Dipartimento di Matematica
Corso di Laurea in Matematica
Tesi di Laurea Magistrale
Approximation in SBV spaces
Candidato:
Relatore:
Giuseppe Giorgio Colabufo
Prof. Aldo Pratelli
Controrelatore:
Prof. Giuseppe Buttazzo
Contents
Introduction 1
Special functions of bounded variation . . . 1
Approximation results in SBV . . . 3
Comparison of the principal results . . . 3
The idea of the proofs . . . 4
Structure of this document . . . 6
1 Preliminaries 8 1.1 Basic terminology and notation . . . 8
1.2 Functions of bounded variation . . . 9
1.3 “Classic” theorems . . . 10 1.4 Compactness in SBVp (Ω, Rm) . . . . 11 2 Strong approximation in SBVp (Ω, Rm) 15 3 Approximation in SBVp(Ω) 21 3.1 Preliminary lemmas . . . 21 3.1.1 Approximation in SBV (Ω) . . . . 22
3.2 The refined approximation theorem for SBVp(Ω) . . . . 28
4 Polyhedral jump sets 37 4.1 Preliminaries . . . 37
4.2 The approximation theorem . . . 43
5 Polyhedral partitions 52
Introduction
The main goal of this master thesis is to compare different approximation results for SBV functions. The document is meant to be a review of the foremost results available to date on the subject.
Special functions of bounded variation
Special functions of bounded variation (SBV ) constitute a subclass of func-tions of bounded variation (BV ) which have been applied by several authors to study Fourier series in several variables, geometric measure theory, calculus of variations, and mathematical physics. This special class was introduced by De Giorgi and Ambrosio in [DGA88] to deal with “free discontinuity” variational problems. Free discontinuity problems are model for several applied questions involving a “free discontinuity set” (i.e. problems in which the discontinuity set is an unknown) such as image segmentation, fracture mechanics and liquid crystal theory which are a wide class of variational problems. A free disconti-nuity problem is characterised by a competition between volume energies and surface energies. The idea of De Giorgi and Ambrosio introducing this special functions was to replace the free discontinuity set mentioned above by the set of “discontinuity points” of these SBV functions, which turns out to be “suffi-ciently regular” so that weak notions of surface area, orientation and traces can be given.
The class of SBV functions is also closely related to sets of finite perimeter (Caccioppoli [Cac52] and De Giorgi [DG54, DG55] defined measure of non-smooth boundaries of sets through BV functions). See [AFP00, Sections 3 and 4.6] for an overview.
In the aforementioned paper [DGA88] the authors focus on the functional
F (u) = ˆ Ω f (x, u, ∇u)dx + ˆ Ju ϕ(x, u+, u−, ν)dHN −1 (1) where Ω ⊆ RN is an open set, HN −1 is the (N − 1)-dimensional Hausdorff
measure, f and ϕ are Borel functions that satisfy some growth and symmetry conditions and u+, u−are the traces of the special function of bounded variation
u on both sides of thejump set Ju. This integral functional was inspired by the
static theory of liquid crystals, where the first integral represents the sum of the internal energies of the fluids and the second integral represents the interface energy of the fluids in the regions of mutual contact.
Since then, a general theory for the study of this new class of variational problems characterized by the minimization of volume and surface energies has
been developed. Problems of this kind are also suggested by some variational models in fracture mechanics, thanks to the strong links with Griffith’s the-ory. Specifically, this framework of a variational approach has been used for building a rigorous theory of quasi-static evolution of cracks. The term “
frac-ture mechanics” refers to a specialization within solid mechanics in which the
presence of a crack is assumed. The objectives in the studies of this field of me-chanics concerned with the study of the propagation of cracks in materials are to find quantitative relations between the crack length, the material’s inherent resistance to crack growth, and the stress at which the crack propagates.
The energy-balance approach employed by Griffith in [Gri21] has become one of the most famous developments in materials science. With this approach, one considers the energy balance: the elastic energy plus the energy dissipated to extend the crack is equal to the work done by the external forces. In the variational model, the crack is a countably HN −1 rectifiable set Γ ⊆ Ω with
finite measure HN −1(Γ) < +∞. The energy necessary to the production of a
crack is supposed to be proportional to the crack surface. One also assumes that it can only grow over time (irreversibility of the crack) but its shape and location are not fixed (hence not known a priori). In this sense, this problem has to do with free discontinuity, as the unknown is a set where the solution may be discontinuous.
Quasi-static models of crack growth can be developed under different as-sumptions on the elastic response of the material and the mechanisms of crack formation. Recent studies aim to develop a model that predicts the crack path as well as the time evolution of the crack along its path, taking into account all inertial effects.
Free discontinuity problems, in general, are still a research theme for many mathematicians.
In view of applications to variational problems, it is important to have suit-able closure and compactness properties to be applied to minimising sequences. Necessary conditions for compactness requires uniform bounds on the approx-imate gradients and on the jumps to prevent the appearance of a Cantor part in the limit and a uniform bound on the L∞ norms of the sequence. Such a compactness theorem was proved for the first time by Ambrosio in [Amb89] fol-lowing an integral-geometric approach relying on the slicing theory, motivated by the discussion of the one-dimensional case. The proof was later simplified by the same author in [Amb95] and by Alberti and Mantegazza in [AM97], exploiting a global criterion for membership to SBV .
We will give more precise definitions in Section 1.2, but for now, just recall that a function u ∈ SBV (Ω) has derivative that can be decomposed into an absolutely continuous part ∇u with respect to the Lebesgue measure LN and a
singular part Dsu concentrated on an (N − 1)-dimensional set J
u. As already
underlined more than once, despite the elementary definition, this space is the natural for several applications. It is relevant to notice that SBV (Ω) is not a closed subspace of BV (Ω) in the strict topology, i.e. the topology induced by strict convergence, see Definition 1.2. For this reason, in many applications, one considers the space SBVp(Ω), for p > 1, as the subspace of SBV (Ω) in which
the Lp norm of the absolutely continuous part of the derivative, as well as the (N − 1)-dimensional Hausdorff measure of the set on which the singular part is concentrated, are finite. Aforementioned compactness theorem immediately extends to this new space and we state this result in Lemma 1.5 and provide a
sketch of its proof. As though SBV or SBVp setting are the natural ones in
many applications, the requirement that the measure of the jump set is finite may be too restrictive in some cases. For this reason, an intermediate space between SBV and SBVp is often taken into account. This space, denoted by
SBV∞p, is the space of SBV functions for which the higher Lp integrability of the absolutely continuous part of the derivative holds, but no constraint on the measure of the jump set is assumed.
Approximation results in SBV
Along with compactness, the crucial tools that one need in order to study in-tegral functionals such as the one in (1) are approximation results. In this document, we will present a few of them, that have different characteristics and their own pros and cons. The main differences consist of the demanded regular-ity of the approximating sequences and the requested structure of the jump set. With the purpose of making use of compactness statements, it is convenient to ask for the approximating sequence (uh)h∈N to be in the same space of u
(i.e. SBV, SBVp or SBVp
∞). But furthermore, one would like the
approxi-mations uh of u to be smooth in the largest possible domain, that is outside
their discontinuity set. Additional requirements can be made on their distri-butional derivatives so that the approximating functions are in some Sobolev space too. Lastly, knowing the shape of the jump sets could be interesting for many purposes. Some possible requirements can be asking for it to be “simple” or “smooth”, where “simple” and “smooth” usually stand for a polyhedral set and a C1 manifold respectively. The latter condition can be sometimes relaxed and replaced by HN −1-rectifiability (see Definition 2.1). The interest in having a polyhedral sets lies in the fact that this reduces problems involving surface energies to the case of a planar interface. Moreover, this simple structure can be easily modelled by computer software for numerical simulations and compu-tations.
Comparison of the principal results
In this section, we comment on the results of approximation that we will present in the next chapters, and we compare them with each other. First of all, let us consider a function u ∈ SBV (Ω): as already pointed out, we aim to write u as a limit in BV of SBV functions uh, each of them having a “nice” jump set Juh
and being smooth in the complement of it. This is the best approximation that one can hope to get. In building the approximation, we would like to guarantee that HN −1(J
uh \ Ju) → 0 so that H
N −1(J
uh∆Ju) → 0 (the convergence of
HN −1(J
u\ Juh) to 0 comes directly from the convergence in the BV norm when
Ju has finite measure). In principle, it would be possible for this quantity to
blow up, for instance, if the functions uh have a very large part of the jump
set where the jump |u+ − u−| is very small. This property is satisfied both
from Theorem 2.1 and Theorem 3.4 but, for example, not from Lemma 3.3. Indeed, in this latter result, we get an approximating sequence in the BV norm, whose jump sets are (contained in) a C1 manifold. However, the convergence
HN −1(J
uh \ Ju) → 0 is missing. Both the aforementioned theorems seems
no information about the possible shape of the jump sets of the functions uh,
except the fact that they are contained in a closed rectifiable set. That’s why the latter, ensuring that these jump sets are compact C1 manifolds, is a useful refinement, especially in cases in which knowing the structure of this jump set is important.
A similar comparison can be done with Theorems 4.2 and 5.1. In both of them the jump sets of the approximating functions are polyhedral (i.e., a finite union of (N − 1)-dimensional simplexes, see Definition 4.1). As we remarked before, this kind of structure is useful in applications but, in general, unrelated to the jump set of u. That’s why the strong BV convergence fails in these cases. We point out a difference between the different structures of the jump sets provided by these four theorems: notice that a polyhedral set is not a
C1 manifold since the different simplexes of which is made of might intersect
with each other and with the boundary ∂Ω. On the contrary, being a C1
manifold means, in particular, that Juh is the finite and disjoint union of C
1
images of (N −1)-dimensional simplexes. Lemma 3.2 allows modifying a function
u ∈ SBVp(Ω) ∩ C1(Ω \ J
u) ∩ W1,∞(Ω \ Ju) to transform its polyhedral jump
set into a C1 manifold.
Another difference between Theorems 2.1, 4.2 and Theorems 3.4, 5.1 is that the first two assume u to be in L∞(Ω). This is not restrictive for a function in SBV (Ω) whose approximate gradient ∇u belongs to Lp(Ω) for some p ≥ 1, as observed in Lemma 3.1. Also, it can be proved that, if Ω is bounded,
SBV (Ω)∩L∞(Ω) is an algebra of functions, i.e. stable under sums and products.
Moreover, if u ∈ L∞(Ω) one can always assume that kuk∞ is an upper bound for kuhk∞ for every h ∈ N (see [DPFP17, Lemma 3.2]).
Finally, a peculiarity of Theorem 5.1 is that u is an SBVloc function from
Ω with values in a finite set Z ⊂ Rm. In this way, we can identify a subset of
SBVloc(Ω, Z) with the set of partition into m := #Z sets of finite perimeter:
if Z = {z1, . . . , zm}, we can associate to u the partition (E1, . . . , Em) given by
Ek = {x ∈ Ω | u(x) = zk} for k = 1, . . . , m. Doing this, we can interpret
Theorem 5.1 as follows: given a general partition (E1, . . . , Em) of Ω ⊆ RN into
m sets of finite perimeter, there exist partitions (E1h, . . . , Emh) of Ω into sets whose boundaries are polyhedral, such that limh→+∞LN(Ekh∆Ek) = 0 and the
Hausdorff measure of the reduced boundary HN −1(∂∗Ekh) → HN −1(∂∗Ek) as
h → +∞ for all k ∈ {1, . . . , m}. This partitions are shortly called polyhedral
approximations of (E1, . . . , Em). For convenience of the reader we recall that
the reduced boundary ∂∗E of a Borel set E ⊂ RN is the collection of all points
x such that the limit
νE(x) := lim ρ↓0 D1E(Bρ(x)) |D1E|(Bρ(x)) ∈ RN exists and |νE(x)| = 1.
The idea of the proofs
We briefly report in this section the ideas used in the proofs of Theorems 2.1, 3.4, 4.2 and 5.1. These proofs are indeed quite long and technical so that it is worthy and useful to have an overall prospect of how they work.
Historically, Theorem 2.1 is one of the earliest results that permit to approx-imate a bounded function u ∈ SBVp(Ω) with a “nice” sequence. It exploits the compactness result of Lemma 1.5 and its original proof. In particular, we reduce to the one-dimensional case and take advantage of the slicing theory for BV and
SBV functions (see for instance [AFP00, Section 3.11] for an overview).
Basi-cally, we construct a sequence of auxiliary minimum problems whose minimizers will be the desired approximating functions for the given u. The existence of each minimizer is guaranteed by compactness and semicontinuity results applied to these auxiliary problems.
Theorem 4.2 is a “classical” result that ensures the polyhedral structure for the jump set of the approximating sequence of functions. First of all, we notice that is enough to prove the limsup condition (4.10)1 assuming that ϕ
is continuous, since every upper semicontinuous function is approximated from above by a decreasing sequence of continuous functions. The proof relies then essentially on two operations: the approximation of u with functions uh whose
jump set is more regular than that of u and the modification of each function uh
around its jump set in order to transform it into a polyhedral set. We perform the preliminary approximation of u using Theorem 2.1 and then work with the sequence (uh)h∈N obtained in this way. We will proceed with the rectification
of the jump set of each uh, modifying uh in a small set around Juh, so that
the old jump set is replaced by a new polyhedral one and without changing too much the L1 norm of the function and the Lp norm of its gradient. In
order to simplify the exposition, we do not explicit the dependence on h in the rest of the paragraph. The rectification construction is based on the covering results given in Lemma 4.1. By construction, the jump set of u is contained in a rectifiable set K, and this lemma allows to find two finite families of cubes, covering, respectively, K ∩ Ju and K \ Ju. On the cubes of the first family, we
carefully perform a reflection around suitable hyperplanes to remove a big part of the jump set of u, which is replaced by (N − 1)-dimensional rectangles. We also make sure that the values of the traces (on the new jump set) remain almost equal to the old ones. On the second family of cubes, we just set the value of u to be 0. The new jump set obtained in this way is almost all polyhedral, the only bad part being eventually given by parts of K \Juwhich may have been reflected
in the previous operations. With a little work, we prove that this bad part is contained in a compact set of small HN −1measure and that we can cover it with a finite number of small cubes on which we change again the value of u to 0. The function v obtained in this way is still not the desired one because, in general, it is not “piecewise smooth” as claimed; its jump set Jv is nevertheless contained
in a polyhedral set. To achieve the smoothness requirement, we regularize it by means of convolutions. This process will introduce an error on condition (4.10) that can be estimated essentially by the HN −1measure of K \ J
u. However, as
a byproduct of Theorem 2.1, this quantity can be chosen to be arbitrarily small, thus concluding the proof.
1That is lim sup h→+∞ ˆ A∩Jwh ϕ(x, w+h, w−h, νwh)dH N −1≤ˆ A∩Ju ϕ(x, u+, u−, νu)HN −1,
for every Ab Ω and every upper semicontinuous function ϕ : Ω × Rm× Rm× SN −1→ [0, +∞)
The proof of Theorem 3.4 takes advantage - among other things - of the out-comes of Theorem 4.2 and improves Theorem 2.1 in the sense described above. Three cases are considered separately: first when u is compactly supported, secondly when its jump set is compactly contained in the domain, Jub Ω, and
finally the general case that can be lead back to the previous ones by means of the partitions of unity and suitable multiplications by smooth cut-off functions that permit to separate the jump set from the boundary ∂Ω. Thus, we are left to prove the first case. The starting point is the approximation result of Lemma 3.3 that puts together two different statements: one for SBV and one for SBVp
∞.
This allows us to obtain a first function u1. The function u1 is “close” to u in
the BV norm but we do not have any meaningful information on its gradient and jump set. Also, we make use of a diffeomorphism Φ which coincides with the identity up to a compact set and that maps, roughly speaking, the jump set into a disjoint union of cubes. We can then modify u1 around the jump set, by
means of the Theorem 4.2 applied to the difference u ◦ Φ−1− u1◦ Φ−1, so that
the jump set of this latter is now polyhedral. Unfortunately, we do not have any a priori estimate on the Lp norm of such modification. This composition with the inverse of Φ let us work in a domain where the “bad set” is covered by a finite number of rectangles. By composition with a suitable affine map, we can keep this function small in BV norm and with polyhedral jump set. Adding this last modification to u1◦ Φ−1 and then smoothing the sum, again through
Theorem 4.2, leads to the desired approximation after a final composition with Φ.
The last statement that we will discuss is Theorem 5.1. Its proof relies on a deformation argument allowed by the rectifiability of the jump set Ju.
The first steps aim to cover most of the jump set of u by disjoint balls, such that in each of them the jump set becomes a C1 graph which is almost flat.
Then the jump set can then be explicitly deformed into a hyperplane in each of these balls. In turn, these portions of hyperplanes can be approximated by (N − 1)-dimensional polyhedral sets. We use special grids to construct a system of N -dimensional polyhedra whose boundaries contains the previous (N − 1)-dimensional ones and in such a way that these bigger polyhedra constitute a decomposition of the ambient space. This construction will permit to build a piecewise-constant SBV function with polyhedral jump set whose gradient is close to the polyhedral measure built up in the aforementioned balls. This will turn out to be the intended approximation of u.
Structure of this document
The thesis is divided into five chapters, organised as follows.
Chapter 1 contains preliminary notions: it briefly fixes notation and recalls the definitions and primary properties of (special) functions of bounded varia-tion. It also includes a list of classical theorems in geometric measure theory that will be used in the proofs of the subsequent chapters.
Chapter 2 reports a first approximation result for SBVpfunctions by Braides
and Chiadò Piat in [BP96] stated as Theorem 2.1.
obtained by De Philippis, Fusco, and Pratelli in [DPFP17], preceded by a couple of prerequisite lemmas, included an approximation statement in SBV (Ω.
Finally, Chapter 4 and Chapter 5 comprehend two approximation results which guarantee a polyhedral structure for the jump sets of the approximating sequence. These were obtained by Cortesani and Toader in [CT99] and Braides, Conti and Garroni in [BCG17] respectively.
Chapter 1
Preliminaries
Before broaching the leading result from Chapter 2 on, we fix and inform the reader on the notation used throughout the document.
1.1
Basic terminology and notation
Warning: terms like positive, negative, increasing, decreasing are always
under-stood in their wide sense, for instance, r positive means r ≥ 0; we use “strictly” for the restricted sense, for instance, “r strictly positive” means r > 0.
Symbols for set-theoretic operations (membership ∈, union ∪, intersection ∩, inclusion ⊂ and difference \) are the usual ones. We use ∆ to denote the symmetric difference of sets, i.e. A∆B = (A \ B) ∪ (B \ A). A line over the set denotes its topological closure E, while ∂E will be the topological boundary of
E. We will use the special symbol b for compact closure sets, i.e. E b F if E ⊂ F is compact.
Open balls with centre x and radius r in metric spaces will be denoted by
Br(x) or with B(x, r). Similarly, Qr(x) will denote an open cube centred at x
and side length 2r.
N, Z, Q, R denotes natural, integer, rational and real numbers respectively, while R := R ∪ {−∞, +∞}. We will use a ∧ b and a ∨ b for the minimum and maximum of a and b respectively.
The ambient space will always be the euclidean N -dimensional space RN while the target space Rm
, for N, m ∈ N fixed, unless specified differently. SN −1⊂ RN denotes the unit N − 1-dimensional sphere.
We usually denote with p, p0 ∈ [1, +∞] Lebesgue exponents such that 1
p +
1
p0 = 1.
A bar on the integral sign fflXf dµ := [µ(X)]−1´
Xf dµ stays for the mean
value of f on X.
Throughout the document B(X) is the σ-algebra of Borel subsets of a topo-logical space X and A(X) is the family of all the open sets of X.
The Lebesgue measure of A ∈ B(X) in RN will be LN(A) or sometimes
|A|, whereas the k-dimensional Hausdorff measure Hk(A). For a general set E,
we define #(∅) = 0, #E as the cardinality of E if it is finite and #E = +∞ otherwise.
Sometimes we writeusc (resp. lsc) functions as shorthands for upper (resp.
lower) semi-continuous functions.
1.2
Functions of bounded variation
Definition 1.1. Given an open set Ω ⊆ RN, the space of the functions of
bounded variation is given by the set BV (Ω, Rm)of all the Rm-valued L1
func-tions over Ω whose distributional derivative Du is a finite Radon measure in Ω, that is, Du = (Diuj)j=1,...,mi=1,...,N is a m × N matrix of measures in Ω such that
ˆ Ω uj∂φ ∂xi dx = ˆ Ω φ dDiuj ∀φ ∈ Cc∞(Ω) ∀i = 1, . . . , N ∀j = 1, . . . , m
We call ∇u the absolutely continuous part with respect to the Lebesgue measure LN of Du, the approximate gradient. Hence, Du can be expressed
through the equality of measures
Du = ∇uLN Ω + Dsu = ∇uLN Ω + Dju + Dcu
= ∇uLN Ω + (u+− u−)νuHN −1 Ju+ Dcu
where Dsu is the singular part (with respect to the Lebesgue measure), given
by the sum of thejump part Dju and theCantor part Dcu of Du. The measure Dju is concentrated on an (N − 1)-dimensional set J
u, which is called thejump
set, and which is countably rectifiable. The vector νu = νu(x) ∈ SN −1 is the
measure theoretical normal to Ju, while u+= u+(x), u−= u−(x) are the traces
of u on both sides of Ju. The traces and the normal vector exist for every x ∈ Ju
and their are characterised by lim
ρ→0 B+
νu(x,r)
|u(y) − u+|dy = lim
ρ→0 B−
νu(x,r)
|u(y) − u−|dy = 0 where we denoted by Bν∓(x, r) the two half-balls
Bν∓
u(x, r) :=y ∈ R
N | |y − x| < r, (y − x) · ν u≶ 0
In particular, the strictly positive quantity |u+(x) − u−(x)| is calledjump.
Remark: The space of the functions of bounded variations contains the Sobolev space W1,1(Ω) properly. Indeed, u ∈ BV (Ω) belongs to W1,1(Ω) if and only if
Dsu = 0.
The space BV (Ω, Rm), endowed with the norm kukBV (Ω):=
ˆ
Ω
|u(x)|dx + |Du|(Ω)
where |Du| is the total variation of the measure, is a Banach space.
We recall the two definitions of weak* convergence and strict convergence in
Definition 1.2. Let u, (uh)h∈N⊂ BV (Ω, Rm). We say that (uh)h weakly*
con-verges in to u if (uh)h converges to u in L1(Ω, Rm) and (Duh)h weakly*
con-verges to Du in Ω, i.e. lim h→+∞ ˆ Ω φDuh= ˆ Ω φDu ∀φ ∈ C0(Ω).
We say that (uh)h strictly converges to u if (uh)h converges to u in L1(Ω, Rm)
and the variations |Duh|(Ω)converge to |Du|(Ω) as h → +∞:
kuh− ukL1(Ω)+ ||Duh|(Ω) − |Du|(Ω)| → 0.
Remark: Weak* convergence is equivalent to require that the sequence (uh)h
is bounded and converging to u in L1
(Ω, Rm).
Definition 1.3. The space SBV (Ω, Rm)of Rm-valued special functions of bounded
variation on the open set Ω is the subset of BV (Ω, Rm) of all integrable
func-tions whose distributional derivative Du has vanishing Cantor part. Thus for u ∈ SBV (Ω, Rm)holds Du = ∇uLN Ω + Dju.
Remark: The space SBV (Ω, Rm) is a proper subspace of BV (Ω, Rm) closed in the strong norm topology but which is not closed in the topology induced by strict convergence. The Sobolev space W1,1(Ω) is strictly included in SBV (Ω)
and we can characterise the belonging to the subspace by
u ∈ W1,1(Ω) ⇐⇒ HN −1(Ju) = 0.
Classical closure and compactness result for SBV (Ω) such as [AFP00, The-orems 4.7 and 4.8] holds for weak* topology.
As announced in the introduction, in many applications, one considers the space SBVp(Ω, Rm), where p > 1 is a real number. This space is defined as the space of SBV functions u for which ∇u belongs to Lp and the jump set
Ju has finite HN −1 measure. This is equivalent to require that the quantity
kukBV (Ω)+ k∇ukLp(Ω)+ HN −1(Ju) < ∞ is finite.
Finally, we denote SBVp
∞(Ω) the space of SBV functions u for which ∇u ∈
Lp but without any constraint on the measure of the jump set is required. This
is an intermediate space between SBV and SBVp.
1.3
“Classic” theorems
For the convenience of the reader, we list in this section the main theorems of geometric measure theory that we regularly use in proving the main results in this document. For their proofs see [AFP00], [EG15] and [Fal86].
Theorem 1.1 (Lusin). Let X be a locally compact and separable metric space
and µ a Borel measure on X. Let u: X → R be a µ-measurable function van-ishing outside of a set with finite measure. Then. for any ε > 0 there exists a continuous function v : X → R such that
kvk∞≤ kuk∞ and µ ({x ∈ X | v(x) 6= u(x)}) < ε.
Theorem 1.2 (Egorov). Assume that µ is a finite measure and that a sequence
of µ-measurable functions (fh)converges to f µ − a.e. in X; then, for any ε > 0
there exists a measurable set Xε such that µ(X \ Xε) < εand fh converges to f
uniformly in Xε.
Theorem 1.3 (Vitali-Besicovitch covering). Let A ⊂ RN be a bounded
Borel set, and let F be a family of closed balls that covers A with the property that for every x ∈ A there exist balls in F centred at x and with arbitrarily small radii. Then, for every positive Radon measure in RN there is a disjoint family
F0 ⊂ F such that
µA \[F0= 0.
Remark: The theorem continues to hold, if the balls making up the Besicovitch covering F are replaced by cubes with faces parallel to the coordinate planes. This also holds for the forthcoming theorem. The present remark is crucial in seen of the proof of theorem 4.2 of chapter 4.
Theorem 1.4. [EG15, Theorem 1.28] Let µ be a Borel measure on RN, and F
any collection of nondegenerate closed balls. Let A denote the set of centers of the balls in F. Assume µ(A) < ∞ and inf{r | B(a, r) ∈ F} = 0 for each a ∈ A. Then for each open set U ⊆ RN, there exists a countable collection G of disjoint
balls in F such that
[ B∈G B ⊆ U and µ (A ∩ U ) \ [ B∈G B ! = 0.
1.4
Compactness in SBV
p(Ω, R
m)
The notion of strong convergence in the space SBVp(Ω, Rm) essentially
con-sists in approximating the functions in L1, their approximate gradients in Lp
and their approximate jump sets in Hausdorff measure. Precisely, we have the following definition.
strongly converges to u ∈ SBVp
(Ω, Rm)if the following happens:
uh→ uin L1(Ω, Rm) ∇uh→ ∇uin Lp(Ω, RN ×m) HN −1(J uh∆Ju) → 0 ˆ Juh∪Ju |u+ h − u +| + |u− h − u −|dHN −1→ 0
where the last term makes sense if we choose the same orientation νuh(x) =
νu(x) for HN −1− a.e. x ∈ Juh∩ Ju.
Remark: Notice that we also have convergence in the strong norm topology: kuh− ukBV (Ω,Rm)→ 0.
Indeed the second and the last condition together allow us to control the ab-solutely continuous part and the jump part of the derivatives. The conclusion follows taking into account the L1convergence. This will be used in the follow-ing (for instance in the statement of Theorem 2.1).
Changing some of the previous requirements yields the definition of weak
convergence in SBVp
(Ω, Rm).
Definition 1.5. Let p > 1. We say that a sequence (uh)h∈N ⊂ SBVp(Ω, Rm)
weakly converges to u ∈ SBVp
(Ω, Rm)if the following happens:
uh→ uin L1(Ω, Rm)
∇uh* ∇u in Lp(Ω, RN ×m)
sup
h∈N
|Duh|(Ω) < +∞
The primary result of next chapter is Theorem 2.1, which states that it is possible to find a sequence of functions in SBVp(Ω, Rm) converging to a desired function in the same space in the (strong) aforementioned sense of Definition 1.4. The central steps in the proof of this statement are based on the compactness result proved by [Amb89] and enunciated as Lemma 1.5.
Lemma 1.5. Let Ω be a bounded open subset of RN and let (uh)h∈Nbe a sequence
in SBVp (Ω, Rm)such that sup h∈N kuhkBV (Ω,Rm)< +∞ (1.1a) sup h∈N ˆ Ω |∇uh|pdx + HN −1(Juh∩ Ω) < +∞ (1.1b)
then there exists a sub-sequence (uhk)k∈N converging weakly in SBV
p
(Ω, Rm).
Moreover, if u is the limit of such a sub-sequence then
HN −1(J
u∩ Ω) ≤ lim inf
k→+∞H
N −1(J
Remark: Requiring that the approximate gradients are bounded in Lp(Ω) in (1.1b) is equivalent to equiintegrability (for instance via [AFP00, Proposi-tion 1.27]). As regard (1.2), we have only lower semicontinuity because there can be more than one jump point of uh that approximate a single jump point
of u or it might occur that the jump of uh vanishes in the limit.
Proof. Assume for simplicity that the sequence uh is uniformly bounded in
L∞. Let M be the least upper bound for kuhk∞ and consider the function
ϕ : [0, +∞) → [0, +∞] given by t 7→ ϕ(t) = tp
. We can find some α ∈ R and
β > 0 such that ϕ(t) ≥ t + α for every t ≥ 0 and βt ≤ 1 for t ∈ (0, 2M ]. Then
|Duh|(Ω) = ˆ Ω |∇uh|dx + ˆ Juh |u+ h − u − h|dH N −1 ≤ ˆ Ω |∇uh|pdx − αLN(Ω) + 1 βH N −1(J uh)
are equibounded thanks to the assumption (1.1b). By compactenss in BV (Ω, Rm)
and since Ω is bounded we can extract a sub-sequence uhk strongly converging
to u in L1(Ω) and because of equibounds we can infer that the limit u ∈ L∞
and |Du|(Ω) < +∞. In particular uhk * u weakly* in BV and by a closure
theorem for the SBV (Ω, Rm) ([AFP00, Theorem 4.7]) and the lower
semiconti-nuity of the Lp norm with respect to weak convergence, the limit u belongs to
SBVp(Ω, Rm). Finally, (1.2) follows from weakly* convergence of the measures HN −1 (J
uhk ∩ Ω) * µ ≥ HN −1 (Ju∩ Ω)
where µ is some finite positive measure on Ω.
Remark: As mentioned in the introduction, this compactness result was origi-nally stated for SBV functions. The reason why it is easy to extend it to SBVp
is beacause requiring higher integrability of the absolutely continuous parts of the gradients prevents the sequence to create a Cantor part in the limit, while the boundedness of the measures of the jump sets prevents the jump parts to create Cantor part in the limit.
From Lemma 1.5 it immediately follows this corollary, that we state as it will be directly useful later on.
Corollary 1.6. The functional ˆ Ω |∇u|pdx + ˆ Ju |u+− u−|dHN −1+ HN −1(J u∩ Ω) defined on SBVp
(Ω, Rm) is lower semicontinuous with respect to L1
conver-gence.
Proof. The first term is clearly lower semicontinuous for the weak convergence ∇uh* ∇u in Lp. The fact that it is indeed lower semicontinuous with respect
lower semicontinuous as the jump part of the measures Duh weakly converges
to the jump part of Du. Finally, the latter term is lower semicontinuous because
Chapter 2
Strong approximation in
SBV
p
(Ω, R
m
)
This chapter presents a first approximation result in the space SBVp
(Ω, Rm)
with respect to strong convergence, and it is mainly based on [BP96]. We recall the definition of HN −1-rectifiable set.
Definition 2.1. Let E ⊆ RN be an Hk-measurable set. We say that E is
count-ably Hk-rectifiable if there exist countably many Lipschitz functions f
i: Rk → RN such that Hk E \[ i∈N fi(Rk) ! = 0.
E is said Hk-rectifiable if E is countably Hk-rectifiable and Hk(E) < +∞.
Remark: Hk-rectifiability can be defined in a general metric space X, but in the special case X = RN Definition 2.1 is equivalent to the following: there
exist countably many compact manifold Mi of class C1 in RN so that
E ⊆[
i∈N
Mi∪ N
for a Hk-negligible set N . We will use this characterization more than once hereafter.
Theorem 2.1. Let Ω ⊆ RN be an open bounded set with Lipschitz boundary ∂Ω.
If u ∈ SBVp
(Ω, Rm) ∩ L∞
(uh)h∈N⊆ SBVp(Ω, Rm)that approximates u in the following sense: kuh− ukBV (Ω,Rm)→ 0 (2.1a) ∇uh→ ∇u in Lp (2.1b) kuhk∞≤ kuk∞∀h ∈ N (2.1c) uh∈ C1(Ω \ Rh, Rm) (2.1d) HN −1(J uh∆Ju) → 0 (2.1e)
where (Rh)h∈N is a sequence of HN −1-rectifiable sets with Rh ⊇ Juh for all
h ∈ N.
Proof. For the sake of clarity, the proof is divided into several steps: we define a sequence of auxiliary minimum problems for which we prove the existence of a minimizer and then we show that the sequence of minimizers is the desired approximated sequence for u.
Step 0: Preliminary observations.
First, notice that we can suppose m = 1 without loss of generality and then argue component-wise. For each h ∈ N, let Kh ⊆ Ju a compact subset of the
approximate jump set of u such that HN −1(J
u\ Kh) ≤
1
h.
Step 1: Auxiliary minimum problems. Consider the following minimum problem:
min
v∈SBVp(Ω,Rm)G(v) (2.2)
where the functional G is defined as
G(v) := ˆ Ω |∇v|pdx + HN −1(J v\ Kh) + HN −1(Kh)+ h ˆ Ω |u − v|pdx + ˆ Kh |v+− u+| + |v−− u−| ∧ 1dHN −1 (2.3)
(the last term is well defined choosing νu = νv on Ju∩ Jv∩ Kh). In solving
such a problem we are looking for a good approximation of u in Lp with a small gradient in Lp norm and close value of the traces. Observe that by a truncation argument we can, without loss of generality, minimize G among the
SBVp functions with kvk∞≤ kuk∞. Call X this space, that is
X := {v ∈ SBVp(Ω, Rm) | kvk∞≤ kuk∞}.
It is easy to show that G is coercive on X. Hence, if we show that the functional
G is lower semicontinuous with respect to weak convergence in SBVp
(Ω, Rm)
we would obtain the existence of a minimizer vh. By a slicing argument we
reduce ourselves to prove the lower semi-continuity of the 1-dimensional section
F (v, I) = ˆ I |v0|dt + #((Jv\ K) ∩ I) + M X j=1 (|v(tj+) − aj| + |v(tj−) − bj|) ∧ 1
where I ⊆ RN is an open interval, v ∈ SBVp(I), K = {t
1, . . . , tM} ⊆ I and
a1, . . . , aM and b1, . . . , bM are real numbers. We denoted by v(t±) the right and
left approximate limits of v at t respectively. Indeed, the lower semi-continuity of F (·, I) entails the lower semi-continuity of G along each “sliced direction” (by integration) and thus the lower semi-continuity of G (as the supremum of a sum of its sliced versions). Moreover, since K is now a finite set, we only need to show that F is semi-continuous with respect to the weak SBVp(I) convergence for K = {0} and I = (−1, 1). The functional becomes then
F (v, I) =
ˆ 1 −1
|v0|dt + #((J
v\ {0}) ∩ (−1, 1)) + (|v(0+) − a| + |v(0−) − b|) ∧ 1.
Let vh* v weakly in SBVp(I), i.e.
vh→ v in L1 sup h |Dvh|(I) < +∞ ∇vh* ∇v in Lp with Jvh = {t h 0 < th1 < · · · < thM} and t h j → tj ∈ [0, 1] for j = 0, . . . , M . There
are two possible scenarios:
(i) 0 6∈ {t1, . . . , tM} so that for h large enough one has vh(0+) = vh(0−) →
v(0) from which we infer the lower semicontinuity
F (v, I) ≤ lim inf
h→+∞F (vh, I)
using Lemma 1.5 and Corollary 1.6, after noticing the continuity of the last term of F .
(ii) 0 ∈ {t1, . . . , tM} with some multiplicity, i.e. tk = tk+1 = · · · = tl = 0
for some k, k + 1, . . . , l. In this case take ε > 0 such that tk−1< −ε and
tl+1> ε. Again by Lemma 1.5 we can infer
F (v, (−1, −ε)) ≤ lim inf
h→+∞F (vh, (−1, −ε)) (2.4a)
F (v, (ε, 1)) ≤ lim inf
h→+∞F (vh, (ε, 1)) (2.4b)
We distinguish now two cases:
(a) If #{h ∈ N | ∃j ∈ {l, . . . , m} s.t. th
j 6= 0} = +∞ then
(|v(0+) − a| + |v(0−) − b|) ∧ 1 ≤ 1 ≤ lim
h→+∞#((Jvh\ {0}) ∩ (−1, 1));
(b) otherwise suppose Jvh∩ [−ε, ε] = {0} and vh(0±) → v(0±). This yields
|vh(0+) − a| + |vh(0−) − b| → |v(0+) − a| + |v(0−) − b|.
In both cases (a) and (b) and by 2.4 we can infer
F (v, (−1, −ε)) + F (v, (ε, 1)) + (|v(0+) − a| + |v(0−) − b|) ∧ 1 ≤ lim inf
h→+∞F (vh, I).
It suffices to let ε → 0 to obtain that F (·, I) is lower semicontinuous:
F (v, I) ≤ lim inf
and therefore there exists a solution to the minimum problem (2.2) with kvhk∞≤
kuk∞.
Step 2: Estimates and convergence.
Since vh is a minimizer for (2.2), we easily get G(vh) ≤ G(u) =
ˆ
Ω
|∇u|pdx + HN −1(J
u) (2.5)
from which, recalling the definition of G in (2.3) we infer that there is a (fixed) constant c > 0 such that ˆ
Ω
|u − v|p≤ c
h (2.6)
that is vh→ u in Lp(Ω). Moreover
|Du|(Ω) ≤ |Dau|(Ω) + |Dsu|(Ω)
≤ C kvhkLp+ 2 kvhk∞HN −1(Jvh) ≤ C kvhkLp+ 2 kvhk∞H N −1(J vh∪ Kh) ≤ C kvhkLp+ 2 kuk∞H N −1(J vh∪ Kh) ≤ C
for some constant C > 0 (changing from line to line) where the last inequality comes from (2.5). Until now we built a sequence that satisfies suph∈NkvhkBV <
+∞ and suph∈N´Ω|∇v|pdx+HN −1(J
uh∩Ω) < +∞ thanks to the last estimates.
But these are exactly the necessary assumptions in Lemma 1.5 that yield weak convergence vh * u in SBVp(Ω, Rm) up to (non relabelled) subsequences. As
a byproduct - again up to subsequences - we can say that lim h→+∞ ˆ Ω |∇vh|pdx ≥ ˆ ω |∇u|pdx (2.7a) lim h→+∞H N −1 (Jvh) ≥ H N −1 (Ju) (2.7b)
from which we obtain ˆ Ω |∇u|pdx + HN −1(J u) ≤ lim h→+∞ ˆ Ω |∇vh|pdx + HN −1(Jvh) ≤ lim h→+∞ ˆ Ω |∇vh|pdx + lim h→+∞H N −1(K h) + lim sup h→+∞ HN −1(J vh\ Kh) + h ˆ Ω |u − vh|pdx + ˆ Kh |v+ h − u +| + |v− h − u −| ∧ 1dHN −1 ≤ ˆ Ω |∇u|pdx + HN −1(Ju)
as the first inequality trivially follows from (2.7), the second holds because we just add positive terms and the latter from the minimality of vh and (2.5).
Hence, the left-hand-side and the right-hand-side in previous inequality being equal, we obtain lim h→+∞ ˆ Ω |∇vh|pdx = ˆ Ω |∇u|pdx (2.8) lim h→+∞H N −1(J vh) = H N −1(J u) (2.9) lim h→+∞H N −1(J vh\ Kh) = 0 (2.10) lim h→+∞ ˆ Kh |v+ h − u +| + |v− h − u −| ∧ 1dHN −1= 0. (2.11)
From (2.8) and ∇vh* ∇u weakly in Lp(Ω, RN ×m) we deduce that the
conver-gence is actually ∇vh→ ∇u strongly in Lp(Ω, RN ×m) (see for instance [Bre10,
Prop. 3.32, p. 78]). Moreover, 0 ≤ lim h→+∞ ˆ Ju∪Jvh |v+ h − u +| + |v− h − u −| dHN −1 ≤ lim h→+∞ 4 kuk∞(HN −1(Jvh\ Kh) + H N −1(J u\ Kh)) + ˆ Kh |v+ h − u +| + |v− h − u −| dHN −1 = 0
where, recalling that we chose Kh⊆ Ju, we decomposed Ju∪ Jvh = (Ju\ Kh) ∪
(Jvh \ Kh) ∪ Kh to split the integral and estimate the jump from above with
4 kuk∞. The last equality follows from (2.9), (2.10) and lim h→+∞ ˆ Kh |v+ h − u +| + |v− h − u −| dHN −1= 0
in turn can be easily deduced from (2.11). To summarize, we now have that
vh→ u in L1(Ω) ∇vh→ ∇u in Lp(Ω) lim h→+∞ ˆ Jvh∪Ju |v+ h − u +| + |v− h − u −| ∧ 1dHN −1= 0 HN −1(J vh∆Ju) = 0 kvhk∞≤ kuk∞
that is part of the thesis of the theorem. We are left to prove the existence of the rectifiable sets Rh.
Step 3: Conclusion.
Notice that, because of (2.5), vhis also a minimizer on Ω \ Khfor the functional
ˆ Ω |∇v|pdx + h ˆ Ω |v − u|pdx + HN −1(J v)
so that we can apply the regularity result of [CL91, Lemma 4.5, p. 344] to establish that vh∈ C1((Ω \ Kh) \ Jvh) = C 1(Ω \ (J vh∪ Kh) and HN −1(((Ω \ K h) ∩ Jvh) \ Jvh) = H N −1((J vh\ Jvh) ∩ (Ω \ Kh) = 0.
Therefore it suffices to set Rh:= Jvh∪ Khto conclude the proof.
Remark: In Step 2, when deducing strong Lp convergence of the sequence of the approximate gradients, we need p > 1. In particular, this result is no more valid in SBV (Ω, Rm); this means that one needs to assume higher integrability of ∇u.
Remark: As a consequence of the definition of the rectifiable set Rh, there
exists a positive constant c ≥ 0 independent from h such that HN −1(R
h) ≤ c ∀h ∈ N.
This uniform upper bound is worthy and we will use it in the proof of The-orem 4.2 to show that the part of the jump set of the approximating func-tions that is not polyhedral has small HN −1-measure. However, this is not
completely satisfactory as we only know HN −1(J
uh∆Ju) → 0 and not that
HN −1(R
h∆Ju) → 0. In particular, we do not have any exact information about
Chapter 3
Approximation in SBV
p
(Ω)
In the present chapter, we present an improvement of Theorem 2.1. In particu-lar, this result fixes the issues remarked at the end of Chapter 2, replacing the rectifiable set Rh with a compact C1 manifold Mh for which we know addition
that HN −1(M
h\ Juh) → 0. This particularly means that the jump sets of the
approximating sequence are contained in tightly-fitted C1manifold.
We start by a section of preliminary lemmas and then we present the refined approximation theorem in Section 3.2.
3.1
Preliminary lemmas
The following lemma states that it is always possible to assume that an SBV function is bounded. We will use this observation later on in the proof of Theorem 3.4.
Lemma 3.1. Let Ω ⊆ RN be an open set, and let u ∈ SBV (Ω) be a function
with ∇u ∈ Lp for some p ≥ 1. Then, for every ε > 0 there exists a function
uε∈ SBV (Ω) ∩ L∞(Ω) such that
ku − uεkBV (Ω)+ k∇u − ∇uεkLp(Ω)≤ ε and Juε ⊆ Ju.
Remark: Asking that ∇u ∈ Lp for some p ≥ 1 does not mean that u ∈
SBVp(Ω), unless we also require HN −1(J
u) < +∞. Instead, this includes both
the cases u ∈ SBV (Ω) and u ∈ SBVp
∞(Ω) if p = 1 or p > 1 respectively. Of
course, the lemma is still valid in the particular case u ∈ SBVp(Ω).
Proof. From the general theory of SBV functions, we know that the jump set
Ju is HN −1-countably rectifiable, so there are an HN −1-negligible set N and
countably many C1 compact manifold Mi for i ∈ N that covers Ju. By the
continuity theorem of trace operator, we know that the traces τi±: BV (Ω) →
L1(Mi) on the two sides of each manifold Mi are continuous. Therefore, there
sets AK := {x ∈ Ω | |u(x)| ≥ K} BK := [ i∈N {x ∈ Mi| |τi+(x)| ∨ |τ − i (x)| ≥ K}, one has kukL1(A K)+ k∇ukL1(AK)+ k∇ukLp(AK)+ |D su|(B K) ≤ ε. (3.1) Now, define uε(x) = sgn(u(x))(K ∧ |u(x)|) ∈ SBV (Ω).
It is clear that ∇uε∈ Lp(Ω) and Juε ⊆ Ju. Moreover, D
su
ε= Dsu on Ju\ BK
and on the other hand |Dsu
ε| ≤ |Dsu| on BK. Then, the claim is immediate
from (3.1):
ku − uεkBV (Ω)+ k∇u − ∇uεkLp(ω)≤ kukL1(AK)+ k∇ukL1(AK)
+ k∇ukLp(A K)+ |D
su|(B K)
≤ ε
and the proof is complete.
The following lemma allows modifying a function u ∈ SBVp(Ω) ∩ C1(Ω \
Ju) ∩ W1,∞(Ω \ Ju) to transform its polyhedral jump set into a C1 manifold.
For its proof refer to [DPFP17, Appendix A].
Lemma 3.2. Let Ω ⊆ RN and let u ∈ SBVp(Ω) ∩ C1(Ω \ Ju) ∩ W1,∞(Ω \ Ju).
Suppose that the jump set Ju is polyhedral (see Definition 4.1). Then, for every
ε > 0 there exists uε∈ SBVp(Ω) ∩ C∞(Ω \ Juε) ∩ W
1,∞(Ω \ J
uε)whose jump
set Juε b Ω is a C
1 manifold with C1 boundary and that is ε-close to u in the
sense that
ku − uεkBV (Ω)≤ ε
k∇u − ∇uεkLp(Ω)≤ ε
HN −1(J
u∆Juε) ≤ ε.
Moreover, if Π is a hyperplane in RN, we can build u
ε in such a way that
dist(Juε\ Π, Π) > 0, that is the part of the jump set outside of the hyperplane
Π has positive distance apart from the hyperplane itself.
3.1.1
Approximation in SBV (Ω)
An essential step in the proof of Theorem 3.4 relies on the following Lemma 3.3 which provides an approximation result for functions in SBV (Ω) whose singular part of the derivative is “almost” concentrated on a C1 manifold. In proving the lemma, we will make use of a mollification argument with variable kernel. Fix then a compact set Kb Ω and let E = K ∪ ∂Ω. We provide the contest for
Figure 3.1: A compact set Kb Ω and the set E from the definition of δ.
this tool in the next few paragraphs. Fix any “regularized distance function” from E, i.e. a function δ : Ω → R of class C∞(Ω \ E) and such that
kDδk∞≤ 1 and dist(x, E)
2 ≤ δ(x) ≤ dist(x, E) ∀x ∈ Ω Take also a function f ∈ C∞([0, +∞)) with the following properties:
0 < f (t) ≤ 1 ∀t ∈ (0, +∞) 0 ≤ f0(t) ≤ 1 ∀t ∈ [0, +∞) f(j)(0) = 0 ∀j ∈ N . (3.2)
We can now define a “generalized translation function” as follows.
Definition 3.1. Let δ be a regularized distance function and let f ∈ C∞([0, +∞))
be a function satisfying (3.2). Given a number 0 < σ < 1 and a vector y ∈ B1(0), we define the “generalized translation” as the function
Tσ,y: Ω → Ω
x 7→ Tσ,y(x) = x − σf (δ(x))y.
Notice that Tσ,y is a bijection from Ω onto itself.
Remark: (Properties of Tσ,y.) Notice that, by definition, Tσ,y is the identity
on E and
DTσ,y(x) = id − σf0(δ(x))y ⊗ Dδ(x)
whose determinant is
because of det(id + a ⊗ b) = 1 + a · b, (3.2), the definition of δ, and the fact that
σ, |y| < 1. In particular Tσ,y is a local diffeomorphism; being bijective, it is also
a global diffeomorphism.
We now have all the ingredients to give the definition of the mollification with variable kernel.
Definition 3.2. Let f, σ as above. Fix a smooth positive function ρ ∈ Cc∞(B(0, 1))
with ´B(0,1)ρ(x)dx = 1. For any function u ∈ L1
loc(Ω)define
uσ(x) =
ˆ
B(0,1)
u(Tσ,y(x))ρ(y)dy =
ˆ
B(0,1)
u(x − σf (δ(x))y)ρ(y)dy.
For any Radon measure µ, let µσ=
ˆ
B(0,1)
(Tσ,y−1)#[det(DTσ,y−1µ]ρ(y)dy
that is, µσ is the unique measure such that
ˆ Ω ϕ(x)dµσ(x) = ˆ B(0,1) ˆ Ω ϕ Tσ,y−1(z) det(DT−1 σ,y(z))dµ(z) ρ(y)dy for all ϕ ∈ Cc(Ω).
Remark: If Tσ,y is replaced by the standard translation Ty(x) = x − y, then uσ
and µσ reduce to the standard mollification of functions and measure
respec-tively.
Remark: It is immediate to see that, if µ LN with density u, then µσ LN
with density uσ.
Remark: Suppose that µ is concentrated on K. Then, Tσ,y being the identity
on K, µσ= µ.
Remark: The mollified function uσis of class C∞(Ω\K). Indeed, for x ∈ Ω\K
uσ(x) = ˆ B(0,1) u(x − σf (δ(x))y)ρ(y)dy = 1 (σf (δ(x)))N ˆ B(x,σf (δ(x))) u(z)ρ x − z σf (δ(x)) dz
and δ is smooth on Ω \ K, f is smooth on [0, +∞) and ρ is smooth on B(0, 1). We now have all the instruments to state and prove the approximation lemma on SBV .
Lemma 3.3. Let Ω ⊆ RN be an open set, let u be a function in SBV (Ω), and let
M be a C1 manifold with C1 boundary such that, for some small ε, |Dsu|(J u\
M ) < ε4. Then, there exists a function v ∈ SBV (Ω) ∩ C∞(Ω \ M )such that ku − vkBV (Ω)< 4ε, HN −1(M \ Jv) = 0,
both the traces of v on the two sides of M are of class C1, and
v ∈ W1,∞(Ω0\ M ) ∀M b Ω0 b Ω.
Moreover, if u has compact support in Ω, then the same is true for v (and then, in particular, v ∈ W1,∞(Ω \ M )).
Proof. The proof is divided into steps, to make it clearer. The last point in the statement comes directly from the construction.
Step 0: Extra assumptions.
Thanks to Lemma 3.1 we can assume u ∈ L∞(Ω). Moreover, without loss of generality, we can also assume that the support of u is bounded, up to multiply
u by a smooth function ˆϕ : RN → [0, 1] such that
ˆ ϕ(x) = 1 |x| ≤ R1 ˆ ϕ(x) = 0 |x| ≥ R2 kD ˆϕk∞≤ 1 for some R2 R1 1.
Step 1: Approximation through mollification with variable kernel.
Given ε > 0 as in the statement of the lemma, in this step, we build a function
u1∈ SBV (Ω) ∩ C∞(Ω \ Ju1) such that
ku − u1kBV (Ω)< ε. (3.4)
To do this, we show that it is enough to take u1= uσ for σ > 0 small enough.
First, notice that, since Juis HN −1-rectifiable, there are a compact, C1manifold
M0 b Ω with C1boundary and a compact set Kε⊆ Ju∩M0with |Du|(Ju\Kε) ≤ ε
4. It is possible to show that uσ ∈ BV (Ω) and there exists some Radon measure
ξσ so that
Duσ= (Du)σ+ σξσ (3.5a)
ξσ K = 0 (3.5b)
|ξσ|(Ω) ≤ 2|Du|(Ω) (3.5c)
Duσ K = (Du)σ K = Du K. (3.5d)
This can be done first assuming u ∈ C∞(Ω), obtaining also that ξσsatisfying the properties (3.5) is a smooth function vanishing on K. Then, taking a sequence (uh)h∈N ⊂ BV (Ω) ∩ C∞(Ω) strictly converging to u in BV and checking that
the properties (3.5) pass to the limit. In particular, being smooth out of Kεas
remarked above, uσ∈ SBV (Ω). We aim to show that
ku − uσkBV (Ω)= ku − uσkL1(Ω)+ |Du − Duσ|(Ω) ≤ ε
for σ > 0 small enough. Recall that Cc(Ω) are dense in Lp(Ω) and Tσ,y → id as
σ → 0 in L∞(Ω). This ensures that, if u ∈ Cc(Ω) is uniformly continuous, then
0 when σ → 0. In particular, this is true for p = 1, so that we are left to show that the |Du − Duσ|(Ω) can be arbitrarily small. This follows from (3.5):
|Du − Duσ|(Ω) =
∇uLN + Du K
ε+ Du (Ju\ Kε)
− (∇u)σLN + Du Kε+ (Du (Ju\ Kε))σ+ σξσ (Ω)
≤ k∇u − (∇u)σkL1(Ω)+ |Du (Ju\ Kε)|(Ω)
+ |(Du (Ju\ Kε)σ|(Ω) + σ|ξσ|(Ω)
≤ k∇u − (∇u)σkL1(Ω)+ 3|Du (Ju\ Kε)|(Ω) + 2σ|Du|(Ω)
≤ k∇u − (∇u)σkL1(Ω)+
3
4ε + 2σ|Du|(Ω) ≤ ε
where, in the last inequality, the first and last term can be arbitrarily small since they tend to 0 as σ → 0.
Step 2: Modify the approximation so to become smooth in Ω \ M0, and in such
a way that its traces on M0 become C1 and coincide on ∂M0.
Consider the manifold M0 from the previous step. If M0 is not connected, by compactness it has anyway a finite number P of connected components Mi.
Consider disjoint, smooth sets Ai with Mi b Ai b Ω for 1 ≤ i ≤ P . For
each i ∈ {1, . . . , P }, there exists a positive constant Ci, depending only on Ai
and Mi, with the following property: for any three functions gi∈ L1(∂Ai) and
gi± ∈ L1(M
i), there exists a function ϕi ∈ W1,1(Ai\ Mi) whose trace on ∂Ai
coincides with gi and whose two traces on the two sides of Miare g±, satisfying
kϕikW1,1(A i\Mi)≤ Ci kgikL1(∂A i)+ g + i L1(Mi)+ g−i L1(Mi) . (3.6)
Apply this property with gi= 0 on ∂Ai and g±i ∈ L1(Mi) such that u±1 + g ±
i ∈
C1(M
i) coincide on ∂Mi and whose norms are bounded by
gi± L1(M i)< ε (Ci+ 1)P . (3.7)
In particular, ϕivanishes on ∂Aiand coincides with gi±on Mi. Moreover, (3.6)
becomes kϕikW1,1(Ai\Mi)≤ Ci gi+ L1(M i)+ g−i L1(M i) ≤ 2εCi (Ci+ 1)P . (3.8) Define ˜u2∈ SBV (Ω) by ˜ u2:= ( u1+ ϕi on Ai, i = 1, . . . , P u1 on Ω \S P i=1Ai
whose jump set is Ju˜2 ⊆ M
0 by construction and whose traces on M0 are of
class C1 and coincide on the boundary ∂M0. By (3.7) and (3.8) we obtain
k˜u2− u1kBV (Ω)= P X i=1 kϕikW1,1(Ai\Mi)+ g+i − gi− L1(M i) ≤ 2ε P P X i=1 Ci Ci+ 1 + 1 Ci+ 1 = 2ε
and thus
k˜u2− ukBV (Ω)≤ ku − u1kBV (Ω)+ k˜u2− u1kBV (Ω)< ε + 2ε = 3ε
using (3.4) from Step 1. In a similar way to what we did in the previous step, using mollifiers, we can find a new function u2∈ SBV (Ω) ∩ C∞(Ω \ M0) that
approximates ˜u2 in the BV norm up to
k˜u2− u2kBV (Ω)< 3ε − k˜u2− ukBV (Ω).
By triangular inequality, we still have
ku2− ukBV (Ω)< 3ε
and the traces of u2 on M0 coincides with those of ˜u2.
Step 3: A decomposition of u2.
We can find ψ ∈ W1,∞(Ω \ M0) vanishing outside a neighbourhood of M0 and
whose traces on M0 coincide with those of u2. Up to regularize ψ by means
of mollifiers, we can suppose ψ ∈ SBV (Ω) ∩ W1,∞(Ω \ M0) ∩ C∞(Ω \ M0). Then, we can decompose the function from Step 2 as v2 = ψ + w where w ∈
SBV (Ω)∩C∞(Ω\M0) by definition with traces on M0equal to 0 by construction. In particular, w ∈ W1,1(Ω) ∩ C∞(Ω \ M0) so by Meyers-Serrin theorem there exists wε∈ W1,1(Ω) ∩ C∞(Ω) such that
kwε− wkBV (Ω)= kwε− wkW1,1(Ω)< ε.
Step 4: The final approximation uε.
Define uε= ψ + wεwhich is by construction in SBV (Ω) ∩ C∞(Ω \ M0) and also
in W1,∞(Ω0\ M0) for all M0 b Ω0b Ω. We have, by construction kuε− u2kBV (Ω)= kϕ + wε− ψ + wkBV (Ω)< ε so that kuε− ukBV (Ω)≤ kuε− u2kBV (Ω)+ ku − u2kBV (Ω)< ε + 3ε = 4ε as claimed.
Step 5: HN −1(M \ Juε) = 0 and conclusion.
We still have to prove the part of the statement concerning the Hausdorff mea-sure of the difference M \ Juε. Until now, we were able to find an approximation
uε∈ SBV (Ω) ∩ C∞(Ω \ M ):
ku − uεkBV (Ω)< 4ε.
From the construction of uεdone in the previous steps, it follows that Juε ⊂ M .
In this step we show how to modify uεin v so that HN −1(M \Jv) = 0, preserving
regularity and increasing the difference in the BV norm from u by less than 2ε. Once this is done, up to changing ε into ε2, we can deduce that both properties
are satisfied. The modification needs to be done in each connected component
Mi. Since these are finitely many, we reduce to show how the proceeding works
in one of them. The changing simply consists in taking v = uε+ ηϕ for some
small η > 0 and with ϕ a function similar to those defined in Step 2. More precisely, fix i = 1, . . . , P and consider ϕ ∈ W1,∞(Ai\ Mi) so that ϕ = 0 on
∂Ai and with traces on the two sides of Mi given by
ϕ−≡ 0 and ϕ+(x) ∈ C1(Mi), ϕ+
(
= 0 x ∈ ∂Mi
> 0 x ∈ M .
From this construction, we deduce that any point x ∈ Mi belongs to the jump
set Jv of v for all η except one. Therefore, #{ε | HN −1(Mi\ Jv) > 0} is at most
countable and then it is possible to find an ε that yields the conclusion.
Remark: The function u1 built in Step 1 does not conclude the proof since we
do not know that ∇u1∈ Lp(Ω) nor we have any information on k∇u − ∇u1kLp(Ω).
3.2
The refined approximation theorem for SBV
p(Ω)
We are now ready to state and prove the new approximation theorem forSBVp(Ω).
Theorem 3.4. Let Ω ⊆ RN be an open set with locally Lipschitz boundary
∂Ω. If u ∈ SBVp
(Ω, Rm) for some p > 1, then there exists a sequence of
functions (uh) ⊆ SBVp(Ω, Rm) that are smooth outside their jump sets, i.e.
uh∈ C∞(Ω \ Juh)and there exists a sequence of compact C
1 manifold with C1
boundary Mh b Ω such that Juh ⊆ Mh∩ Ju but H
N −1(M
h\ Juh) = 0and the
sequence (uh)h approximates u in the following sense:
kuh− ukBV → 0 (3.9a)
∇uh→ ∇uin Lp(Ω) (3.9b)
HN −1(J
uh∆Ju) → 0. (3.9c)
Proof. Since the proof is quite long, we divide it into some steps. In Step 1 we will consider the case when u has compact support in Ω, then we will consider the case when the jump set Ju b Ω has compact closure and finally we will
approach the general case first reducing ourselves to Ω = RN+ and closing by
means of the partitions of unity. Step 1: Case supp(u) ⊆ Ω is compact. Step 1.1: A first approximation.
We fix ε > 0 and we aim at building an ε-approximation in the sense of (3.9). Choose a C1 manifold with C1boundary M
0b Ω such that HN −1(J u∆M0) < ε 3 (3.10a) |Du|(Ju\ M0) < ε 5 (3.10b)
