ON FORMULAE DECOUPLING THE TOTAL VARIATION OF
BV FUNCTIONS
AUGUSTO C. PONCE AND DANIEL SPECTOR
To Nicola Fusco, a master of BV, on the occasion of his sixtieth birthday
A BSTRACT. In this paper we prove several formulae that enable one to capture the
singular portion of the measure derivative of a function of bounded variation as a
limit of non-local functionals. One special case shows that rescalings of the fractional Laplacian of a function u ∈ SBV converge strictly to the singular portion
of Du.
1. I NTRODUCTION AND M AIN R ESULTS
Let Ω ⊂ RN be an open, bounded and smooth subset (or all of RN ) and u be a
function of bounded variation in Ω, i.e. u ∈ L1 (Ω) and its distributional derivative
Du is a Radon measure with finite total variation
ˆ
(1.1)
u div Φ.
|Du|(Ω) :=
sup
Φ∈Cc1 (Ω;RN ),
kΦkL∞ (Ω;RN ) ≤1
Ω
The space of such functions BV (Ω) contains the Sobolev space W 1,1 (Ω), with strict
inclusion, since in general for u ∈ BV (Ω) one has the decomposition of the measure
Du = ∇u LN + Ds u,
where ∇u is the Radon-Nikodym density of Du with respect to the N -dimensional
Lebesgue measure LN and Ds u is singular with respect to LN . In particular, u ∈
W 1,1 (Ω) precisely when Ds u ≡ 0.
While the computation of the total variation (1.1) through the theory of distributions is classical, in recent years there has been an interest in its — and other related energies — approximation through the asymptotics of non-local functionals
[1, 4–7, 9, 11, 12, 16–18, 20, 22–26]. For example, Bourgain, Brezis and Mironescu [7]
had shown that for u ∈ W 1,p (Ω), 1 ≤ p < +∞, one has
ˆ ˆ
ˆ
|u(x) − u(y)|p
lim− (1 − α)
(1.2)
dy
dx
=
C
|∇u|p .
p,N
N +αp
|x
−
y|
α→1
Ω Ω
Ω
Here, for p = 1,
ˆ
|e · v| dHN −1 (v) = 2ωN −1 ,
C1,N =
SN −1
where e ∈ RN is any unit vector and ωN −1 is the volume of the unit ball in RN −1 .
Formula (1.2) expresses the fact that appropriately scaled Gagliardo semi-norms
tend to the total variation as the differential parameter tends to one. Their result
includes the case u ∈ W 1,1 (Ω), while for general u ∈ BV (Ω) it was proved by
Dávila [14, Theorem 1] that
ˆ ˆ
|u(x) − u(y)|
(1.3)
lim− (1 − α)
dy dx = C1,N |Du|(Ω).
N +α
α→1
Ω Ω |x − y|
1
2
AUGUSTO C. PONCE AND DANIEL SPECTOR
The Gagliardo semi-norms can be thought heuristically as the energy of derivatives of fractional order α ∈ (0, 1), while the space of p-integrable functions for
which they are finite coincides with the W α,p space obtained by real interpolation
of Lp and W 1,p with parameter α. Indeed, it was subsequently shown by Milman [24] that the convergence (1.2) can be alternatively deduced from the theory
of interpolation.
In the complex interpolation between Lp and W 1,p when Ω = RN , one classically
sees the fractional Laplacian
ˆ
u(x) − u(y)
α/2
dy
(−∆) u(x) := cα
N +α
RN |x − y|
filling the role of the differential object of order α ∈ (0, 1), and even in a more
extended range, though with a different formula. The constant
cα =
2α−1 αΓ( N +α
2 )
N
π 2 Γ( 2−α
2 )
\
α/2 u(ξ) =
ensures that the Fourier transform of the fractional Laplacian satisfy (−∆)
α
(2π|ξ|) u
b(ξ).
As integer powers of the Laplacian can also be understood from the framework
of the functional calculus, the natural limits here are α = 0 and α = 2. This
observation is supported by the trivial convergence, as α tends to 1, of the energies
p
ˆ ˆ
u(x) − u(y) (1.4)
lim (1 − α)
dy
N |x − y|N +α
dx = 0,
α→1−
N
R
R
whenever u ∈ W 1,p (RN ) and p ≥ 1. The case p > 1 follows from the Lp boundedness of the Riesz transforms that yields the equivalence between the quantities
k∇ukLp (RN ) and k(−∆)1/2 ukLp (RN ) ; see e.g. [19, Lemma 3.6]. The case p = 1 requires a different argument that can be found in [27, 28], based on the approximation of u by smooth functions. Alternatively, one can still obtain the expected
energy, in the spirit of (1.2), via a non-local gradient approach [23].
What is perhaps surprising is that the convergence (1.4) is no longer true for
general u ∈ BV (Ω) and p = 1. For example, in the case where u = χA is the
characteristic function of a set of finite perimeter, Dávila’s formula (1.3) and the
straightforward integral identity
ˆ ˆ
ˆ ˆ
χA (x) − χA (y) |χA (x) − χA (y)|
(1.5)
dy dx =
dy dx,
N
+α
|x − y|
|x − y|N +α
N
N
N
N
R
R
imply that
(1.6)
R
ˆ
lim (1 − α)
α→1−
RN
ˆ
RN
R
χA (x) − χA (y) dy dx = C1,N |Ds χA |(RN ).
|x − y|N +α
Here we observe that the measure DχA is singular with respect to LN , and thus
DχA = Ds χA . More generally, in this paper we are interested in several formulae for the asymptotics of functionals capturing the singular portion of the measure
derivative of u ∈ BV (Ω). Let us first restrict our attention to the setting of special functions of bounded variation SBV (Ω). This space has been introduced by
De Giorgi and Ambrosio [15], and consists of those BV functions whose singular
part of the derivative is supported by an (N − 1)-dimensional rectifiable set; see
Section 3 below. Then our first result is
Theorem 1.1. For every u ∈ SBV (Ω), we have
ˆ ˆ
u(x) − u(y)
lim
ρ (x − y) dy dx = K1,N |Ds u|(Ω),
→0 Ω Ω
|x − y|
ON FORMULAE DECOUPLING THE TOTAL VARIATION OF BV FUNCTIONS
3
where K1,N = C1,N /HN −1 (SN −1 ) and HN −1 (SN −1 ) denotes the (N − 1)-dimensional
Hausdorff measure of the unit sphere SN −1 .
We assume throughout the paper that (ρ )>0 ⊂ L1 (RN ) is a family of nonnegative radial functions that satisfies
ˆ
(1.7)
lim
ρ (h) dh = 1,
→0 |h|<δ
ˆ
(1.8)
lim
ρ (h) dh = 0,
→0
|h|>δ
for every δ > 0. Taking in particular
ρ (h) =
χBR (0) (h)
,
HN −1 (SN −1 ) |h|N −
for any fixed R > 0, one finds an analogue to the convergence (1.2) for p = 1.
Moreover, as c1 C1,N = 2/π, one can show by a slight variation of the argument of
Theorem 1.1 the following
Corollary 1.2. For every u ∈ SBV (RN ), we have
lim− (1 − α)k(−∆)α/2 ukL1 (RN ) =
α→1
2 s
|D u|(RN ).
π
In Section 4 below, we show that the standard Cantor function admits a positive
lower bound for some families (ρ )>0 , including ρ = χ(−,) /2. This motivates
Open Problem 1.3. Do these identities hold for every u ∈ BV (RN )?
The answer is not clear. For example, Fusco, Moscariello and Sbordone [18]
have recently studied a related functional introduced by Ambrosio, Bourgain,
Brezis and Figalli [1, Section 4.2]. In their case, the contribution of the singular part of the derivative cannot be expressed solely in terms of the total mass
|Ds u|(RN ) [18, Example 2.2 and Theorem 3.3]. In Section 5, we discuss some connections between these various functionals.
We next turn our attention to the entire space of functions of bounded variation.
In this broader setting, we have the following result, an extension of the recent
Taylor characterization by the second author [27, 28] to this regime, which bears
some analogy with formula (1.6):
Theorem 1.4. For every u ∈ BV (Ω), we have that
ˆ ˆ
|u(x) − u(y) − ∇u(x) · (x − y)|
lim
ρ (x − y) dy dx = K1,N |Ds u|(Ω).
→0 Ω Ω
|x − y|
Remark 1.5. Theorem 1.4 combined with the results of [7] or [23] yields the following
characterization of the Sobolev space W 1,1 : u ∈ W 1,1 (Ω) if and only if
ˆ ˆ
|u(x) − u(y) − F (x) · (x − y)|
lim
ρ (x − y) dy dx = 0,
→0 Ω Ω
|x − y|
for some function F ∈ L1 (Ω; RN ), and in that case F = ∇u almost everywhere in Ω.
In fact we prove slightly stronger results than the convergence of the energies
in Theorems 1.1 and 1.4 and Corollary 1.2, the so-called strict convergence of the
measures defined by the integrands; see Propositions 2.1 and 3.2 below. The organization of the paper is as follows: as we utilize the result of Theorem 1.4 in
the proof of Theorem 1.1, we first prove Theorem 1.4 in Section 2. In Section 3
we prove Theorem 1.1 and Corollary 1.2. Next in Section 4 we give the example
of the non-trivial lower bound for the Cantor function. Finally, in Section 5 we
4
AUGUSTO C. PONCE AND DANIEL SPECTOR
discuss some of the connections between the functionals we consider and those
of Bourgain, Brezis and Mironescu [7], Ambrosio, Bourgain, Brezis and Figalli [1]
and Fusco, Moscariello and Sbordone [18].
2. P ROOF OF T HEOREM 1.4
Given u ∈ BV (RN ), define the function R1 u : RN → R by
ˆ
|u(x + h) − u(x) − ∇u(x) · h|
R1 u(x) =
ρ (h) dh.
|h|
N
R
We are interested in the convergence of kR1 ukL1 (RN ) as tends to zero. Observe
that if u ∈ Cc∞ (RN ), then
R1 u → 0 in L1 (RN ).
This property still holds if u ∈ C ∞ (RN ) and ∇u ∈ L1 (RN ; RN ).
Proposition 2.1. Let u ∈ BV (RN ). For every bounded continuous function ϕ : RN →
R, we have
ˆ
ˆ
R1 u ϕ = K1,N
ϕ d|Ds u|.
lim
→0
RN
RN
We first prove a couple of lemmas. Given u ∈ BV (RN ) and a family of mollifiers (ψδ )δ>0 in Cc∞ (RN ), we use the notation uδ = u ∗ ψδ , (∇u)δ = ∇u ∗ ψδ ,
...
Lemma 2.2. For every u ∈ BV (RN ), we have
K1,N |Ds u|(RN ) ≤ lim inf kR1 ukL1 (RN ) .
→0
Proof. By comparison of integrals and Fubini’s theorem, for every x ∈ RN we have
ˆ
|uδ (x + h) − uδ (x) − (∇u)δ (x) · h|
ρ (h) dh ≤ (R1 u ∗ ψδ )(x) = (R1 u)δ (x).
|h|
RN
Integration with respect to x yields
ˆ
ˆ ˆ
|uδ (x + h) − uδ (x) − (∇u)δ (x) · h|
ρ (h) dh dx ≤
(R1 u)δ (x) dx = kR1 ukL1 (RN ) .
|h|
RN
RN RN
On the other hand, since
ˆ h s
K1,N |(Ds u)δ (x)| =
ρ (h) dh
(D u)δ (x) ·
|h|
RN
and
(∇u)δ + (Ds u)δ = ∇(uδ ),
by the triangle inequality we have
ˆ
|uδ (x + h) − uδ (x) − (∇u)δ (x) · h|
s
ρ (h) dh − K1,N |(D u)δ (x)| ≤ R1 (uδ )(x).
N
|h|
R
Hence,
ˆ
K1,N
ˆ
s
|(D u)δ (x)| dx ≤
RN
RN
ˆ
R1 (uδ )(x) dx
ˆ
+
RN
RN
|uδ (x + h) − uδ (x) − (∇u)δ (x) · h|
ρ (h) dh dx
|h|
≤ kR1 (uδ )kL1 (RN ) + kR1 ukL1 (RN ) .
Since uδ ∈ C ∞ (RN ) and ∇(uδ ) ∈ L1 (RN ; RN ) for every δ > 0, the first term in the
right-hand side converges to zero as tends to zero, and we get
ˆ
K1,N
|(Ds u)δ (x)| dx ≤ lim inf kR1 ukL1 (RN ) .
RN
→0
Letting δ tend to zero, the conclusion follows from the lower semicontinuity of the
norm.
ON FORMULAE DECOUPLING THE TOTAL VARIATION OF BV FUNCTIONS
5
Lemma 2.3. Let u ∈ BV (RN ). For every non-negative bounded continuous function
ϕ : RN → R, we have
ˆ
ˆ
R1 u ϕ ≤ K1,N
lim sup
ϕ d|Ds u|.
→0
RN
RN
Proof. We prove that
ˆ
ˆ
(2.1)
R1 u ϕ ≤ K1,N
ϕ d|Ds u|
N
N
R
R
ˆ 1ˆ ˆ
+
|ϕ(x + th) − ϕ(x)|ρ (h) dh d|Ds u|(x) dt
0
RN
RN
1
ˆ
ˆ
ˆ
|∇u(x + th) − ∇u(x)| dx ρ (h) dh dt.
+ kϕkL∞ (RN )
0
RN
RN
Since ϕ is bounded and continuous and ∇u ∈ L1 (RN ; RN ), the second and third
terms converge to zero as tends to zero. It thus suffices to establish this estimate.
For this purpose, we proceed by approximation using the functions uδ . By the
Fundamental Theorem of Calculus and the triangle inequality, we have that
ˆ 1
ˆ 1
s
|uδ (x+h)−uδ (x)−(∇u)δ (x)·h| ≤
|(D u)δ (x+th)·h| dt+
|(∇u)δ (x+th)−(∇u)δ (x)|·|h| dt.
0
0
By the triangle inequality and the change of variables z = x + th, we also have that
ˆ
|(Ds u)δ (x + th) · h|ϕ(x) dx
RN
ˆ
ˆ
s
≤
|(D u)δ (x + th) · h|ϕ(x + th) dx +
|(Ds u)δ (x + th) · h| · |ϕ(x + th) − ϕ(x)| dx
RN
RN
ˆ
ˆ
=
|(Ds u)δ (z) · h|ϕ(z) dz + |h|
|(Ds u)δ (z)| · |ϕ(z) − ϕ(z − th)| dz.
RN
RN
By Fubini’s theorem, we deduce that
ˆ ˆ ˆ 1
h s
dt ρ (h) dh ϕ(x) dx
(D u)δ (x + th) ·
|h|
RN RN
0
ˆ
ˆ 1ˆ ˆ
s
≤ K1,N
|(D u)δ (z)|ϕ(z) dz+
|ϕ(z)−ϕ(z−th)|ρ (h) dh |(Ds u)δ (z)| dz dt.
RN
RN
0
RN
In the second term we can replace the variable h by −h and switch the letter z to
x. Hence,
ˆ ˆ
|uδ (x + h) − uδ (x) − (∇u)δ (x) · h|
ρ (h) dh ϕ(x) dx
|h|
RN
RN
ˆ
≤ K1,N
|(Ds u)δ (x)|ϕ(x) dx
RN
ˆ 1ˆ ˆ
+
|ϕ(x + th) − ϕ(x)|ρ (h) dh |(Ds u)δ (x)| dx dt
0
RN
RN
1
ˆ
ˆ
ˆ
+ kϕkL∞ (RN )
|(∇u)δ (x + th) − (∇u)δ (x)| dx ρ (h) dh dt.
0
RN
RN
Since |(Ds u)δ | ≤ |Ds u| ∗ ψδ , letting δ tend to zero we deduce (2.1).
Proof of Proposition 2.1. We may assume that ϕ is non-negative. Since ϕ is bounded,
there exists M ≥ 0 such that ϕ ≤ M in RN . We now write
ˆ
ˆ
ˆ
R1 u ϕ = M
R1 u −
R1 u (M − ϕ).
RN
RN
RN
6
AUGUSTO C. PONCE AND DANIEL SPECTOR
By Lemmas 2.2 and 2.3, we get
ˆ
ˆ
lim inf
R1 u ϕ ≥ M K1,N |Ds u|(RN )−K1,N
→0
RN
ˆ
(M −ϕ) d|Ds u| = K1,N
RN
ϕ d|Ds u|.
RN
Since the reverse inequality holds for the limsup, the conclusion follows.
Corollary 2.4. Let u ∈ BV (RN ). For every open set W ⊂ RN , we have
ˆ
ˆ
s
1
K1,N |D u|(W ) ≤ lim inf
R u ≤ lim sup
R1 u ≤ K1,N |Ds u|(W ).
→0
→0
W
W
Proof. We prove the last inequality; the first one can be proved along the same idea
using Proposition 2.1 instead of Lemma 2.3. For this purpose, take a sequence of
continuous functions (ϕn )n∈N such that 0 ≤ ϕn ≤ 1 in RN , ϕn = 1 on W and
converging pointwisely to 0 in RN \ W . For every > 0 and n ∈ N, we have
ˆ
ˆ
1
R u ≤
R1 u ϕn .
RN
W
Letting tend to zero, by Lemma 2.3 we have that
ˆ
ˆ
lim sup
R1 u ≤ K1,N
ϕn d|Ds u|.
→0
RN
W
The conclusion follows from the Dominated Convergence Theorem as n tends to
infinity.
Proof of Theorem 1.4. The case Ω = RN follows from Proposition 2.1 applied to the
test function ϕ = 1. We may thus assume that Ω is an open, bounded and smooth
subset of RN . Given u ∈ BV (Ω), we may extend u as a function in RN , still
denoted by u, such that u ∈ BV (RN ) and |Ds u|(∂Ω) = 0 [3, Proposition 3.21]. We
now consider a function f : RN × RN → R such that, for every x 6= y,
f (x, y) =
|u(x) − u(y) − ∇u(x) · (x − y)|
ρ (x − y).
|x − y|
By an affine change of variables, we have
ˆ
R1 u(x) =
f (x, y) dy.
RN
s
Since |D u|(∂Ω) = 0, it thus follows from Corollary 2.4 that
ˆ ˆ
ˆ ˆ
f ≤ lim
f = K1,N |Ds u|(Ω).
lim sup
→0
Ω
→0
Ω
Ω
RN
To obtain the reverse inequality for the liminf, take an open set V c Ω. We estimate
ˆ ˆ
ˆ ˆ
ˆ ˆ
ˆ ˆ
f =
f +
f +
f
Ω RN
Ω Ω
Ω V \Ω
Ω RN \V
ˆ ˆ
ˆ
ˆ
ˆ ˆ
(2.2)
f +
f .
≤
f +
Ω
Ω
V \Ω
RN
Ω
RN \V
Taking r > 0 such that d(RN \ V, Ω) > r, we have
ˆ ˆ
ˆ
2
f ≤
kukL1 (RN ) + k∇ukL1 (RN )
ρ .
r
Ω RN \V
RN \Br (0)
Thus, as tends to zero in estimate (2.2), by Corollary 2.4 we get
ˆ ˆ
K1,N |Ds u|(Ω) ≤ lim inf
f + K1,N |Ds u|(V \ Ω).
→0
Ω
Ω
ON FORMULAE DECOUPLING THE TOTAL VARIATION OF BV FUNCTIONS
7
Minimizing the right-hand side with respect to V , it follows from the Monotone
Set Lemma that
ˆ ˆ
K1,N |Ds u|(Ω) ≤ lim inf
f .
→0
Ω
Ω
3. P ROOFS OF T HEOREM 1.1 AND C OROLLARY 1.2
Given u ∈ BV (RN ), define the function S u : RN → R by
ˆ
u(x + h) − u(x)
(3.1)
S u(x) = ρ (h) dh.
|h|
N
R
Using a regularization argument and the Fundamental Theorem of Calculus (as in
the proof of Lemma 2.1 in [23], for example), one can show that
ˆ ˆ
|u(x + h) − u(x)|
(3.2)
kS ukL1 (RN ) ≤
ρ (h) dh dx ≤ |Du|(RN ),
|h|
RN RN
so that S u is well-defined and uniformly bounded in L1 (RN ) with respect to .
h
ρ (h) is odd, and so we additionally
Moreover, as ρ is even, the function h 7→ |h|
have the pointwise estimate
0 ≤ S u ≤ R1 u.
(3.3)
Let us recall that, for any u ∈ BV (Ω), the singular part Ds u of the distributional
derivative satisfies the property that, for every measurable subset E ⊂ RN such
that HN −1 (E) = 0, one has |Ds u|(E) = 0. We thus have a Lebesgue decomposition
of the measure Ds u of the form
Ds u = Dj u + Dc u,
where the jump part Dj u is carried by a measurable set which is σ-finite with
respect to the Hausdorff measure HN −1 , and Dc u is the Cantor part. We say that
u belongs to the class of functions of special bounded variation SBV (Ω) when
Dc u = 0.
This jump–Cantor terminology comes from the fact that, by the Federer-Vol’pert
decomposition of Dj u [3, Theorem 3.78], there exists an (N − 1)-dimensional rectifiable set Ju such that there exists a triple (u+ , u− , νu ) ∈ R × R × SN −1 of Borel
functions defined on Ju which satisfies
(3.4)
Dj u = (u+ − u− ) ⊗ νu HN −1 bJu .
In dimension one, Dj u is a countable combination of Dirac masses, each one indicating a jump discontinuity of the function. The distributional derivative of the
Cantor function contains only the Cantor part, and is supported on the standard
Cantor set.
In what follows, we need some additional properties on the jump set Ju . Since
Ju is rectifiable, it is contained in a countable union of graphs of Lipschitz functions, and for HN −1 -almost every x0 ∈ Ju , the following properties hold:
(i) [21, Theorem 10.2] there exists an approximate tangent plane Tx0 (Ju ) at x0
and
HN −1 (Ju ∩ Br (x0 ))
lim
= 1,
r→0
ωN −1 rN −1
and [3, Theorem 3.78] νu (x0 ) is orthogonal to Tx0 (Ju );
(ii) [21, Corollary 6.5 and Proposition 10.5] up to a rotation, there exists a Lipschitz function γ : RN −1 → R such that x0 is contained in the graph gr γ, the
approximate tangent planes Tx0 (gr γ) and Tx0 (Ju ) exist and coincide,
HN −1 (gr γ ∩ Br (x0 ))
= 1,
r→0
ωN −1 rN −1
lim
8
AUGUSTO C. PONCE AND DANIEL SPECTOR
and
HN −1 ((gr γ4Ju ) ∩ Br (x0 ))
= 0,
r→0
rN −1
where A4B = (A \ B) ∪ (B \ A) denotes the symmetric difference between
the sets A and B;
(iii) [21, Theorem 5.16] x0 is a Lebesgue point of u+ and u− with respect to the
measure HN −1 bJu , and the precise representatives are u+ (x0 ) and u− (x0 ),
respectively.
Given κ > 0, denote by Tκ : R → R the truncation function at levels ±κ:


if s > κ,
κ
Tκ (s) = s
if −κ ≤ s ≤ κ,


−κ if s < −κ.
lim
For every u ∈ BV (RN ), by Lipschitz continuity of Tκ one deduces using a smooth
approximation of u that Tκ (u) ∈ BV (RN ) and |D(Tκ (u))|(RN ) ≤ |Du|(RN ). It
follows from the Federer-Vol’pert chain rule formula for BV functions [2, Corollary 3.1; 3, Theorem 3.96] that JTκ (u) ⊂ Ju and
Dj (Tκ (u)) = (Tκ (u+ ) − Tκ (u− )) ⊗ νu HN −1 bJu .
(3.5)
From the Dominated Convergence Theorem, we then have that
lim |Dj (Tκ (u)) − Dj u|(RN ) = 0.
(3.6)
k→∞
We now have the ingredients to establish
Lemma 3.1. Let u ∈ BV (RN ). For every non-negative continuous function with compact support ϕ : RN → R, we have
ˆ
ˆ
K1,N
ϕ d|Dj u| ≤ lim inf
S u ϕ.
RN
→0
RN
N
Proof. Since u ∈ BV (R ), the inequality (3.2) implies that the family (S u)>0 is
bounded in (C0 (RN ))0 , where C0 (RN ) denotes the Banach space of continuous
real functions in RN that converge uniformly to zero at infinity. Given ϕ, take a
sequence (n )n∈N converging to zero such that
ˆ
ˆ
lim
Sn u ϕ = lim inf
S u ϕ
n→∞
RN
→0
RN
∗
and Sn u * λ for some finite Borel measure λ in RN . The existence of λ follows
from the Riesz Representation Theorem [21, Theorem 4.7].
We prove that λ ≥ K1,N |Dj u|. Given x0 ∈ Ju that satisfies Assertions (i)–(iii)
and is contained in the graph gr γ of a Lipschitz function as above, we partition
the space RN in two parts:
N −1
RN
× R : t > γ(x)}
+ = {(x, t) ∈ R
N −1
and RN
× R : t < γ(x)}.
− = {(x, t) ∈ R
Let g : RN → R be the function defined by
(
u+ (x0 ) if x ∈ RN
+,
g(x) =
−
u (x0 ) if x ∈ RN
−.
We have that g ∈ BVloc (RN ) and
Dg = (u+ (x0 ) − u− (x0 )) ⊗ νγ HN −1 bgr γ ,
where νγ is the unit normal vector with respect to the graph of γ. Even though g
does not belong to BV (RN ), nor to L1 (RN ), we may define S g and S (u − g) as
ON FORMULAE DECOUPLING THE TOTAL VARIATION OF BV FUNCTIONS
9
in the formula (3.1) and they both belong to L1loc (RN ). To see this, let BR (0) be the
ball centered at zero with radius R > 0. Then one has the estimate
ˆ
ˆ
ˆ
|g(x + h) − g(x)|
|S g(x)| dx ≤
ρ (h) dh dx
|h|
BR (0)
BR (0) B1 (0)
ˆ
ˆ
+
2kgkL∞ (RN ) ρ (h) dh dx
BR (0)
RN \B1 (0)
≤ |Dg|(BR+1 (0)) + 2ωN RN kgkL∞ (RN ) .
Thus, as S u, S g ∈ L1loc (RN ) so also does S (u − g).
By the triangle inequality, we have
S u ≥ S g − S (u − g).
(3.7)
Applying the pointwise estimate (3.3) and Lemma 2.3 to u − g, we have that
ˆ
ˆ
(3.8)
lim sup
S (u − g)ϕ ≤
ϕ d|Ds (u − g)|.
→0
RN
RN
Moreover, since the image of g has only two elements, by a straightforward computation in the spirit of (1.5) we have the identity
ˆ
ˆ ˆ
|g(x) − g(y)|
ρ (x − y)ϕ(x) dy dx.
S g ϕ =
|x − y|
N
N
N
R
R
R
Hence, by the local version of Dávila’s result [14, Lemma 2] we get
ˆ
ˆ
(3.9)
lim
S g ϕ = K1,N
ϕ d|Dg|.
→0
RN
RN
∗
Recalling that Sn u * λ and combining equations (3.7)–(3.9), we deduce that
ˆ
ˆ
ˆ
ϕ dλ ≥ K1,N
ϕ d|Dg| − K1,N
ϕ d|Ds (u − g)|.
RN
RN
RN
N
Hence, on every compact subset of R , we have
λ ≥ K1,N |Dg| − K1,N |Ds (u − g)|,
and then, by inner regularity of finite Borel measures, this inequality holds on
every Borel subset of RN .
Denote by λj the measure λbJu . Then restricting the previous inequality to Ju
and using the Federer-Vol’pert decomposition formula (3.4), we have the following relation between measures:
1
λj ≥ |Dj g| − |Dj (u − g)|
K1,N
= |u+ (x0 ) − u− (x0 )|HN −1 bgr γ
− (u+ − u+ (x0 )) − (u− − u− (x0 ))HN −1 bgr γ∩J
j
+
−
− |D u|bJu \gr γ −|u (x0 ) − u (x0 )|H
N −1
u
bgr γ\Ju .
For any given κ > 0, by the triangle inequality and the structure of the measure
Dj Tκ (u) given by the chain rule (3.5), we have
|Dj u|bJu \gr γ ≤ |Dj (Tκ (u))|bJu \gr γ +|Dj (Tκ (u)) − Dj u|bJu \gr γ
≤ 2κHN −1 bJu \gr γ +|Dj (Tκ (u)) − Dj u|bJu .
The last measure in the right-hand side does not depend on x0 . Denoting
µκ =
1
λj + |Dj (Tκ (u)) − Dj u|bJu ,
K1,N
10
AUGUSTO C. PONCE AND DANIEL SPECTOR
we thus have that
µκ ≥ |u+ (x0 ) − u− (x0 )|HN −1 bgr γ
− (u+ − u+ (x0 )) − (u− − u− (x0 ))HN −1 bgr γ∩Ju
− 2κHN −1 bJu \gr γ −|u+ (x0 ) − u− (x0 )|HN −1 bgr γ\Ju .
By Properties (i)–(iii) satisfied by the point x0 , we deduce that
lim inf
r→0
µκ (Br (x0 ))
≥ |u+ (x0 ) − u− (x0 )|.
ωN −1 rN −1
Since this relation holds for HN −1 -almost every point x0 ∈ Ju , it follows from the
Besicovitch differentiation theorem [21, Theorem 6.4] that
1
λj + |Dj (Tκ (u)) − Dj u|bJu = µκ ≥ |u+ − u− |HN −1 bJu .
K1,N
As κ tends to infinity, we deduce from the strong convergence (3.6) that
1
λj ≥ |u+ − u− |HN −1 bJu = |Dj u|.
K1,N
Therefore,
ˆ
lim inf
→0
ˆ
ˆ
j
ϕ dλ ≥
S u ϕ =
RN
ˆ
RN
ϕ d|Dj u|.
ϕ dλ ≥ K1,N
RN
RN
Proof of Theorem 1.1. Let u ∈ SBV (RN ). We first prove that, for every bounded
continuous function ϕ : RN → R (not necessarily with compact support), we have
ˆ
ˆ
(3.10)
K1,N
ϕ d|Dj u| ≤ lim inf
S u ϕ.
RN
→0
RN
Taking this estimate from granted, we may combine it with the pointwise inequality (3.3) and Lemma 2.3 to get
ˆ
ˆ
ˆ
ˆ
j
K1,N
ϕ d|D u| ≤ lim inf
S u ϕ ≤ lim sup
S u ϕ ≤ K1,N
ϕ d|Ds u|.
RN
s
→0
RN
→0
RN
RN
j
Since D u = D u, we deduce that
ˆ
ˆ
(3.11)
lim
S u ϕ = K1,N
→0
RN
ϕ d|Ds u|.
RN
Choosing ϕ = 1, we have the conclusion when Ω = RN . In the case when Ω is an
open, bounded, smooth subset of RN , we can proceed by extension to RN along
the lines of the proof of Theorem 1.4.
To prove (3.10), it suffices to consider the case of a non-negative function ϕ.
Given a sequence of continuous functions with compact support (ψn )n∈N converging pointwise to 1 and such that 0 ≤ ψn ≤ 1 in RN , we have
ˆ
ˆ
S u ϕψn ≤
S u ϕ.
RN
RN
Letting tend to zero, we deduce from Lemma 3.1 that
ˆ
ˆ
K1,N
ϕψn d|Dj u| ≤ lim inf
S u ϕ.
RN
→0
RN
The conclusion follows from the Dominated Convergence Theorem as n tends to
infinity.
In the previous proof, we have established the strict convergence of (S u)>0 .
This conclusion still holds under a weaker assumption on the family (ρ )>0 :
ON FORMULAE DECOUPLING THE TOTAL VARIATION OF BV FUNCTIONS
11
Proposition 3.2. Let (ρ )>0 ⊂ L1loc (RN ) be a family of non-negative radial functions
in RN that satisfies (1.7) and
ˆ
ρ (h)
(3.12)
lim
dh = 0,
→0 |h|>δ |h|
for every δ > 0. Then, for every u ∈ SBV (RN ) and every bounded continuous function
ϕ : RN → R, we have
ˆ
ˆ
S u ϕ = K1,N
ϕ d|Ds u|.
lim
→0
RN
RN
Proof. The function S u is still well-defined in this case since u ∈ L1 (RN ). Given
R > 0, let
ˆ
u(x + h) − u(x)
S,R u(x) = ρ (h)χBR (0) (h) dh.
|h|
RN
By the triangle inequality, for every x ∈ RN we have
ˆ
|u(x + h) − u(x)|
|S u(x) − S,R u(x)| ≤
ρ (h) dh.
|h|
N
R \BR (0)
By Fubini’s theorem, we thus get
ˆ
kS u − S,R ukL1 (RN ) ≤ 2kukL1 (RN )
RN \BR (0)
ρ (h)
dh.
|h|
In particular, for every bounded continuous function ϕ : RN → R, we have
ˆ
ˆ
ˆ
ρ (h)
∞
N
1
N
≤
2kϕk
kuk
dh.
S
u
ϕ
−
S
u
ϕ
,R
L (R )
L (R )
N |h|
N
N
R
R \BR (0)
R
Letting tend to zero, the conclusion follows from formula (3.11) applied to the
family (ρ χBR (0) )>0 .
Proof of Corollary 1.2. It suffices to apply Proposition 3.2 to
ρ (h) =
1
,
HN −1 (SN −1 ) |h|N −
and then take = 1 − α. Since c1 C1,N = 2/π, we have the conclusion.
4. A COMPUTATION INVOLVING THE C ANTOR FUNCTION
The goal of this section is to motivate why one should expect a non-trivial contribution from the Cantor part of the derivative of a BV function for the functional
in Theorem 1.1. We focus our attention on the standard Cantor function: this is the
unique fixed point of the contraction map T : X → X defined by


if 0 ≤ x ≤ 1/3,
v(3x)/2
T (v)(x) = 1/2
if 1/3 < x < 2/3,


(v(3x − 2) + 1)/2 if 2/3 ≤ x ≤ 1,
where X is the complete metric space formed by all continuous functions v :
[0, 1] → R such that v(0) = 0 and v(1) = 1, equipped with the uniform distance.
Since the Cantor function is non-decreasing, it belongs to BV (0, 1).
Proposition 4.1. If u : [0, 1] → R is the standard Cantor function, then, for every
0 < < 1/6 and every non-negative even function ρ ∈ L1 (R) with supp ρ ⊂ [−, ],
we have
ˆ 1 ˆ 1
ˆ u(x) − u(y)
1
ρ
(x
−
y)
dy
dx
≥
ρ .
|x − y|
36 3/4
0
0
12
AUGUSTO C. PONCE AND DANIEL SPECTOR
In the case where ρ = χ(−,) /2, the right-hand side is constant and positive,
and we deduce that
ˆ 1 ˆ 1
u(x) − u(y)
dy ≥ 1 .
lim inf
ρ
(x
−
y)
dx
→0
|x
−
y|
36 · 8
0
0
We begin with the following identity based on Fubini’s theorem:
Lemma 4.2. Let u ∈ BV (0, 1)∩C 0 [0, 1] be a non-decreasing function, and let ρ ∈ L1 (R)
be a non-negative even function with compact support. For every measurable subset A ⊂
(0, 1) such that A − supp ρ ⊂ (0, 1), we have
ˆ ˆ 1
ˆ 1 ˆ (z − h, z + h) ∩ A
|u(x) − u(y)|
ρ(x−y) dy dx =
2ρ(h) dh d|Du|(z).
|x − y|
2h
0
A
0
R+
Proof. We first extend u as a continuous function in R, which we still denote by
u, by taking u(x) = u(1) for x > 1 and u(x) = u(0) for x < 0. In particular, the
measure Du is supported on [0, 1]. Since A − supp ρ ⊂ (0, 1), we have
ˆ ˆ 1
ˆ ˆ
|u(x) − u(y)|
|u(x) − u(y)|
ρ(x − y) dy dx =
ρ(x − y) dy dx.
|x − y|
|x − y|
0
R
A
A
Denote this common quantity by I(u). Making the change of variable y = x + h
with respect to y, we get
ˆ ˆ
|u(x + h) − u(x)|
ρ(h) dh dx.
I(u) =
|h|
A
R
Since u is continuous and non-decreasing, the measure Du does not contain atoms
and is non-negative. For every h > 0, we then have
ˆ x+h
|u(x + h) − u(x)| = u(x + h) − u(x) =
dDu = |Du|(x, x + h).
x
A similar computation holds for h < 0. Since ρ is even, we get
ˆ ˆ
ˆ ˆ
|Du|(x − h, x + h)
|Du|(x − h, x + h)
I(u) =
ρ(h) dh dx =
dx ρ(h) dh.
h
h
A
R+
R+
A
We apply again Fubini’s theorem to reach the conclusion. Indeed, for every h > 0
we have
ˆ
ˆ ˆ
|Du|(x − h, x + h) dx =
d|Du|(z) dx
A
A
x−h<z<x+h
ˆ
ˆ
=
dx d|Du|(z)
A+(−h,h)
x∈A, z−h<x<z+h
ˆ
(z − h, z + h) ∩ A d|Du|(z).
=
A+(−h,h)
Note that for z 6∈ A + (−h, h), the set (z − h, z + h) ∩ A is empty. We thus get
ˆ
ˆ
ˆ 1
(z−h, z+h)∩A d|Du|(z).
|Du|(x−h, x+h) dx = (z−h, z+h)∩A d|Du|(z) =
A
R
The proof of the lemma is complete.
0
Proof of Proposition 4.1. Given 0 < < 1/6, let n ∈ N∗ be the greatest integer such
that 2 ≤ 1/3n . Taking An to be the union of the constant sections of the Cantor
ON FORMULAE DECOUPLING THE TOTAL VARIATION OF BV FUNCTIONS
13
function of length 1/3n , we have that
ˆ 1 ˆ 1
ˆ ˆ 1
u(x) − u(y)
u(x) − u(y)
ρ (x − y) dy dx ≥
ρ (x − y) dy dx
|x − y|
|x − y|
0
0
An 0
ˆ ˆ 1
|u(x) − u(y)|
=
ρ (x − y) dy dx.
|x − y|
0
An
Indeed, the properties supp ρ ⊂ [−, ] and 2 ≤ 1/3n ensure that, for each x ∈ An ,
the function
u(x) − u(y)
y ∈ (0, 1) 7−→
ρ (x − y)
|x − y|
has constant sign. It thus follows from the previous lemma that
(4.1)
ˆ 1 ˆ ˆ 1ˆ 1
(z − h, z + h) ∩ An u(x) − u(y)
ρ (x−y) dy dx ≥
ρ (h) dh d|Du|(z).
|x − y|
h
0
0
R+
0
In the construction of the Cantor set, let In be the collection of closed intervals
of length 1/3n that contain the Cantor set. For example, I1 = {[0, 1/3], [2/3, 1]}.
Let Jn+2 be the subcollection of intervals in In+2 that intersect the closure of an
interval of length 1/3n that is removed during the construction of the Cantor set. For
example, J3 = {[1/3 − 1/27, 1/3], [2/3, 2/3 + 1/27]}, and the number of elements
of Jn+2 is 2n .
Given J ∈ Jn+2 and z ∈ J, we have that the set (z − h, z + h) ∩ An is non-empty
for every h > 1/3n+2 . In particular, for every 1/3n+2 ≤ h ≤ 1/3n we have
(z − h, z + h) ∩ An h − 1/3n+2
1/3n+2
≥
=1−
.
h
h
h
Since 1/3n+1 < 2 ≤ 1/3n , for every 3/4 ≤ h ≤ we deduce that
(z − h, z + h) ∩ An 2/3
1
≥1−
= .
h
3/4
9
Hence, for every z ∈ J we have
ˆ ˆ ˆ
(z − h, z + h) ∩ An (z − h, z + h) ∩ An 1 ρ (h) dh ≥
ρ (h) dh ≥
ρ .
h
h
9 3/4
3/4
R+
Writing J = [ai , bi ] ∈ Jn+2 , we also have
|Du|(J) = u(bi ) − u(ai ) =
We then deduce that
ˆ ˆ (z − h, z + h) ∩ An J
h
R+
1
.
2n+2
ρ (h) dh d|Du|(z) ≥
1
2n+2
·
1
9
ˆ
ρ .
3/4
Since the family Jn+2 has 2n elements, using estimate (4.1) we get
ˆ 1 ˆ
0
0
≥
1
u(x) − u(y)
ρ (x − y) dy dx
|x − y|
X ˆ ˆ (z − h, z + h) ∩ An J∈Jn+2
J
R+
This gives the conclusion.
h
ˆ 1
ρ (h) dh d|Du|(z) ≥
ρ .
36 3/4
14
AUGUSTO C. PONCE AND DANIEL SPECTOR
5. C ONNECTION WITH A BMO- TYPE NORM
Inspired from [8], in the paper [1, Section 4.3] the authors associate to u ∈
L1loc (RN ) a semi-norm
X
dx,
u(x) −
κ (u) = N −1 sup
u(y)
dy
Q
Q0 ∈Q
Q0
Q0
where the supremum is taken over all families Q of disjoint cubes with side
length , with arbitrary orientation and cardinality. In particular, for u = χA ,
where A ⊂ RN is a subset of finite perimeter, they show that one has the convergence [1, Eq. (4.4)]
lim κ (χA ) =
→0
1
|DχA |(RN ).
2
This result has been extended by Fusco, Moscariello and Sbordone [18, Theorem 3.3] to u ∈ SBV (RN ) such that either ∇u ≡ 0 or |J u | = 0, where they prove
that for such u one has
ˆ
1
1
|∇u| + |Ds u|(RN ).
lim κ (u) =
→0
4 RN
2
It is not difficult to see that any limit of the family (κ (u))>0 is comparable to
the total variation of Du:
Proposition 5.1. For every u ∈ BV (RN ), we have
1
1
|Du|(RN ) ≤ lim inf κ (u) ≤ lim sup κ (u) ≤ |Du|(RN ).
→0
4
2
→0
Proof. For the upper bound one can apply the Poincaré inequality for functions of
bounded variation to obtain
X 1
X
1
dx ≤
u(x) −
N −1
|Du|(Q0 ) ≤ |Du|(RN ).
u(y)
dy
2
2
0
0
Q
Q
Q0 ∈Q
Q0 ∈Q
For the lower bound, mollification lowers the energy following an observation by
E. Stein, as for the functionals introduced by Bourgain, Brezis and Mironescu [7];
cf. [10]. Indeed, letting uδ = u ∗ ψδ , by Fubini’s theorem one has
ˆ
dx dz,
uδ (x) −
u
(y)
dy
dx
≤
ψ
(z)
u(x)
−
u(y)
dy
δ
δ
Q0
Q0
RN
and this implies that
X
Q0 ∈Q
N −1
uδ (x) −
0
Q
Q0
Q0 −z
Q0 −z
ˆ
uδ (y) dy dx ≤ κ (u)
ψδ (z) dz = κ (u).
RN
Hence,
κ (uδ ) ≤ κ (u),
so that first applying the result of Fusco, Moscariello and Sbordone [18, Theorem 3.3] to the W 1,1 (RN ) (smooth) function uδ and then using the lower semicontinuity of the total variation with respect to the strict convergence of the family
(uδ )δ>0 in BV (RN ), one deduces the lower bound
1
|Du|(RN ) ≤ lim inf κ (u).
→0
4
ON FORMULAE DECOUPLING THE TOTAL VARIATION OF BV FUNCTIONS
15
We can also show that, up to an affine change in , the energies κ are comparable at every scale to a special case of those originally introduced by Bourgain,
Brezis and Mironescu [7]. To this end, we first introduce an equivalent energy to
κ , the functional
X
κ0 (u) = N −1 sup
|u(x) − u(y)| dy dx.
Q
Q0 ∈Q
Q0
Q0
As is the case with semi-norms of functions with bounded mean oscillation (BMO) [13],
one has
κ (u) ≤ κ0 (u) ≤ 2κ (u),
and so it suffices to establish the equivalence of κ0 . We now make the choice of the
family of mollifiers (ρ )>0 ,
(5.1)
ρ (h) =
CN
|h|χB (0) (h),
N +1
which gives
(5.2)
ˆ ˆ
ˆ |u(x) − u(y)|
CN ωN
ρ (x − y) dy dx =
|x − y|
RN RN
RN
|u(x) − u(y)| dy dx.
B (x)
Then, the equivalence we assert is
Proposition 5.2. Let u ∈ L1loc (RN ). For every > 0, we have
ˆ 1
0 0
|u(x) − u(y)| dy dx ≤ C 00 κ03 (u),
C κ √ (u) ≤
N
RN
B (x)
for some constants C 0 , C 00 > 0 depending on N .
Proof. Given a cube Q0 with side length √N , for every x ∈ Q0 we have that Q0 ⊂
B (x). Hence,
ˆ ˆ
ˆ ˆ
|u(x) − u(y)| dy dx ≤
|u(x) − u(y)| dy dx.
Q0
Q0
Q0
It thus follows that
N −1 X
√
N
Q0 ∈Q √
B (x)
|u(x) − u(y)| dy dx ≤
Q0
Q0
C1
N
≤
C1
ˆ
X
Q0
Q0 ∈Q √
|u(x) − u(y)| dy dx
B (x)
N
ˆ
RN
|u(x) − u(y)| dy dx.
B (x)
Taking the supremum with respect to all families of cubes Q √ , we deduce the
N
first inequality.
To prove the second inequality, write RN as a union of disjoint cubes Q3 with
side length 3. Divide each cube Q0 ∈ Q3 in 3N cubes with disjoint interiors and
side length . Denote by Q00 the inner cube; this is the only cube that does not
intersect ∂Q0 . For every x ∈ Q00 , we have B (x) ⊂ Q0 . Hence,
ˆ ˆ
ˆ ˆ
|u(x) − u(y)| dy dx ≤
|u(x) − u(y)| dy dx,
Q00
which implies that
ˆ 1 X
Q0 ∈Q
3
Q00
Q0
B (x)
B (x)
Q0
|u(x) − u(y)| dy dx ≤ C2 (3)N −1
X
Q0 ∈Q3
Q0
Q0
|u(x) − u(y)| dy dx ≤ C2 κ03 (u).
16
AUGUSTO C. PONCE AND DANIEL SPECTOR
Proceeding by suitable translations of the family Q3 , we can make sure that each
of the subcubes of Q0 is the inner cube of some cube in a translated family. Summing over the 3N translated families of cubes, by additivity of the integral we then
get
ˆ 1
|u(x) − u(y)| dy dx ≤ 3N C2 κ03 (u).
RN
B (x)
For the same choice of mollifiers (5.1) in the functional of Theorem 1.1, one finds
ˆ ˆ
ˆ CN ωN
u(x) − u(y)
dx.
(5.3)
dx
=
ρ
(x−y)
dy
u(x)−
u(y)
dy
|x − y|
RN RN
RN
B (x)
Now, according to our results the energy (5.3) cannot be comparable to κ , κ0 or (5.2)
on all scales for arbitrary BV functions. Indeed, Theorem 1.1 implies that (5.3)
tends to zero for u ∈ W 1,1 (RN ) while κ , κ0 and (5.2) tend to a constant times the
total variation of the weak derivative ∇u as tends to zero. They are nevertheless
all comparable for BV functions of the form u = χA in view of the identity
ˆ ˆ
ˆ ˆ
|χA (x) − χA (y)|
χA (x) − χA (y)
dx
=
ρ
(x−y)
dy
ρ (x−y) dy dx.
N
|x − y|
|x − y|
N
N
N
R
R
R
R
A CKNOWLEDGEMENTS
The authors would like to thank the referee for his or her detailed reading
and comments that have improved the quality of the presentation of the paper.
This work was initiated while the first author (ACP) was visiting the Department
of Applied Mathematics of the National Chiao Tung University (Research Grant
104W986). He warmly thanks the Department for the invitation and hospitality;
he has also been supported by the Fonds de la Recherche scientifique–FNRS under
research grant J.0026.15. The second author (DS) is supported by the Taiwan Ministry of Science and Technology under research grants 103-2115-M-009-016-MY2
and 105-2115-M-009-004-MY2.
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A UGUSTO C. P ONCE
U NIVERSITÉ CATHOLIQUE DE L OUVAIN
I NSTITUT DE R ECHERCHE EN M ATHÉMATIQUE ET P HYSIQUE
C HEMIN DU CYCLOTRON 2, L7.01.02
1348 L OUVAIN - LA -N EUVE
B ELGIUM
E-mail address: [email protected]
D ANIEL S PECTOR
N ATIONAL C HIAO T UNG U NIVERSITY
D EPARTMENT OF A PPLIED M ATHEMATICS
H SINCHU , TAIWAN
E-mail address: [email protected]