter 3. “Vertical” functionals
Step 1.1 (p = 1). In this case we have
Iδ,[0,1](uδ) = ˆ tδ 1 0 dx ˆ 1 tδ 2 dy δ |x − y|2 + ˆ tδ 2 tδ 1 dx ˆ 1 tδ 3 dy δ |x − y|2 + ˆ tδ Nδ tδ Nδ−1 dx ˆ tδ Nδ−2 0 dy δ |x − y|2 + ˆ 1 tδ Nδ dx ˆ tδ Nδ−1 0 dy δ |x − y|2 + Nδ−2 X k=2 ˆ tδ k+1 tδ k dx ˆ tδ k−1 0 dy δ |x − y|2 + ˆ 1 tδ k+2 dy δ |x − y|2 ! ≤ δ log t δ 2 tδ 2− tδ1 ! + δ log t δ 3− tδ1 tδ 3− tδ2 ! + δ log t δ Nδ − t δ Nδ−2 tδN δ−1− t δ Nδ−2 ! + δ log 1 − t δ Nδ−1 tδN δ − t δ Nδ−1 ! + Nδ−2 X k=2 δ log t δ k+1− tδk−1 tδk− tδ k−1 ! + log t δ k+2− tδk tδk+2− tδ k+1 !! = δ log t δ 2 tδ 2− tδ1 ! + δ log 1 − t δ Nδ−1 tδ Nδ − t δ Nδ−1 ! + Nδ−2 X k=1 δ log t δ k+2− tδk tδk+1− tδ k ! + log t δ k+2− tδk tδk+2− tδ k+1 !! .
Now, by Lagrange’s mean value theorem (and ˙u ≥ m > 0), we have
tδk+1− tδ k = u(tδk+1) − u(tδk) ˙ u(cδ k) = δ ˙ u(cδ k) , for some cδk∈ (tδ
k, tδk+1) for every k ∈ {0, . . . , Nδ− 1} and
1 − tδN δ = u(1) − u(tδN δ) ˙ u(cδ Nδ) ≤ δ ˙ u(cδ Nδ) ,
for some cNδ ∈ (tNδ, 1). Then Iδ,[0,1](uδ) ≤ δ log 1 + ˙ u(cδ1) ˙ u(cδ0) ! + δ log 1 +u(c˙ δ Nδ−1) ˙ u(cδN δ) ! + Nδ−2 X k=1 δ log 1 + u(c˙ δ k) ˙ u(cδ k+1) ! + log 1 +u(c˙ δ k+1) ˙ u(cδ k) !! .
Now, let us fix ε > 0. Since ˙u is uniformly continuous, there exists η > 0 such that
|x − y| ≤ η ⇒ | ˙u(x) − ˙u(y)| ≤ εm.
Then, if δ < mη/2, for every k ∈ {0, . . . , Nδ− 1} we have that tδk+2− tδk ≤ η, that implies
cδk+1− cδ k≤ η. It follows that ˙ u(cδk+1) ˙ u(ck)δ ∨ u(c˙ δ k) ˙ u(cδk+1) ≤ 1 + ε, 28
3.4. Limsup inequality
for every k. Then
Iδ,[0,1](uδ) ≤ δ log(2 + ε) + δ log(2 + ε) + Nδ−1 X k=1 δ (log(2 + ε) + log(2 + ε)) = 2Nδδ log(2 + ε). Therefore lim sup δ→0 Iδ,[0,1](uδ) ≤ lim sup δ→0 δbδ−1c2 log(2 + ε) = 2 log(2 + ε). Since ε is arbitrary, the conclusion follows.
Step 1.2 (p > 1). In this case, Iδ,[0,1](uδ) is equal to
ˆ tδ 1 0 dx ˆ 1 tδ 2 dy δ p |x − y|1+p+ ˆ tδ 2 tδ 1 dx ˆ 1 tδ 3 dy δ p |x − y|1+p + ˆ tδ Nδ tδ Nδ−1 dx ˆ tδ Nδ−2 0 dy δ p |x − y|1+p + ˆ 1 tδ Nδ dx ˆ tδ Nδ−1 0 dy δ p |x − y|1+p + Nδ−2 X k=2 ˆ tδ k+1 tδ k dx ˆ tδ k−1 0 dy δ p |x − y|1+p+ ˆ 1 tδ k+2 dy δ p |x − y|1+p ! ≤ δ p p(p − 1) 1 (tδ 2− tδ1)p−1 − 1 (tδ 2)p−1 + 1 (tδ 3− tδ2)p−1 − 1 (tδ 3− tδ1)p−1 + 1 (tδN δ−1− t δ Nδ−2) p−1 − 1 (tδN δ− t δ Nδ−2) p−1 + 1 (tδN δ − t δ Nδ−1) p−1 − 1 (1 − tδN δ−1) p−1 ! + Nδ−2 X k=2 δp p(p − 1) 1 (tδk− tδ k−1)p−1 − 1 (tδk+1− tδ k−1)p−1 + 1 (tδ k+2− tδk+1)p−1 − 1 (tδ k+2− tδk)p−1 ! ≤ δ p p(p − 1) 1 (tδ2− tδ 1)p−1 − 1 (tδ2)p−1 + 1 (tδN δ − t δ Nδ−1) p−1 − 1 (1 − tδN δ−1) p−1 ! + Nδ−2 X k=1 δp p(p − 1) 1 (tδ k+1− tδk)p−1 + 1 (tδ k+2− tδk+1)p−1 − 2 (tδ k+2− tδk)p−1 ! .
As before, applying Lagrange’s mean value theorem, we find cδk∈ (tδ
k, tδk+1) such that tδk+1− tδk = δ ˙ u(cδ k) ,
for every k ∈ {0, . . . , Nδ− 1} and
1 − tδN δ ≤ δ ˙ u(cδ Nδ) .
Chapter 3. “Vertical” functionals It follows that Iδ,[0,1](uδ) ≤ δp p(p − 1) ˙ u(cδ1)p−1 δp−1 − ˙ u(cδ0)p−1u(c˙ δ1)p−1 δp−1( ˙u(cδ 0) + ˙u(cδ1))p−1 + u(c˙ δ Nδ−1) p−1 δp−1 − ˙ u(cδNδ−1)p−1u(c˙ δNδ)p−1 δp−1( ˙u(cδ Nδ−1) + ˙u(c δ Nδ)) p−1 ! + Nδ−2 X k=1 δp p(p − 1) ˙ u(cδk)p−1 δp−1 + ˙ u(cδk+1)p−1 δp−1 − 2 ˙ u(cδk+1)p−1u(c˙ δk)p−1 δp−1( ˙u(cδ k+1) + ˙u(cδk))p−1 ! = δ p(p − 1) u(c˙ δ 1)p−1− ˙ u(cδ 0)p−1u(c˙ δ1)p−1 ( ˙u(cδ 0) + ˙u(cδ1))p−1 + ˙u(cδNδ−1)p−1− ˙ u(cδN δ−1) p−1u(c˙ δ Nδ) p−1 ( ˙u(cδN δ−1) + ˙u(c δ Nδ)) p−1 ! + Nδ−2 X k=1 δ p(p − 1) u(c˙ δ k)p−1+ ˙u(cδk+1)p−1− 2 ˙ u(cδk+1)p−1u(c˙ δk)p−1 ( ˙u(cδ k+1) + ˙u(cδk))p−1 ! ≤ δ p(p − 1) 2k ˙ukp−1∞ + Nδ−2 X k=1 2δ p(p − 1)u(c˙ δ k)p−1 1 2 + 1 2 ˙ u(cδk+1) ˙ u(cδk) !p−1 − 1 1 + u(c˙ δ k) ˙ u(cδ k+1) p−1 .
Now, let us fix ε > 0. Since ˙u is uniformly continuous, there exists η > 0 such that, for
every δ < mη/2, we have ˙ u(cδk+1) ˙ u(ck)δ ∨ u(c˙ δ k) ˙ u(cδk+1) ≤ 1 + ε, for every k. Then
Iδ,[0,1](uδ) ≤ 2δk ˙uk p−1 ∞ p(p − 1) + Nδ−2 X k=1 2δ p(p − 1)u(c˙ δ k)p−1 1 2+ 1 2(1 + ε) p−1− 1 (2 + ε)p−1 = 2δk ˙uk p−1 ∞ p(p − 1) + Nδ−2 X k=1 2 p(p − 1)u(c˙ δ k)p δ ˙ u(cδk) 1 2 + 1 2(1 + ε) p−1− 1 (2 + ε)p−1 = 2δk ˙uk p−1 ∞ p(p − 1) + 2 p(p − 1) 1 2 + 1 2(1 + ε) p−1− 1 (2 + ε)p−1 Nδ−1 X k=0 ˙ u(cδk)p(tδk+1− tδ k).
Now, we notice that sum at the right hand side is a Riemann sum for´01u(x)˙ pdx. So,
since ˙up is continuous, it follows that
lim sup δ→0 Iδ,[0,1](uδ) ≤ 2 p(p − 1) 1 2 + 1 2(1 + ε) p−1− 1 (2 + ε)p−1 k ˙ukpp. 30
