BMO-type norms related to the perimeter of sets
aa r X i v : . [ m a t h . F A ] J u l BM O -TYPE NORMS RELATED TO THE PERIMETER OF SETS
LUIGI AMBROSIO, JEAN BOURGAIN, HAIM BREZIS, AND ALESSIO FIGALLI
Abstract.
In this paper we consider an isotropic variant of the
BM O -type norm re-cently introduced in [6]. We prove that, when considering characteristic functions ofsets, this norm is related to the perimeter. A byproduct of our analysis is a new char-acterization of the perimeter of sets in terms of this norm, independent of the theory ofdistributions. Introduction
Let Q = (0 , n be the unit cube in R n , n > . In a very recent paper [6], the secondand third author, in collaboration with P. Mironescu, introduced a new function space B ⊂ L ( Q ) based on the following seminorm, inspired by the celebrated BM O space ofJohn-Nirenberg [14]: k f k B := sup ǫ ∈ (0 , [ f ] ǫ , where [ f ] ǫ := ǫ n − sup G ǫ X Q ′ ∈G ǫ − Z Q ′ (cid:12)(cid:12)(cid:12) f ( x ) − − Z Q ′ f (cid:12)(cid:12)(cid:12) dx, (1.1)and G ǫ denotes a collection of disjoint ǫ -cubes Q ′ ⊂ Q with sides parallel to the coordinateaxes and cardinality not exceeding ǫ − n ; the supremum in (1.1) is taken over all suchcollections. In addition to k f k B , it is also useful to consider its infinitesimal version,namely [ f ] := lim sup ǫ ↓ [ f ] ǫ ≤ k f k B , and the space B := { f ∈ B : [ f ] = 0 } .Their main motivation was the search of a space X on the one hand sufficiently largeto include V M O , W , , and the fractional Sobolev spaces W /p,p with < p < ∞ , onthe other hand sufficiently small (i.e., with a sufficiently strong seminorm) to provide theimplication f ∈ X and Z -valued = ⇒ f = k L n -a.e. in Q , for some k ∈ Z (1.2)an implication known to be true in the spaces V M O , W , , and W /p,p .One of the main results in [6] asserts that (1.2) holds with X = B . The principalingredient in the proof concerns the case f = A , where A ⊂ Q is measurable. For such special functions it is proved in [6] that (cid:13)(cid:13)(cid:13)(cid:13) f − − Z Q f (cid:13)(cid:13)(cid:13)(cid:13) L n/ ( n − ( Q ) ≤ C [ f ]; (1.3)here and in what follows we denote by C a generic constant depending only on n . Estimate(1.3) suggests a connection with Sobolev embeddings and isoperimetric inequalities; recalle.g. that (cid:13)(cid:13)(cid:13)(cid:13) f − − Z Q f (cid:13)(cid:13)(cid:13)(cid:13) L n/ ( n − ( Q ) ≤ C | Df | ( Q ) ∀ f ∈ BV ( Q ) . (1.4)When f = A , (1.4) takes the form (cid:13)(cid:13)(cid:13)(cid:13) A − − Z Q A (cid:13)(cid:13)(cid:13)(cid:13) L n/ ( n − ( Q ) ≤ C P ( A, Q ) , (1.5)where P ( A, Q ) denotes the perimeter of A relative to Q . Combining (1.5) with the obviousinequality (cid:13)(cid:13)(cid:13)(cid:13) A − − Z Q A (cid:13)(cid:13)(cid:13)(cid:13) L n/ ( n − ( Q ) ≤ yields (cid:13)(cid:13)(cid:13)(cid:13) A − − Z Q A (cid:13)(cid:13)(cid:13)(cid:13) L n/ ( n − ( Q ) ≤ C min { , P ( A, Q ) } . (1.6)In view of (1.3) and (1.6) it is natural to ask whether there exists a relationship between [ A ] and min { , P ( A, Q ) } . The aim of this paper is to answer positively to this question.Since the concept of perimeter is isotropic, it is better to make also the main object of[6] isotropic, by considering I ǫ ( f ) := ǫ n − sup F ǫ X Q ′ ∈F ǫ − Z Q ′ (cid:12)(cid:12)(cid:12) f ( x ) − − Z Q ′ f (cid:12)(cid:12)(cid:12) dx, (1.7)where F ǫ denotes a collection of disjoint ǫ -cubes Q ′ ⊂ R n with arbitrary orientation andcardinality not exceeding ǫ − n .The main result of this paper is the following: Theorem 1.1.
For any measurable set A ⊂ R n one has lim ǫ → I ǫ ( A ) = 12 min (cid:8) , P ( A ) (cid:9) . (1.8) In particular, lim ǫ I ǫ ( A ) < / implies that A has finite perimeter and P ( A ) = 2 lim ǫ I ǫ ( A ) . We present the proof of Theorem 1.1 in Section 3. Although we confine ourselves tothe most interesting case n > throughout this paper, we point out in Section 3.4 thatTheorem 1.1 still holds in the case n = 1 and we present a brief proof; since in this case P ( A ) is an integer, we infer that lim ǫ I ǫ ( A ) < / implies that either A or R \ A areLebesgue negligible. MO -TYPE NORMS RELATED TO THE PERIMETER OF SETS 3 Returning to the case n > , we can also understand better the role of the upper boundon cardinality with the formula lim ǫ ↓ sup F ǫ,M ǫ n − X Q ′ ∈F ǫ,M − Z Q ′ (cid:12)(cid:12)(cid:12) A ( x ) − − Z Q ′ A (cid:12)(cid:12)(cid:12) dx = min { M, P ( A ) } ∀ M > , (1.9)where F ǫ,M denotes a collection of ǫ -cubes with arbitrary orientation and cardinality notexceeding M ǫ − n . The proof of (1.9) can be achieved by a scaling argument. Indeed,setting ρ = M − / ( n − , it suffices to apply (1.8) to ˜ A = ρA , noticing that min { , P ( ˜ A ) } = 1 M min { M, P ( A ) } = ρ n − min { M, P ( A ) } , that ǫ − n = M ( ǫ/ρ ) − n , and finally that the transformation x x/ρ maps ˜ A to A , aswell as ǫ -cubes to ǫ/ρ -cubes.In Section 4.1 we return to the framework of measurable subsets A of a Lipschitz domain Ω , and we establish that lim ǫ → I ǫ ( A , Ω) = 12 min (cid:8) , P ( A, Ω) (cid:9) , (1.10)where I ǫ ( A , Ω) is a localized version of (1.7) where we restrict the supremum over cubescontained in Ω .Also, going back to the setting of [6], in Section 4.1 we prove that [ A ] ≤
12 min { , P ( A, Q ) } ≤ C [ A ] for every measurable subset of Q .Then, in Section 4.2 we discuss how removing the bound on the cardinality allows usto obtain a new characterization both of sets of finite perimeter and of the perimeter,independent of the theory of distributions. In a somewhat different direction, see also [7],[8, Corollary 3 and Equation (46)], and [9].We conclude this introduction with a few more words on the strategy of proof. Asillustrated in Remark 3.1, using the canonical decomposition in cubes, it is not too difficultto show the existence of dimensional constants ξ n , η n > satisfying lim sup ǫ ↓ I ǫ ( A ) < ξ n = ⇒ P ( A ) ≤ η n lim sup ǫ ↓ I ǫ ( A ) (1.11)for any measurable set A ⋐ Q . This idea can be very much refined, leading to the proof ofthe inequality lim inf ǫ I ǫ ( A ) ≥ / whenever P ( A ) = + ∞ . Since I ǫ ( A ) ≤ / , see (3.1)below, this proves our main result for sets of infinite perimeter. For sets of finite perime-ter, the inequality ≤ in (1.8) relies on the relative isoperimetric inequality in the cubewith sharp constant, see (2.2) below, while the inequality ≥ relies on a blow-up argument.This paper originated from a meeting in Naples in November 2013, dedicated to CarloSbordone’s 65th birthday, where three of us (LA, HB, and AF) met. On that occasion HBpresented some results from [6] and formulated a conjecture which became the motivation L. AMBROSIO, J. BOURGAIN, H. BREZIS, AND A. FIGALLI and the main result of the present paper. For this and many other reasons, we are happyto dedicate this paper to Carlo Sbordone.2.
Notation and preliminary results
Throughout this paper we assume n ≥ . We denote by F the cardinality of a set F , by A c the complement of A , by | A | the Lebesgue measure of a (Lebesgue) measurableset A ⊂ R n , by H n − the Hausdorff ( n − -dimensional measure. For δ > , we say that Q ′ is a δ -cube if Q ′ is a cube obtained by rotating and translating the standard δ -cube (0 , δ ) n .2.1. BV functions, sets of finite perimeter, and relative isoperimetric inequal-ities. Given Ω ⊂ R n open and f ∈ L (Ω) , we define | Df | (Ω) := sup (cid:26)Z f ( x ) div φ ( x ) dx : φ ∈ C c (Ω; R n ) , | φ | ≤ (cid:27) . (2.1)By construction, f
7→ | Df | (Ω) ∈ [0 , ∞ ] is lower semicontinuous w.r.t. the L (Ω) con-vergence. By Riesz theorem, whenever | Df | (Ω) is finite the distributional derivative Df = ( D f, . . . , D n f ) of f is a vector-valued measure with finite total variation, therefore f ∈ BV loc (Ω) . In addition, the total variation of Df coincides with the supremum in(2.1) (thus, justifying our notation).We will mostly apply these concepts when f = A is a characteristic function of ameasurable set A ⊂ R n . In this case we use the traditional and more convenient notation P ( A, Ω) = | D A | (Ω) , P ( A ) = P ( A, R n ) . A key property of the perimeter is the so-called relative isoperimetric inequality: for anybounded open set Ω ⊂ R n with Lipschitz boundary one has | E | · | Ω \ E | ≤ c (Ω) P ( E, Ω) for any measurable set E ⊂ Ω .In the case when Ω is the unit cube Q , we will need the inequality with sharp constant: | E | (1 − | E | ) ≤ P ( E, Q ) for any measurable set E ⊂ Q . (2.2)This inequality is originally due to H. Hadwiger [13] for polyhedral subsets of the cube.Far reaching variants appeared subsequently in the literature (see e.g. S.G. Bobkov [4, 5],D. Bakry and M. Ledoux [2], F. Barthe and B. Maurey [3], and their references). Howeverwe could not find (2.2) stated in the required generality used here (it is often formulatedwith the Minkowski content instead of the perimeter, so that some extra approximationargument is anyhow needed). For this reason, and for the reader’s convenience, we haveincluded in the appendix a proof of (2.2) based on the results of [3], in any number ofspace dimensions. Recall that f ∈ BV ( U ) if f ∈ L ( U ) and | Df | ( U ) < ∞ . MO -TYPE NORMS RELATED TO THE PERIMETER OF SETS 5 Fine properties of sets of finite perimeter.
In §3.3 we will need finer propertiesof sets of finite perimeter A in an open set Ω . In [10], De Giorgi singled out a set F A of finite H n − -measure, called reduced boundary, on which | D A | is concentratedand A is asymptotically close to a half-space. More precisely | D A | = H n − F A , i.e., | D A | ( E ) = H n − ( E ∩ F A ) for any Borel set E ⊂ Ω . De Giorgi also proved that | D A | -almost all of the reduced boundary can be covered by a sequence of C hypersurfaces Γ i (the so-called rectifiability property). A few years later, Federer in [11] extended theseresults to the so-called essential boundary, namely the complement of density 0 and density1 sets: ∂ ∗ A := (cid:26) x ∈ R n : lim sup r → + | B r ( x ) ∩ A || B r ( x ) | > and lim sup r → + | B r ( x ) \ A || B r ( x ) | > (cid:27) . (2.3)Federer also slightly strenghtned the rectifiability result, by replacing | D A | with H n − .We collect in the next theorem the results we need on sets of finite perimeter. Theorem 2.1 (De Giorgi-Federer) . Let A be a set of finite perimeter in Ω . Then thefollowing properties hold: (i) | D A | ( E ) = H n − ( E ∩ ∂ ∗ A ) for any Borel set E ⊂ Ω ; (2.4)(ii) there exist embedded C hypersurfaces Γ i ⊂ R n satisfying H n − (cid:18) Ω ∩ ∂ ∗ A \ ∞ [ i =1 Γ i (cid:19) = 0; (2.5)(iii) if Γ i are as in (2.5) , for H n − -a.e. x ∈ Ω ∩ ∂ ∗ A ∩ Γ i there exists a half-space H A ( x ) with inner normal ν A ( x ) orthogonal to Γ i at x such that ( A − x ) /r → H A ( x ) in L ( R n ) as r → + ; (iv) if F is any other set with finite perimeter in Ω , H A ( x ) = ± H F ( x ) H n − -a.e. in Ω ∩ ∂ ∗ A ∩ ∂ ∗ F .Proof. For the first three properties, see [10], [11], or [1, Theorems 3.59 and 3.61]. Taking(iii) into account, the last assertion (iv) follows from the elementary property
Tan(Γ , x ) = Tan(˜Γ , x ) for H n − -a.e. x ∈ Γ ∩ ˜Γ whenever Γ and ˜Γ are C embedded hypersurfaces. (cid:3) Since BV loc (Ω) ∩ L ∞ (Ω) is easily seen to be an algebra with | D ( uv ) | ≤ k u k ∞ | Du | + k v k ∞ | Dv | , it turns out that the class of sets of finite perimeter in an open set Ω is stable underrelative complement, union, and intersection. We need also the following property: H n − (cid:0) Ω ∩ ∂ ∗ ( E ∆ F ) \ ( ∂ ∗ E ∆ ∂ ∗ F ) (cid:1) = 0 whenever E, F have finite perimeter in Ω .(2.6) L. AMBROSIO, J. BOURGAIN, H. BREZIS, AND A. FIGALLI
In order to prove it, we first notice that ∂ ∗ is invariant under complement and ∂ ∗ ( E ∪ F ) ⊂ ∂ ∗ E ∪ ∂ ∗ F , ∂ ∗ ( E ∩ F ) ⊂ ∂ ∗ E ∪ ∂ ∗ F , hence it follows that ∂ ∗ ( E ∆ F ) ⊂ ∂ ∗ E ∪ ∂ ∗ F . Then,take x ∈ Ω ∩ ∂ ∗ ( E ∆ F ) and assume (possibly permuting E and F ) that x ∈ ∂ ∗ E . Byproperty (iii) of Theorem 2.1, possibly ignoring a H n − -negligible set, we can also assumethat ( E − x ) /r converges as r → + to a half-space H E ( x ) . Still ignoring another H n − -negligible set, we have then three possibilities for F : either x is a point of density 1, ora point of density , or there exists a half-space H F ( x ) such that ( F − x ) /r → H F ( x ) as r → + . In the first two cases it is clear that x ∈ ∂ ∗ E \ ∂ ∗ F and we are done. In thethird case, we know by property (iv) of Theorem 2.1 that H E ( x ) = ± H F ( x ) for H n − -a.e. x ∈ Ω ∩ ∂ ∗ E ∩ ∂ ∗ F . But H E ( x ) = H F ( x ) implies that x is a point of density 0 for E ∆ F and H E ( x ) = − H F ( x ) implies that x is a point of density 1 for E ∆ F , so the third casecan occur only on a H n − -negligible set.3. Proof of Theorem 1.1
The proof of Theorem 1.1 is quite involved and will take all of this section. Notice thatsince − Z Q ′ (cid:12)(cid:12)(cid:12) A ( x ) − − Z Q ′ A (cid:12)(cid:12)(cid:12) dx = 2 | Q ′ ∩ A | · | Q ′ \ A || Q ′ | ≤ (3.1)for any ǫ -cube Q ′ , we clearly have I ǫ ( A ) ≤ / .We now prove the theorem is three steps: first we show that I ǫ ( A ) ≤ P ( A ) / for all ǫ > , which proves that I ǫ ( f ) ≤ min { , P ( A ) } . Then we prove that lim inf ǫ → + I ǫ ( f ) ≥ min { , P ( A ) } first when P ( A ) = ∞ (the non-rectifiable case) and finally when A hasfinite perimeter (the rectifiable case).3.1. Upper bound.
We prove that I ǫ ( A ) ≤ P ( A ) / for all ǫ > . For this, we mayobviously assume P ( A ) < ∞ , hence f = A ∈ BV loc ( R n ) . By the additivity of P ( A, · ) , itsuffices to show that if Q ′ is an ǫ -cube, then | Q ′ ∩ A | · | Q ′ \ A || Q ′ | ≤ ǫ − n P ( A, Q ′ ) . After rescaling, this inequality reduces to (2.2), which proves the desired result.3.2.
Lower bound: the non-rectifiable case.
Here we assume that P ( A ) = ∞ andwe prove, under this assumption, that lim inf ǫ I ǫ ( f ) ≥ / . Before coming to the actualproof we sketch in the next remark the proof of (1.11), announced in the introduction. Remark 3.1.
Let us consider the canonical subdivision (up to a Lebesgue negligible set)of (0 , n in hn cubes Q i,h with length side − h . We define on the scale ǫ = 2 − h anapproximate interior Int h ( A ) of A by considering the set I h := (cid:26) i ∈ { , . . . , hn } : − Z Q i,h A > (cid:27) MO -TYPE NORMS RELATED TO THE PERIMETER OF SETS 7 and taking the union of the cubes Q i,h , i ∈ I h . Analogously we define a set of indices E h and the corresponding approximate exterior Ext h ( A ) = Int h ( Q \ A ) . We denote by F h the complement of I h ∪ E h and by Bdry h ( Q ) the union of the corresponding cubes.Since Int h ( A ) → A in L as h → ∞ , by the lower semicontinuity of the perimeter itsuffices to give a uniform estimate on P (Int h ( A )) = H n − ( ∂ Int h ( A )) as h → ∞ under asmallness assumption on lim sup h I − h ( A ) .Since − R Q i,h | A ( x ) − − R Q i,h A | dx ≥ / for all i ∈ F h (by definition of F h ), we obtainthat I − h ( A ) <
14 = ⇒ F h ≤ I − h ( A ) (2 − h ) − n < (2 − h ) − n , (3.2)which provides a uniform estimate on H n − ( ∂ Bdry h ( A )) . Hence, to control H n − ( ∂ Int h ( A )) it suffices to bound the number of faces F ⊂ Q common to a cube Q i,h and a cube Q j,h ,with i ∈ I h and j ∈ E h . For this, notice that if ˜ Q is any cube with side length − h containing Q i,h ∪ Q j,h , it is easily seen that − Z ˜ Q (cid:12)(cid:12)(cid:12) A ( x ) − − Z ˜ Q A (cid:12)(cid:12)(cid:12) dx ≥ − − n and this leads once more to an estimate of the number of these cubes with (2 − h ) − n provided I − h ( A ) < − − n . Combining this estimate with the uniform estimate on H n − ( ∂ Bdry h ( A )) leads to (1.11).We now refine the strategy above to prove: Lemma 3.2.
Let
K > and A ⊂ R n measurable with P ( A ) = ∞ . Then there exists δ = δ ( K, A ) > with the following property: for all δ ∈ (0 , δ ] it is possible to find adisjoint collection U δ of δ -cubes satisfying: (a) − n − < | Q ′ ∩ A | / | Q ′ | < − − n − for all Q ′ ∈ U δ ; (b) U δ > Kδ − n +1 ; (c) if U δ = { Q δ ( x i ) } i ∈ I , the homothetic cubes { Q δ ( x i ) } i ∈ I are pairwise disjoint.Proof. In this proof we tacitly assume that all cubes have sides parallel to a fixed systemof coordinates. Partition canonically R n in a family { Q i } i ∈ Z n of δ/ -cubes and set V δ/ := (cid:26) Q i : | Q i ∩ A || Q i | > (cid:27) , A δ/ := [ Q i ∈V δ/ Q i . Since A δ/ → A locally in measure as δ → + , it follows from the lower semicontinuity of P that lim inf δ → + P ( A δ/ ) ≥ P ( A ) = ∞ . We define δ = δ ( K, A ) > by requiring that P ( A δ/ ) > n +2 nK for all δ ∈ (0 , δ ] .Fixing now δ ∈ (0 , δ ] and defining ˜ V := (cid:8) Q i ∈ V δ/ : H n − ( ∂Q i ∩ ∂A δ/ ) > (cid:9) L. AMBROSIO, J. BOURGAIN, H. BREZIS, AND A. FIGALLI as the subset of “boundary cubes” (see Figure 1) we can estimate n +2 nK < P ( A δ/ ) ≤ n (cid:18) δ (cid:19) n − V , so that V > n Kδ − n +1 . (3.3) Q i v Q i ‘ Q i ‘‘ ~ Figure 1.
Let Q i ∈ ˜ V and let Q ′ i be a δ/ -cube sharing a face σ ⊂ ∂Q i ∩ ∂A δ/ with Q i (see Figure1). Since obviously Q ′ i / ∈ V δ/ we obtain | Q ′ i ∩ A | ≤ | Q ′ i | / . Hence, if Q ′′ i is any δ -cubecontaining Q i ∪ Q ′ i we have | Q ′′ i ∩ A | ≥ | Q i ∩ A | > | Q i | = 12 n +1 | Q ′′ i | , | Q ′′ i ∩ A c | ≥ | Q ′ i ∩ A c | ≥ | Q ′ i | = 12 n +1 | Q ′′ i | . It then suffices to consider a maximal subfamily V ∗ ⊂ ˜ V of δ/ cubes with centers atmutual distance (along at least one of the coordinate directions) larger or equal than δ/ and define U δ := { Q ′′ i : Q i ∈ V ∗ } . It is easy to check that U δ is a family of δ -cubes whose homothetic enlargements by afactor 2 along their centers are disjoint, so that (c) holds, and that (a) holds as well. Inorder to check (b), we notice that the union of the enlargements by a factor of all cubesin V ∗ contains ˜ V , by the maximality of V ∗ . Hence, from (3.3) we get V ∗ ≥ − n V > Kδ − n +1 . (cid:3) MO -TYPE NORMS RELATED TO THE PERIMETER OF SETS 9 Lemma 3.3.
Let c ∈ (0 , / and A ⊂ (0 , n = Q measurable, with c < | A | < − c . (3.4) Then, there exists ǫ = ǫ ( c , A ) > with the following property: for ǫ ∈ (0 , ǫ ) thereexists a disjoint collection G ǫ of ǫ -cubes contained in (0 , n and satisfying | V ∩ A || V | = 12 ∀ V ∈ G ǫ , (3.5) G ǫ > c ǫ − n +1 , (3.6) with c > depending only on c .Proof. First we choose ǫ ∗ = ǫ ∗ ( c , n ) ∈ (0 , / such that the sets A := A ∩ ( ǫ ∗ , − ǫ ∗ ) n ⊂ A and A := A ∪ [(0 , n \ ( ǫ ∗ , − ǫ ∗ ) n ] ⊃ A satisfy c < | A | , | A | < − c . (3.7)We now extend the set A by periodicity: ˜ A := [ h ∈ Z n A + h, ˜ A c = R n \ ˜ A . Then (3.7) implies Z Q Z Q A ( x ) ˜ A c ( x + z ) dx dz = | A | (1 − | A | ) > c . Hence, we can find a nonzero vector z ∈ Q satisfying | A ∩ ( ˜ A c − z ) | > c . Set now e := z/ | z | , H a := z ⊥ + ae , ˆ A := A ∩ ( ˜ A c − z ) , and A δ := (cid:26) x ∈ ˆ A : | Q r ( y ) ∩ A || Q r ( y ) | > ∀ y ∈ Q r ( x ) , r ∈ (0 , δ ) (cid:27) . Since A δ monotonically converge as δ ↓ to a set containing the set of points of density1 of ˆ A , it follows that | A δ | > c / for δ small enough. Hence, because | A δ | ≤ Z √ n/ −√ n/ H n − ( A δ ∩ H a ) da, we can find a ∈ ( −√ n/ , √ n/ satisfying H n − ( A δ ∩ H a ) > c √ n . (3.8) z H a Q α +z ‘ Q α ‘ Q α Figure 2.
For ǫ ≤ δ , let us consider a canonical division { R α } α ∈ Z n − of H a in ǫ -cubes of dimension n − , and select those cubes R α that satisfy H n − ( A δ ∩ R α ) > H n − ( R α ) / , to build afamily R ǫ . Since [ R α ∈R ǫ R α → A δ ∩ H a in H n − -measure as ǫ → + , we obtain from (3.8) lim sup ǫ → ǫ n − R ǫ > c √ n . (3.9)Out of R ǫ we can build a disjoint collection G ′ ǫ of ǫ -cubes Q ′ α centered at points x α ∈ H a with faces either orthogonal or parallel to z , such that | Q ′ α ∩ ˆ A | > | Q ′ α | , (3.10) G ′ ǫ > c √ n ǫ − n +1 . (3.11)Indeed, (3.10) follows from the definition of A δ , while (3.11) follows by (3.9). It followsfrom (3.10) and the definition of ˆ A that | Q ′ α ∩ A | ≥ | Q ′ α ∩ A | > | Q ′ α | and | ( Q ′ α + z ) ∩ A c | ≥ | ( Q ′ α + z ) ∩ ˜ A c | > | Q ′ α | . Since A does not intersect (0 , n \ ( ǫ ∗ , − ǫ ∗ ) n , if ǫ < ǫ ∗ / √ n we obtain that Q ′ α ⊂ (0 , n .Analogously, since A contains (0 , n \ ( ǫ ∗ , − ǫ ∗ ) n , if ǫ < ǫ ∗ / √ n we obtain that Q ′ α + z ∩ ∂Q = ∅ , which implies that there exists a vector h in Z n such that Q ′ α + z + h ⊂ (0 , n (to be precise, h is of the form − γ e + . . . − γ n e n with γ i ∈ { , } ).Hence, by a continuity argument there exists t α ∈ (0 , such that, setting Q α := Q ′ α + t α ( z + h ) , one has Q α ⊂ (0 , n and | Q α ∩ A | = | Q α | / (see Figure 2, that correspondsto the case h = 0 ). Then we can define G ǫ as the collection of the cubes Q α , which isdisjoint by construction (since their projections on H a are disjoint). (cid:3) MO -TYPE NORMS RELATED TO THE PERIMETER OF SETS 11 We can now prove that lim inf ǫ I ǫ ( f ) ≥ / . Set c = 2 − n − , let c be given byLemma 3.3, and set K := 1 /c . If δ = δ ( A, K ) is given by Lemma 3.2, we can ap-ply Lemma 3.2 to obtain a finite disjoint family U δ of δ -cubes with U δ > c − δ − n and c < | Q ′ ∩ A || Q ′ | < − c for all Q ′ ∈ U δ .Since U δ is finite, for < ǫ ≪ δ and all Q ′ ∈ U δ we can apply Lemma 3.3 to a rescaled copyby a factor δ − of Q ′ and A ∩ Q ′ to obtain a disjoint family G ǫ ( Q ′ ) of ǫ -cubes containedin Q ′ and satisfying | V ∩ A || V | = 12 ∀ V ∈ G ǫ ( Q ′ ) , (3.12) G ǫ ( Q ′ ) > c (cid:18) δǫ (cid:19) n − . (3.13)Now, by construction, the family G ǫ := [ Q ′ ∈U δ G ǫ ( Q ′ ) of ǫ -cubes is disjoint (taking into account condition (c) of Lemma 3.2) and | V ∩ A | / | V | =1 / for each V in the family. In addition, its cardinality can be estimated from below asfollows: G ǫ ≥ c (cid:18) δǫ (cid:19) n − U δ > c (cid:18) δǫ (cid:19) n − c δ − n = ǫ − n . Extracting from G ǫ a subfamily F ǫ with F ǫ = [ ǫ − n ] we get I ǫ ( f ) ≥ ǫ − n X V ∈F ǫ − Z V − Z V | f ( x ) − f ( y ) | dx dy = 2 ǫ n − X V ∈F ǫ | V ∩ A | · | V \ A || V | = 12 ǫ n − [ ǫ − n ] . By taking the limit as ǫ → + the conclusion is achieved.3.3. Lower bound: the rectifiable case.
The heuristic idea of the proof is to choosecubes well adapted to the local geometry of ∂A , as in Figure 3 below. Although it iseasy to make this argument rigorous if ∂A is smooth, when A has merely finite perimeterthe argument becomes much less obvious. Still, the rectifiability of ∂ ∗ A and a suitablelocalization/blow-up argument allow us to prove the result in this general setting.Let A ⊂ R n be measurable and Ω ⊂ R n open. We localize I ǫ ( A ) to Ω and, at the sametime, we impose a scale-invariant bound on the cardinality of the families by defining J ǫ ( A, Ω) := ǫ n − sup F ǫ X Q ′ ∈F ǫ − Z Q ′ (cid:12)(cid:12)(cid:12) A ( x ) − − Z Q ′ A (cid:12)(cid:12)(cid:12) dx, where the supremum runs, this time, among all collections of disjoint families of ǫ -cubescontained in Ω , with arbitrary orientation and cardinality not exceeding P ( A, Ω) ǫ − n . Figure 3.
Notice that J ǫ has a nice scaling property, namely J ǫ ( A/r,
Ω) = r − n J rǫ ( A, r Ω) ∀ r > . (3.14)In addition, the additivity of P ( A, · ) shows that J ǫ is superadditive, namely J ǫ ( A, Ω ∪ Ω ) ≥ J ǫ ( A, Ω ) + J ǫ ( A, Ω ) whenever Ω ∩ Ω = ∅ . (3.15)Then, the lower bound ǫ I ǫ ( A ) ≥ min { , P ( A ) } (3.16)is a direct consequence of Theorem 3.4 below, choosing Ω = R n . Indeed, since P ( A ) ≤ implies J ǫ ( A, R n ) ≤ I ǫ ( A ) we obtain (3.16) when P ( A ) ≤ . If P ( A ) > , let k = [ P ( A )] ≥ be its integer part and split any disjoint family F ǫ of ǫ -cubes with maximal cardinalitywhich enters in the definition of J ǫ ( A, R n ) into k subfamilies with cardinality [ ǫ − n ] anda remainder subfamily of cardinality not exceeding P ( A ) ǫ − n − k [ ǫ − n ] ≤ P ( A ) ǫ − n − k ( ǫ − n − . Since F ǫ is arbitrary, recalling (3.1) we see that J ǫ ( A, R n ) ≤ k I ǫ ( A ) + 12 (cid:0) P ( A ) − k (cid:1) + kǫ n − . Applying once more Theorem 3.4 with
Ω = R n yields ǫ → I ǫ ( A ) ≥ P ( A ) k − ( P ( A ) − k ) k = 1 = min { , P ( A ) } since P ( A ) > . Theorem 3.4.
For any measurable set A ⊂ R n with finite perimeter in Ω one has lim ǫ → + J ǫ ( A, Ω) = 12 P ( A, Ω) . The proof of the upper bound lim sup ǫ J ǫ ( A, Ω) ≤ P ( A, Ω) / can be obtained exactlyas in §3.1, so we focus on the lower bound. To this aim, it will be convenient to introducethe function J − ( A, Ω) := lim inf ǫ → + J ǫ ( A, Ω) . MO -TYPE NORMS RELATED TO THE PERIMETER OF SETS 13 Because of (3.14) we get J − ( A/r,
Ω) = r − n J − ( A, r Ω) ∀ r > . (3.17)In addition, the superadditivity of J ǫ ( A, · ) and of the lim inf give J − ( A, Ω ∪ Ω ) ≥ J − ( A, Ω ) + J − ( A, Ω ) whenever Ω ∩ Ω = ∅ . (3.18)In the first lemma we consider (local) subgraphs of C functions. Lemma 3.5.
Let E be the subgraph of a C function in a neighbourhod of . Then lim inf r → + J − ( E/r, B ) ≥ ω n − . Proof.
The proof is elementary, just choosing the canonical division in ǫ -cubes, if B r ∩ ∂E is contained in a hyperplane for r > small enough. In the general case we use thefact that E/r is bi-Lipschitz equivalent to a half-space in B , with bi-Lipschitz constantsconverging to as r ↓ . (cid:3) In the second lemma we provide a sort of modulus of continuity for E J ǫ ( E, Ω) . Lemma 3.6.
Let
E, F ⊂ Ω be sets of finite perimeter in Ω . Then J ǫ ( F, Ω) ≤ J ǫ ( E, Ω) + 12 ǫ n − + H n − (cid:0) ( ∂ ∗ F ∆ ∂ ∗ E ) ∩ Ω (cid:1) ∀ ǫ > . (3.19) Proof.
The inequality min { z, − z } ≤ z (1 − z ) in [0 , combined with (2.2) yields therelative isoperimetric inequality min {| L | , | Q ′ \ L |} ≤ ǫ P ( L, Q ′ ) for any ǫ -cube Q ′ with L ⊂ Q ′ . (3.20)Let now F ǫ be a family of ǫ -cubes contained in Ω with cardinality less than ǫ − n P ( F, Ω) .For any Q ′ ∈ F ǫ , adding and subtracting E we have − Z Q ′ − Z Q ′ | F ( x ) − F ( y ) | dx dy ≤ − Z Q ′ − Z Q ′ | E ( x ) − E ( y ) | dx dy + ǫ − n | Q ′ ∩ ( F ∆ E ) | . Analogously, adding and subtracting Q ′ \ E and using E c ( x ) − E c ( y ) = E ( y ) − E ( x ) ,we have − Z Q ′ − Z Q ′ | F ( x ) − F ( y ) | dx dy ≤ − Z Q ′ − Z Q ′ | E ( x ) − E ( y ) | dx dy + ǫ − n | Q ′ ∩ ( F ∆ E c ) | . Since F ∆ E = Ω \ ( F ∆ E c ) , we can apply (3.20) with L = Q ′ ∩ ( F ∆ E ) , single out from F ǫ a maximal subfamily with cardinality less than ǫ − n P ( E, Ω) , and use (3.1) and thedefinition of J ǫ ( E, Ω) to get ǫ n − X Q ′ ∈F ǫ − Z Q ′ − Z Q ′ | F ( x ) − F ( y ) | ≤ J ǫ ( E, Ω) + 12 ǫ n − + 12 (cid:0) P ( F, Ω) − P ( E, Ω) (cid:1) + + 12 X Q ′ ∈F ǫ P ( F ∆ E, Q ′ ) . Then, we use the additivity of P ( F ∆ E, · ) and take the supremum in the left hand side toobtain J ǫ ( F, Ω) ≤ J ǫ ( E, Ω) + 12 ǫ n − + 12 (cid:0) P ( F, Ω) − P ( E, Ω) (cid:1) + + 12 P ( F ∆ E, Ω) , and we conclude using (2.4) and (2.6). (cid:3) Notice that, in particular, the previous lemma gives J − ( F, Ω) ≤ J − ( E, Ω) + H n − (cid:0) ( ∂ ∗ F ∆ ∂ ∗ E ) ∩ Ω (cid:1) . (3.21)In the third lemma we prove a density lower bound for J − ( E, · ) by comparing E onsmall scales with the subgraph of a C function. Lemma 3.7. If E has locally finite perimeter in Ω , then lim inf r → + J − ( E, B r ( x )) ω n − r n − ≥ for | D E | -a.e. x ∈ Ω . (3.22) Proof.
Recall that | D E | = H n − ∂ ∗ E on Borel sets of Ω . In view of (3.21), the scalingproperty (3.17) of J − , and Lemma 3.5, it suffices to show that for H n − -a.e. x ∈ ∂ ∗ E there exists a set F which is the subgraph of a C function in the neighbourhood of x ,with H n − (cid:0) ( ∂ ∗ F ∆ ∂ ∗ E ) ∩ B r ( x )) = o ( r n − ) . (3.23)To this aim, we use the representation (2.5), we fix i and consider a point x ∈ Γ i ∩ ∂ ∗ E where H n − (cid:0) (Γ i \ ∂ ∗ E ) ∩ B r ( x ) (cid:1) = o ( r n − ) and H n − (cid:0) ( ∂ ∗ E \ Γ i ) ∩ B r ( x ) (cid:1) = o ( r n − ) . In this way we obtain that (3.23) holds for H n − -a.e. x ∈ Γ i ∩ ∂ ∗ E and the statement isproved, since i is arbitrary and Γ i is a C hypersurface. (cid:3) We can now prove the missing part J − ( E, Ω) ≥ P ( E, Ω) / of Theorem 3.4. Let S ⊂ Ω be the set where the lim inf in (3.22) is greater or equal than / , and notice thatLemma 3.7 shows that | D E | Ω is concentrated on S , so that H n − ( S ) ≥ P ( E, Ω) . If J − ( E, · ) =: µ ( · ) were a σ -additive measure, then the well-known implication lim inf r → + µ ( B r ( x )) ω n − r n − ≥ ∀ x ∈ S = ⇒ µ (Ω) ≥ H n − ( S ) (3.24)would provide us with the needed inequality (see for instance [1, Theorem 2.56] for aproof of (3.24)). However, the traditional proof of (3.24) works also when µ is only asuperadditive set function defined on open sets, as we illustrate below. In particular (3.24)is applicable to J − ( E, · ) in view of (3.18), which concludes the proof of Theorem 3.4. (cid:3) Here we use (first with S = Γ i , then with S = ∂ ∗ E ) the property that H n − ( S ∩ B r ( x )) = o ( r n − ) for H n − -a.e. x ∈ R n \ S whenever S has locally finite H n − -measure, see for instance [1, pag. 79, Eq.(2.41)]. MO -TYPE NORMS RELATED TO THE PERIMETER OF SETS 15 We now give a sketch of proof of (3.24) in the superadditive case, writing k = n − for convenience. We can assume without loss of generality S ⋐ Ω . To prove (3.24) we fix δ ∈ (0 , and consider all the open balls C centered at points of L and with diameter d C strictly less than δ , such that µ ( C ) ≥ (1 − δ ) ω k d k C / k . By applying Besicovitch coveringtheorem (see for instance [1, Theorem 2.17]) we obtain families F , . . . , F ξ (with ξ = ξ ( n ) dimensional constant) with the following properties:(a) each family F i , ≤ i ≤ ξ , is disjoint;(b) ∪ ξ S F i contains S .In particular, using the superadditivity of µ we can estimate from above the pre-Hausdorffmeasure H kδ ( L ) as follows: H kδ ( S ) ≤ ξ X i =1 X C∈F i ω k k d k C ≤ − δ ξ X i =1 X C∈F i µ ( C ) ≤ ξ − δ µ (Ω) . By letting δ ↓ we obtain that H k ( S ) ≤ ξ µ (Ω) < ∞ . Using this information we canimprove the estimate, now applying Besicovitch–Vitali covering theorem to the abovementioned fine cover of S , to obtain a disjoint family {C i } which covers H k -almost all(hence H kδ -almost all) of S . As a consequence H kδ ( S ) ≤ X i ω k k d k C i ≤ X i − δ µ ( C i ) ≤ − δ µ (Ω) . Letting δ ↓ we finally obtain H k ( S ) ≤ µ (Ω) , as desired. (cid:3) Proof of Theorem 1.1 in the case n = 1 . Note that I ǫ ( A ) := sup I − Z I (cid:12)(cid:12)(cid:12) A ( x ) − − Z I A (cid:12)(cid:12)(cid:12) dx, (3.25)and I runs among all intervals with length ǫ . Recall (see for instance [1]) that in the1-dimensional case any set of finite and positive perimeter is equivalent to a finite disjointunion of closed intervals or half-lines, and the perimeter is the number of the endpoints;in addition P ( A ) = 0 if and only if either | A | = 0 or | R \ A | = 0 .The inequalities I ǫ ( A ) ≤ / and I ǫ ( A ) ≤ P ( A ) follow by (3.1) and (2.2) respectively,as in the case n > , hence ǫ I ǫ ( A ) ≤ min { , P ( A ) } . It remains to prove ǫ → I ǫ ( A ) ≥ min { , P ( A ) } and, since P ( A ) is always a natural number (possibly infinite), we need only to show that P ( A ) ≥ implies ǫ I ǫ ( A ) ≥ . We will prove the stronger implication P ( A ) > ⇒ ǫ → I ǫ ( A ) ≥ . (3.26)To prove (3.26), notice that P ( A ) > implies that both A and R \ A have nontrivialmeasure, so there exist distinct points x , x ∈ R such that x is a density point of A and x is a density point of R \ A . Hence, for ǫ sufficiently small we have then − Z ( x − ǫ/ ,x + ǫ/ A > and − Z ( x − ǫ/ ,x + ǫ/ A < We can then use a continuity argument to find, for ǫ sufficiently small, a point x ǫ suchthat − Z ( x ǫ + ǫ/ ,x ǫ + ǫ/ A = 12 , so that − Z ( x ǫ − ǫ/ ,x ǫ + ǫ/ (cid:12)(cid:12)(cid:12) A ( x ) − − Z ( x ǫ − ǫ/ ,x ǫ + ǫ/ A (cid:12)(cid:12)(cid:12) dx = 12 . Hence P ( A ) > implies I ǫ ( A ) ≥ for ǫ small enough, so in particular ǫ I ǫ ( A ) ≥ , as desired. (cid:3) Variants
A localized version of Theorem 1.1.
Let Ω ⊂ R n be a bounded domain withLipschitz boundary and consider the quantity I ǫ ( f, Ω) := ǫ n − sup F ǫ X Q ′ ∈F ǫ − Z Q ′ (cid:12)(cid:12)(cid:12) f ( x ) − − Z Q ′ f (cid:12)(cid:12)(cid:12) dx, (4.1)where F ǫ denotes a collection of disjoint ǫ -cubes Q ′ ⊂ Ω with arbitrary orientation andcardinality not exceeding ǫ − n .In analogy with Theorem 1.1, we can also prove the following result. Theorem 4.1.
For any measurable set A ⊂ R n one has lim ǫ → I ǫ ( A , Ω) = 12 min (cid:8) , P ( A, Ω) (cid:9) . Proof.
We begin by noticing that both the upper and the lower bound in the rectifiablecase are local, so that parts of the proof go throughout without any essential modification.Hence, we only need to discuss the lower bound in the non-rectifiable case.If P ( A, Ω) = ∞ , we can find an open smooth subset Ω ′ ⋐ Ω such that P ( A, Ω ′ ) isarbitrarily large (the largeness will be fixed later). We set c = 2 − n − , consider c begiven by Lemma 3.3, and set K := 1 /c . Then, by looking at the proof of Lemma 3.2it is immediate to check that the same result still holds with K = 1 /c and consideringonly δ -cubes which intersect Ω ′ (this ensures that, if δ is sufficiently small, all cubes arecontained inside Ω ) provided P ( A ; Ω ′ ) > n +2 nK . Thanks to this fact, the proof at theend of Section 3.2 now goes through without modifications: first we apply Lemma 3.2 tofind a disjoint family U δ of δ -cubes intersecting Ω ′ with U δ > c − δ − n and c < | Q ′ ∩ A || Q ′ | < − c for all Q ′ ∈ U δ , MO -TYPE NORMS RELATED TO THE PERIMETER OF SETS 17 and then we apply Lemma 3.3 with ǫ ≪ δ to obtain, for each Q ′ ∈ U δ , a disjoint family G ǫ ( Q ′ ) of ǫ -cubes satisfying | V ∩ A || V | = 12 ∀ V ∈ G ǫ ( Q ′ ) , G ǫ ( Q ′ ) > c (cid:18) δǫ (cid:19) n − . We then conclude as in Section 3.2. (cid:3)
Next, we return to the quantity [ f ] defined in [6]. As announced in the introduction,we establish the following result: Corollary 4.2.
For any measurable set A ⊂ Q one has [ A ] ≤
12 min (cid:8) , P ( A, Q ) (cid:9) ≤ C [ A ] . Proof.
The result is an immediate consequence of Theorem 4.1 and Lemma 4.3 below,where we compare the quantities I ǫ ( f ) with their anisotropic counterparts [ f ] ǫ as definedin [6]. (cid:3) Lemma 4.3.
For ǫ > and f ∈ L ( Q ) measurable, let I ǫ ( f, Q ) be defined as in (1.7) withcubes Q ′ ⊂ Q and let [ f ] ǫ be defined by (1.1) . Then [ f ] ǫ ≤ I ǫ ( f, Q ) ≤ C [ f ] √ nǫ ∀ ǫ ∈ (0 , / √ n ) . (4.2)In the proof of this result, we shall to use the elementary inequalities − Z Q (cid:12)(cid:12)(cid:12) f ( x ) − − Z Q f (cid:12)(cid:12)(cid:12) dx ≤ − Z Q − Z Q | f ( x ) − f ( y ) | dx dy ≤ − Z Q (cid:12)(cid:12)(cid:12) f ( x ) − − Z Q f (cid:12)(cid:12) dx. (4.3) Proof.
The first inequality in (4.2) is obvious. In order to prove the second one, let F ǫ = { Q i } i ∈ I be a disjoint family of ǫ -cubes in Q with cardinality of I less than ǫ − n . Foreach cube Q i in F ǫ we can find a √ nǫ -cube Q ′ i containing Q i , contained in Q , and withsides parallel to the coordinate axes. Since F ǫ is disjoint, the family of the correspondingcubes Q ′ i has bounded overlap, more precisely for each cube Q ′ i the cardinality of theset { j : Q ′ j ∩ Q ′ i = ∅} does not exceed c n . Hence, by an exhaustion procedure, we canpartition the index set I in families I , . . . , I N , with N ≤ c n , in such a way that thefamilies G ′ j := { Q ′ i : i ∈ I j } j = 1 , . . . , N are disjoint. Since G ′ j ≤ ǫ − n for each j = 1 , . . . , N , splitting the family G ′ j in at most n ( n − / + 1 subfamilies with cardinality less than ( √ nǫ ) − n , we have ( √ nǫ ) − n X Q ′ ∈G ′ j − Z Q ′ (cid:12)(cid:12)(cid:12) f ( x ) − − Z Q ′ f (cid:12)(cid:12)(cid:12) dx ≤ (cid:0) n ( n − / + 1 (cid:1) [ f ] √ nǫ . On the other hand, if G j , j = 1 , . . . , N , denote the corresponding families of originalcubes, since − Z Q − Z Q | f ( x ) − f ( y ) | dx dy ≤ ( √ n ) n − Z Q ′ − Z Q ′ | f ( x ) − f ( y ) | dx dy, using (4.3) we readily obtain X Q ∈G j − Z Q (cid:12)(cid:12)(cid:12) f ( x ) − − Z Q f (cid:12)(cid:12)(cid:12) dx ≤ n n X Q ′ ∈G ′ j − Z Q ′ (cid:12)(cid:12)(cid:12) f ( x ) − − Z Q ′ f (cid:12)(cid:12) dx. Hence, since F ǫ = G ∪ · · · ∪ G N , adding with respect to j and using the fact that F ǫ isarbitrary, we obtain the second inequality in (4.2) with C := 2 c n ( n ( n − / + 1) n (3 n − / . (cid:3) Remark 4.4.
Notice that Corollary 4.2 could be refined by adapting the argument used inSection 3.3 to the setting of [6] where the cubes are forced to be parallel to the coordinateaxes: more precisely, if we denote by K n the largest possible intersection of a hyperplanewith the unit cube, i.e., K n := sup H ⊂ R n hyperplane H n − ( H ∩ Q ) , then [ A ] ≥
12 min n , K n P ( A, Q ) o . A new characterization of the perimeter.
Theorem 1.1 provides a characteri-zation of sets of finite perimeter only when lim ǫ → I ǫ ( A ) < / . This critical thresholdcould be easily tuned by modifying the upper bound on the cardinality of the families F ǫ in (1.7), as (1.9) shows. As a consequence a byproduct of our results is a characterizationof the perimeter free of truncations: lim ǫ ↓ sup H ǫ ǫ n − X Q ′ ∈H ǫ − Z Q ′ (cid:12)(cid:12)(cid:12) A ( x ) − − Z Q ′ A (cid:12)(cid:12)(cid:12) dx = P ( A ) , (4.4)where now H ǫ denotes a collection of disjoint ǫ -cubes Q ′ ⊂ R n with arbitrary orientationbut no constraint on cardinality.In order to prove (4.4) we notice that the argument in Section 3.1, based on the relativeisoperimetric inequality, easily gives ǫ n − X Q ′ ∈H ǫ − Z Q ′ (cid:12)(cid:12)(cid:12) A ( x ) − − Z Q ′ A (cid:12)(cid:12)(cid:12) dx ≤ P ( A ) . MO -TYPE NORMS RELATED TO THE PERIMETER OF SETS 19 On the other hand, we can use (1.9) to get lim inf ǫ ↓ sup H ǫ ǫ n − X Q ′ ∈H ǫ − Z Q ′ (cid:12)(cid:12)(cid:12) A ( x ) − − Z Q ′ A (cid:12)(cid:12)(cid:12) dx ≥ sup M> min { M, P ( A ) } = P ( A ) , proving (4.4).Notice that the formulation given in Theorem 1.1 is stronger than (4.4), because itshows that a cardinality constrained maximization is sufficient to provide finiteness ofperimeter, under the critical threshold.It is also worth noticing that (4.4) can be extended to general Z -valued functions:indeed, the argument in Section 3.3 can be easily adapted to prove that if f ∈ BV ( R n ; Z ) then lim inf r → J − ( f, B r ( x )) | Df | ( B r ( x )) ≥ for | Df | -a.e. x ,where J − ( f ; B r ( x )) is defined analogously to J − ( E ; B r ( x )) in Section 3.3, thus showingthat lim inf ǫ ↓ sup H ǫ ǫ n − X Q ′ ∈H ǫ − Z Q ′ (cid:12)(cid:12)(cid:12) f ( x ) − − Z Q ′ f (cid:12)(cid:12)(cid:12) dx ≥ | Df | ( R n ) , while the converse inequality follows by writing f = X k> { f ≥ k } − X k> { f ≤− k } , which gives sup H ǫ ǫ n − X Q ′ ∈H ǫ − Z Q ′ (cid:12)(cid:12)(cid:12) f ( x ) − − Z Q ′ f (cid:12)(cid:12)(cid:12) dx ≤ X k> sup H ǫ ǫ n − X Q ′ ∈H ǫ − Z Q ′ (cid:12)(cid:12)(cid:12) { f ≥ k } ( x ) − − Z Q ′ { f ≥ k } (cid:12)(cid:12)(cid:12) dx + X k> sup H ǫ ǫ n − X Q ′ ∈H ǫ − Z Q ′ (cid:12)(cid:12)(cid:12) { f ≤− k } ( x ) − − Z Q ′ { f ≤− k } (cid:12)(cid:12)(cid:12) dx ≤ X k> P ( { f ≥ k } ) + X k> P ( { f ≤ − k } ) = | Df | ( R n ) . These facts provide a new characterization both of sets of finite perimeter and of theperimeter of sets, independent of the theory of distributions. Heuristically, given ǫ > ,any maximizing family F ǫ provides a sort of boundary on scale ǫ of A , and some proofs(in particular the one of Lemma 3.2, see also Remark 3.1) make more rigorous this idea.It is interesting also to compare this result with another non-distributional character-ization of sets of finite perimeter due to H. Federer, see [12, Theorem 4.5.11]: P ( A ) isfinite if and only if the essential boundary ∂ ∗ A , namely the set in (2.3) of points of den-sity neither 0 nor 1, has finite H n − -measure, and then P ( A ) = H n − ( ∂ ∗ A ) . However,Federer’s characterization and the one provided by this paper seem to be quite different. Approximation of the total variation.
Motivated by the results in Section 4.2,for f ∈ L ( R n ) we may define K ǫ ( f ) := sup H ǫ ǫ n − X Q ′ ∈H ǫ − Z Q ′ (cid:12)(cid:12)(cid:12) f ( x ) − − Z Q ′ f (cid:12)(cid:12)(cid:12) dx where, once more, H ǫ denotes a collection of disjoint ǫ -cubes Q ′ ⊂ R n with arbitraryorientation but no constraint on cardinality. Then, the result of the previous section canbe read as follows: lim ǫ → K ǫ ( f ) = | Df | ( R n ) for any Z -valued function f .For general BV loc functions f , the asymptotic analysis of K ǫ seems to be more difficultto grasp. By considering smooth functions and functions with a jump discontinuity alonga hyperplane, one is led to the conjecture that lim ǫ → K ǫ ( f ) = 14 | D a f | ( R n ) + 12 | D s f | ( R n ) (4.5)for all f ∈ SBV loc ( R n ) , where (see [1]) SBV loc (Ω) is the vector space of all f ∈ BV loc (Ω) whose distributional derivative is the sum of a measure D a f absolutely continuous w.r.t. L n and a measure D s f concentrated on a set σ -finite w.r.t. H n − . For functions f ∈ BV loc \ SBV loc , having the so-called Cantor part of the derivative, it might possibly happenthat K ǫ ( f ) oscillates as ǫ → between | Df | ( R n ) and | Df | ( R n ) .Notice that all functionals K ǫ are L ( R n ) -lower semicontinuous. On the other hand,it is natural to expect that the Γ -limit of K ǫ w.r.t. the L ( R n ) topology exists and that Γ − lim ǫ → K ǫ ( f ) = 14 | Df | ( R n ) . Appendix: proof of (2.2)In this section we prove the relative isoperimetric inequality in the cube, in the sharpform provided by (2.2), in any Euclidean space R n , n ≥ . To this aim, we introduce theGaussian isoperimetric function I : (0 , → (0 , / √ π ] defined by I ( t ) := ϕ ◦ Φ − ( t ) with ϕ ( x ) = e − x / √ π , Φ( x ) := Z x −∞ ϕ ( y ) dy, x ∈ R . We extend I by continuity to [0 , setting I (0) = I (1) = 0 . Notice that I (1 /
2) = ϕ (0) =1 / √ π and it is also easy to check that I ( t ) = I (1 − t ) . Lemma 5.1.
The function K ( t ) := √ πI ( t ) − t (1 − t ) is nonnegative in [0 , and K ( t ) = 0 if and only if t ∈ { , / , } .Proof. Let us record an additional property of I : I ∈ C ∞ (0 , and I ′′ = − I on (0 , . (5.1) MO -TYPE NORMS RELATED TO THE PERIMETER OF SETS 21 Indeed, from I ◦ Φ = ϕ and Φ ′ = φ we obtain ( I ′ ◦ Φ( x )) ϕ ( x ) = ϕ ′ ( x ) = − xϕ ( x ) , so that I ′ ◦ Φ( x ) = − x . By differentiating once more we get I ′′ ( t ) = − / Φ ′ ◦ Φ − ( t ) = − /I ( t ) in (0 , , as desired.Since I attains its maximum at / , from (5.1) we obtain that I ′ > in (0 , / , hencethere exists a unique t ∈ (0 , / such that I ( t ) = √ π/ .Since K vanishes on { , / , } and it inherits from I the symmetry property K ( t ) = K (1 − t ) , it suffices to check that K is strictly positive in (0 , / . Differentiating K weget K ′ ( t ) = √ πI ′ ( t ) − t, (5.2)In particular, K ′ (1 /
2) = I ′ (1 /
2) = 0 . Differentiating once more and using (5.1) we get K ′′ ( t ) = √ πI ′′ ( t ) + 8 = − √ πI ( t ) + 8 , (5.3)so that K ′′ < in (0 , t ) , K ′′ > in ( t , / . (5.4)In particular K ′ (1 /
2) = 0 gives K ′ < in [ t , / and therefore K (1 /
2) = 0 gives
K > in [ t , / . (5.5)For the interval [0 , t ] we use K ( t ) > , K (0) = 0 and the concavity of K in (0 , t ) ,ensured by (5.4), to get K > in (0 , t ] . (5.6)The conclusion follows by (5.5) and (5.6). (cid:3) Now, let us prove (2.2). Combining [3, Proposition 5 and Theorem 7] we obtain theinequality I (cid:18)Z (0 , n f dx (cid:19) ≤ Z (0 , n (cid:2) I ( f ) + 1 √ π |∇ f | (cid:3) dx (5.7)for any locally Lipschitz function f : (0 , n → [0 , . Then, the BV version of theMeyers-Serrin approximation theorem (due to Anzellotti-Giaquinta, see for instance [1,Theorem 3.9]) enables us to approximate in L ((0 , n ) any function f ∈ BV ((0 , n ) byfunctions f k ∈ C ∞ ((0 , n ) in such a way that lim k →∞ Z (0 , n |∇ f k | dx = | Df | (cid:0) (0 , n (cid:1) . In addition, if f : (0 , n → [0 , , a simple truncation argument provides approximatingfunctions f k with the same property. It then follows from (5.7) that I (cid:18)Z (0 , n f dx (cid:19) ≤ Z (0 , n I ( f ) dx + 1 √ π | Df | (cid:0) (0 , n (cid:1) for all f : (0 , n → [0 , with bounded variation. Since I (0) = I (1) = 0 , choosing f = E gives I ( | E | ) ≤ √ π P (cid:0) E, (0 , n (cid:1) . We conclude using Lemma 5.1.
Acknowledgements.
The first author (LA) was partially supported by the ERC ADGproject GeMeThNES, the second author (JB) was partially supported by NSF grant DMS-1301619, the third author (HB) was partially supported by NSF grant DMS-1207793 andby grant number 238702 of the European Commission (ITN, project FIRST), the fourthauthor (AF) was partially supported by NSF grant DMS-1262411.The authors thank B. Kawohl and F. Barthe for very useful information about therelative isoperimetric inequality (2.2).
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