Fractional Perimeters from a Fractal Perspective

Luca Lombardini

doi:10.1515/ans-2018-2016

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Fractional Perimeters from a Fractal Perspective

Published/Copyright: June 13, 2018

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From the journal Advanced Nonlinear Studies Volume 19 Issue 1

Abstract

The purpose of this paper consists in a better understanding of the fractional nature of the nonlocal perimeters introduced in [L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 2010, 9, 1111–1144]. Following [A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math. 8 1991, 2, 175–201], we exploit these fractional perimeters to introduce a definition of fractal dimension for the measure theoretic boundary of a set. We calculate the fractal dimension of sets which can be defined in a recursive way, and we give some examples of this kind of sets, explaining how to construct them starting from well-known self-similar fractals. In particular, we show that in the case of the von Koch snowflake S⊆ℝ2 this fractal dimension coincides with the Minkowski dimension. We also obtain an optimal result for the asymptotics as s→1- of the fractional perimeter of a set having locally finite (classical) perimeter.

Keywords: 28A80; 35R11; 49Q05

MSC 2010: Nonlocal Minimal Surfaces; Fractal Dimensions; Fractional Operators

1 Introduction and Main Results

The s-fractional perimeter and its minimizers, the s-minimal sets, were introduced in [5] in 2010 and since then they have attracted a lot of interest, especially concerning the regularity theory of the boundaries of the s-minimal sets, which are the so-called nonlocal minimal surfaces. We refer the interested reader to the recent survey [11] and the references cited therein.

Even if finding the optimal regularity of nonlocal minimal surfaces is still an engaging open problem, it is known that nonlocal minimal surfaces are (n-1)-rectifiable. More precisely, they are smooth, except possibly for a singular set of Hausdorff dimension at most equal to n-3 (see [5, 20, 15]). In particular, an s-minimal set has (locally) finite perimeter (in the sense of De Giorgi and Caccioppoli).

On the other hand, the boundary of a generic set E having finite s-perimeter can be very irregular and indeed it can be “nowhere rectifiable”, like in the case of the von Koch snowflake.

Actually, the s-perimeter can be used (following the seminal paper [22]) to define a “fractal dimension” for the measure theoretic boundary

∂ - ⁡ E := { x ∈ ℝ n ∣ 0 < | E ∩ B r ⁢ ( x ) | ⁢ < ω n ⁢ r n ⁢ for every ⁢ r > ⁢ 0 }

of a set E⊆ℝn.

Before going on, it is useful to recall the definition of the s-perimeter. Given a fractional parameter s∈(0,1), we define the interaction

ℒ s ⁢ ( A , B ) := ∫ A ∫ B 1 | x - y | n + s ⁢ 𝑑 x ⁢ 𝑑 y

for every couple of disjoint sets A,B⊆ℝn. Then the s-fractional perimeter of a set E⊆ℝn in an open set Ω is defined by

P s ⁢ ( E , Ω ) := ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ E ∩ Ω ) + ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ E ∖ Ω ) + ℒ s ⁢ ( E ∖ Ω , 𝒞 ⁢ E ∩ Ω ) .

We observe that we can rewrite the s-perimeter as

(1.1) P s ⁢ ( E , Ω ) = 1 2 ⁢ ∫ ∫ ℝ 2 ⁢ n ∖ ( 𝒞 ⁢ Ω ) 2 | χ E ⁢ ( x ) - χ E ⁢ ( y ) | | x - y | n + s ⁢ 𝑑 x ⁢ 𝑑 y .

Formula (1.1) shows that the fractional perimeter is, roughly speaking, the Ω-contribution to the Ws,1-seminorm of the characteristic function χE.

This functional is nonlocal, in that we need to know the set E in the whole of ℝn even to compute its s-perimeter in a small bounded domain Ω (contrary to what happens with the classical perimeter or the ℋn-1 measure, which are local functionals). Moreover, the s-perimeter is “fractional”, in the sense that the Ws,1-seminorm measures a fractional order of regularity.

The main purpose of this paper consists in clarifying and better understanding the “fractional” nature of the s-perimeter.

In 1991, Visintin [22] suggested using the index s of the fractional seminorm [χE]Ws,1⁢(Ω) (and more general continuous families of functionals satisfying appropriate generalized coarea formulas) as a way to measure the codimension of the measure theoretic boundary ∂-⁡E of the set E in Ω. He proved that the fractal dimension obtained in this way,

Dim F ⁡ ( ∂ - ⁡ E , Ω ) := n - sup ⁡ { s ∈ ( 0 , 1 ) ∣ [ χ E ] W s , 1 ⁢ ( Ω ) < ∞ }

is less than or equal to the (upper) Minkowski dimension.

The relationship between the Minkowski dimension of the boundary of E and the fractional regularity (in the sense of Besov spaces) of the characteristic function χE was investigated also in [21] in 1999. In particular, in [21, Remark 3.10], Sickel proved that the dimension DimF of the von Koch snowflake S coincides with its Minkowski dimension, exploiting the fact that S is a John domain.

The Sobolev regularity of a characteristic function χE was further studied in [14], in 2013, where Faraco and Rogers considered the case in which the set E is a quasiball. Since the von Koch snowflake S is a typical example of a quasiball, the authors were able to prove that the dimension DimF of S coincides with its Minkowski dimension.

In this paper, we compute the dimension DimF of the von Koch snowflake S in an elementary way, using only the roto-translation invariance and the scaling property of the s-perimeter and the “self-similarity” of S.

The proof can be extended in a natural way to all sets which can be defined in a recursive way similar to that of the von Koch snowflake. As a consequence, we compute the dimension DimF of all such sets, without having to require them to be John domains or quasiballs.

Furthermore, we show that we can easily obtain a lot of sets of this kind by appropriately modifying well-known self-similar fractals like, e.g., the von Koch snowflake, the Sierpinski triangle and the Menger sponge. An example is depicted in Figure 1.

$Figure 1 Example of a “fractal” set constructed by exploiting thestructure of the Sierpinski triangle (seen at the fourth iterative step).$

Figure 1

Example of a “fractal” set constructed by exploiting thestructure of the Sierpinski triangle (seen at the fourth iterative step).

The previous discussion shows that the s-perimeter of a set E with an irregular, eventually fractal, boundary can be finite for s below some threshold s<σ, and infinite for s∈(σ,1). On the other hand, it is well known that sets with a regular boundary have finite s-perimeter for every s and actually their s-perimeter converges, as s tends to 1, to the classical perimeter, both in the classical sense (see [6]) and in the Γ-convergence sense (see [2, 19] for related results).

In this paper, we exploit [7, Theorem 1] to prove an optimal version of this asymptotic property for a set E having finite classical perimeter in a bounded open set with Lipschitz boundary. More precisely, we prove that if E has finite classical perimeter in a neighborhood of Ω, then

lim s → 1 ⁡ ( 1 - s ) ⁢ P s ⁢ ( E , Ω ) = ω n - 1 ⁢ P ⁢ ( E , Ω ¯ ) .

We observe that we lower the regularity requested in [6], where Caffarelli and Valdinoci required the boundary ∂⁡E to be C1,α, to the optimal regularity (requiring E to have only finite perimeter). Moreover, we do not have to require E to intersect ∂⁡Ω “transversally”, i.e. we do not require

ℋ n - 1 ⁢ ( ∂ * ⁡ E ∩ ∂ ⁡ Ω ) = 0 ,

with ∂*⁡E denoting the reduced boundary of E.

Indeed, we prove that the “nonlocal part” of the s-perimeter converges to the perimeter on the boundary of Ω, i.e. we prove that

lim s → 1 ⁡ ( 1 - s ) ⁢ P s N ⁢ L ⁢ ( E , Ω ) = ω n - 1 ⁢ ℋ n - 1 ⁢ ( ∂ * ⁡ E ∩ ∂ ⁡ Ω ) ,

which is, to the best of the author’s knowledge, a new result.

Now we give precise statements of the results obtained, starting with the fractional analysis of fractal dimensions.

1.1 Fractal Boundaries

We observe that we can split the fractional perimeter as the sum

P s ⁢ ( E , Ω ) = P s L ⁢ ( E , Ω ) + P s N ⁢ L ⁢ ( E , Ω ) ,

where

P s L ⁢ ( E , Ω ) := ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ E ∩ Ω ) = 1 2 ⁢ [ χ E ] W s , 1 ⁢ ( Ω ) ,

P s N ⁢ L ⁢ ( E , Ω ) := ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ E ∖ Ω ) + ℒ s ⁢ ( E ∖ Ω , 𝒞 ⁢ E ∩ Ω ) .

We can think of PsL⁢(E,Ω) as the local part of the fractional perimeter, in the sense that if |(E⁢Δ⁢F)∩Ω|=0, then PsL⁢(F,Ω)=PsL⁢(E,Ω).

We sometimes refer to PsN⁢L⁢(E,Ω) as the nonlocal part of the s-perimeter.

We say that a set E has locally finite s-perimeter if it has finite s-perimeter in every bounded open set Ω⊆ℝn.

When Ω=ℝn, we simply write

P s ⁢ ( E ) := P s ⁢ ( E , ℝ n ) = 1 2 ⁢ [ χ E ] W s , 1 ⁢ ( ℝ n ) .

First of all, we prove in Section 3.1 that in some sense the measure theoretic boundary ∂-⁡E is the “right definition” of boundary for working with the s-perimeter.

To be more precise, we show that

∂ - ⁡ E = { x ∈ ℝ n ∣ P s L ⁢ ( E , B r ⁢ ( x ) ) > 0 ⁢ for all ⁢ r > 0 }

and that if Ω is a connected open set, then

P s L ⁢ ( E , Ω ) > 0 if and only if ∂ - ⁡ E ∩ Ω ≠ ∅ .

This can be thought of as an analogue in the fractional framework of the fact that for a Caccioppoli set E we have ∂-⁡E=supp⁡|D⁢χE|.

Now the idea of the definition of the fractal dimension consists in using the index s of PsL⁢(E,Ω) to measure the codimension of ∂-⁡E∩Ω,

Dim F ⁡ ( ∂ - ⁡ E , Ω ) := n - sup ⁡ { s ∈ ( 0 , 1 ) ∣ P s L ⁢ ( E , Ω ) < ∞ } .

As shown in [22, Propositions 11 and 13], the fractal dimension DimF defined in this way is related to the (upper) Minkowski dimension (whose precise definition we recall in Definition 3.4) by

(1.2) Dim F ⁡ ( ∂ - ⁡ E , Ω ) ≤ Dim ¯ ℳ ⁢ ( ∂ - ⁡ E , Ω ) .

For the convenience of the reader we provide a proof of inequality (1.2) in Proposition 3.6.

If Ω is a bounded open set with Lipschitz boundary, (1.2) means that

P s ⁢ ( E , Ω ) < ∞ for every ⁢ s ∈ ( 0 , n - Dim ¯ ℳ ⁢ ( ∂ - ⁡ E , Ω ) )

since the nonlocal part of the s-perimeter of any set E⊆ℝn is

P s N ⁢ L ⁢ ( E , Ω ) ≤ 2 ⁢ P s ⁢ ( Ω ) < ∞ for every ⁢ s ∈ ( 0 , 1 ) .

We show that for the von Koch snowflake (1.2) is actually an equality.

Namely, we prove the following theorem.

Theorem 1.1 (Fractal Dimension of the von Koch Snowflake).

Let S⊆R2 be the von Koch snowflake. Then

(1.3) P s ⁢ ( S ) < ∞ for all ⁢ s ∈ ( 0 , 2 - log ⁡ 4 log ⁡ 3 ) ,

and

(1.4) P s ⁢ ( S ) = ∞ for all ⁢ s ∈ [ 2 - log ⁡ 4 log ⁡ 3 , 1 ) .

Therefore,

Dim F ⁡ ( ∂ ⁡ S ) = Dim ℳ ⁡ ( ∂ ⁡ S ) = log ⁡ 4 log ⁡ 3 .

Actually, exploiting the self-similarity of the von Koch curve, we have

Dim F ⁡ ( ∂ ⁡ S , Ω ) = log ⁡ 4 log ⁡ 3

for every Ω such that ∂⁡S∩Ω≠∅. In particular, this is true for every Ω=Br⁢(p) with p∈∂⁡S and r>0 as small as we want.

We remark that this represents a deep difference between the classical and the fractional perimeter. Indeed, if a set E has (locally) finite perimeter, then by De Giorgi’s structure theorem we know that its reduced boundary ∂*⁡E is locally (n-1)-rectifiable. Moreover, ∂*⁡E¯=∂-⁡E, so the reduced boundary is, in some sense, a “big” portion of the measure theoretic boundary.

On the other hand, we have seen that there are (open) sets, like the von Koch snowflake, which have a “nowhere rectifiable” boundary (meaning that ∂-⁡E∩Br⁢(p) is not (n-1)-rectifiable for every p∈∂-⁡E and r>0) and still have finite s-perimeter for every s∈(0,σ0).

1.1.1 Self-similar Fractal Boundaries

Our argument for the von Koch snowflake is quite general and can be adapted to compute the dimension DimF of all sets which can be constructed in a similar recursive way.

To be more precise, we start with a bounded open set T0⊆ℝn with finite perimeter P⁢(T0)<∞, which is, roughly speaking, our basic “building block”.

Then we go on inductively by adding roto-translations of a scaling of the building block T0, i.e. sets of the form

T k i = F k i ⁢ ( T 0 ) := ℛ k i ⁢ ( λ - k ⁢ T 0 ) + x k i ,

where λ>1, k∈ℕ, 1≤i≤a⁢bk-1, with a,b∈ℕ, ℛki∈SO⁢(n) and xki∈ℝn. We require that these sets do not overlap, i.e.

| T k i ∩ T h j | = 0 whenever ⁢ i ≠ j ⁢ or ⁢ k ≠ h .

Then we define

(1.5) T k := ⋃ i = 1 a ⁢ b k - 1 T k i and T := ⋃ k = 1 ∞ T k .

The final set E is either

E := T 0 ∪ ⋃ k ≥ 1 ⋃ i = 1 a ⁢ b k - 1 T k i or E := T 0 ∖ ( ⋃ k ≥ 1 ⋃ i = 1 a ⁢ b k - 1 T k i ) .

For example, the von Koch snowflake is obtained by adding pieces. Examples obtained by removing the Tki are the middle Cantor set E⊆ℝ, the Sierpinski triangle E⊆ℝ2 and the Menger sponge E⊆ℝ3.

We will consider just the set T and exploit the same argument used for the von Koch snowflake to compute the fractal dimension related to the s-perimeter. However, we observe that the Cantor set, the Sierpinski triangle and the Menger sponge are such that |E|=0, i.e. |T0⁢Δ⁢T|=0. Therefore, neither the perimeter nor the s-perimeter can detect the fractal nature of the (topological) boundary of T, and indeed, since

P ⁢ ( T ) = P ⁢ ( T 0 ) < ∞ ,

we have Ps⁢(T)<∞ for every s∈(0,1).

For example, in the case of the Sierpinski triangle, T0 is an equilateral triangle and ∂-⁡T=∂⁡T0, even if ∂⁡T is a self-similar fractal.

The reason of this situation is that the fractal object is the topological boundary of T, while the s-perimeter “measures” the measure theoretic boundary, which is regular. Roughly speaking, the problem is that in these cases there is not room enough to find a small ball Bki=Fki⁢(B)⊆𝒞⁢T near each piece Tki.

Therefore, we will make the additional assumption that

(1.6) there exists ⁢ S 0 ⊆ 𝒞 ⁢ T ⁢ such that ⁢ | S 0 | > 0 ⁢ and ⁢ S k i := F k i ⁢ ( S 0 ) ⊆ 𝒞 ⁢ T ⁢ for all ⁢ k , i .

We remark that it is not necessary to require that these sets do not overlap.

Theorem 1.2.

Let T⊆Rn be a set which can be written as in (1.5). If log⁡blog⁡λ∈(n-1,n) and (1.6) holds true, then

P s ⁢ ( T ) < ∞ for all ⁢ s ∈ ( 0 , n - log ⁡ b log ⁡ λ ) ,

and

P s ⁢ ( T ) = ∞ for all ⁢ s ∈ [ n - log ⁡ b log ⁡ λ , 1 ) .

Thus,

Dim F ⁡ ( ∂ - ⁡ T ) = log ⁡ b log ⁡ λ .

Furthermore, we show how to modify self-similar sets like the Sierpinski triangle, without altering their “structure”, to obtain new sets which satisfy the hypothesis of Theorem 1.2 (see Remark 3.10 and the final part of Section 3.4). An example is given in Figure 1 above.

However, we also remark that the measure theoretic boundary of such a new set will look quite different from the original fractal (topological) boundary, and in general it will be a mix of smooth parts and unrectifiable parts.

The most interesting examples of this kind of sets are probably represented by bounded sets because in this case the measure theoretic boundary does indeed have, in some sense, a “fractal nature” (see Remark 3.11). Indeed, if T is bounded, then its boundary ∂-⁡T is compact. Nevertheless, it has infinite (classical) perimeter and actually ∂-⁡T has Minkowski dimension strictly greater than n-1, thanks to (1.2).

However, even unbounded sets can have an interesting behavior. Indeed, we obtain the following result.

Proposition 1.3.

Let n≥2. For every σ∈(0,1) there exists a Caccioppoli set E⊆Rn such that

P s ⁢ ( E ) < ∞ for all ⁢ s ∈ ( 0 , σ ) 𝑎𝑛𝑑 P s ⁢ ( E ) = ∞ for all ⁢ s ∈ [ σ , 1 ) .

Roughly speaking, the interesting thing about this proposition is the following. Since E has locally finite perimeter, χE∈BVloc⁢(ℝn), it also has locally finite s-perimeter for every s∈(0,1), but the global perimeter Ps⁢(E) is finite if and only if s<σ<1.

1.2 Asymptotics as s→1-

In Section 1.1, we have shown that sets with an irregular, eventually fractal, boundary can have finite s-perimeter.

On the other hand, if the set E is “regular”, then it has finite s-perimeter for every s∈(0,1). Indeed, if Ω⊆ℝn is a bounded open set with Lipschitz boundary (or Ω=ℝn), then BV⁢(Ω)↪Ws,1⁢(Ω). As a consequence of this embedding, we obtain that

P ⁢ ( E , Ω ) < ∞ implies P s ⁢ ( E , Ω ) < ∞ for every ⁢ s ∈ ( 0 , 1 ) .

Actually, we can be more precise and obtain a sort of converse, using only the local part of the s-perimeter and adding the condition

lim inf s → 1 - ⁡ ( 1 - s ) ⁢ P s L ⁢ ( E , Ω ) < ∞ .

Indeed, one has the following result, which is a combination of [4, Theorem 3’] and [7, Theorem 1], restricted to characteristic functions.

Theorem 1.4.

Let Ω⊆Rn be a bounded open set with Lipschitz boundary. Then E⊆Rn has finite perimeter in Ω if and only if PsL⁢(E,Ω)<∞ for every s∈(0,1), and

(1.7) lim inf s → 1 ⁡ ( 1 - s ) ⁢ P s L ⁢ ( E , Ω ) < ∞ .

In this case, we have

(1.8) lim s → 1 ⁡ ( 1 - s ) ⁢ P s L ⁢ ( E , Ω ) = n ⁢ ω n 2 ⁢ K 1 , n ⁢ P ⁢ ( E , Ω ) .

We briefly show how to get this result (and in particular why the constant looks like that) from the two theorems cited above. Then we compute the constant K1,n in an elementary way, proving that

n ⁢ ω n 2 ⁢ K 1 , n = ω n - 1 .

Moreover, we show the following remark.

Remark 1.5.

Condition (1.7) is necessary. Indeed, there exist bounded sets (see Example 1.6) having finite s-perimeter for every s∈(0,1) which do not have finite perimeter. This also shows that in general the inclusion

BV ⁢ ( Ω ) ⊆ ⋂ s ∈ ( 0 , 1 ) W s , 1 ⁢ ( Ω )

is strict.

Example 1.6.

Let 0<a<1 and consider the open intervals Ik:=(ak+1,ak) for every k∈ℕ. Define

E := ⋃ k ∈ ℕ I 2 ⁢ k ,

which is a bounded (open) set. Due to the infinite number of jumps, we have χE∉BV⁢(ℝ). However, it can be proved that E has finite s-perimeter for every s∈(0,1). We postpone the proof to Section A.

Remark 1.7.

For completeness, we also mention a related result contained in [9, Example 2.10], where Dipierro, Figalli, Palatucci and Valdinoci provide an example of a bounded set E⊆ℝ which does not have finite s-perimeter for any s∈(0,1). In particular, this example proves that in general the inclusion

⋃ s ∈ ( 0 , 1 ) W s , 1 ⁢ ( Ω ) ⊆ L 1 ⁢ ( Ω )

is strict.

The main result of Section 2 is the following theorem, which extends the asymptotic convergence of (1.8) to the whole s-perimeter.

Theorem 1.8 (Asymptotics).

Let Ω⊆Rn be an open set and let E⊆Rn. Then E has locally finite perimeter in Ω if and only if E has locally finite s-perimeter in Ω for every s∈(0,1) and

lim inf s → 1 ⁡ ( 1 - s ) ⁢ P s L ⁢ ( E , Ω ′ ) < ∞ for all ⁢ Ω ′ ⋐ Ω .

If E has locally finite perimeter in Ω, then

lim s → 1 ⁡ ( 1 - s ) ⁢ P s ⁢ ( E , 𝒪 ) = ω n - 1 ⁢ P ⁢ ( E , 𝒪 ¯ )

for every open set O⋐Ω with Lipschitz boundary. More precisely,

lim s → 1 ⁡ ( 1 - s ) ⁢ P s L ⁢ ( E , 𝒪 ) = ω n - 1 ⁢ P ⁢ ( E , 𝒪 )

and

(1.9) lim s → 1 ⁡ ( 1 - s ) ⁢ P s N ⁢ L ⁢ ( E , 𝒪 ) = ω n - 1 ⁢ P ⁢ ( E , ∂ ⁡ 𝒪 ) = ω n - 1 ⁢ ℋ n - 1 ⁢ ( ∂ * ⁡ E ∩ ∂ ⁡ 𝒪 ) .

The proof of Theorem 1.8 relies only on [4, Theorem 3’], [7, Theorem 1] and on an appropriate estimate of what happens in a neighborhood of ∂⁡𝒪. The main improvement of the known asymptotics results is the convergence (1.9).

1.3 Notation and Assumptions

We write A⋐B to mean that the closure of A is compact in ℝn and A¯⊆B.
In ℝn we will usually write |E|=ℒn⁢(E) for the n-dimensional Lebesgue measure of a set E⊆ℝn.
We write ℋd for the d-dimensional Hausdorff measure for any d≥0.
We define the dimensional constants

ω d := π d 2 Γ ⁢ ( d 2 + 1 ) , d ≥ 0 .

In particular, we remark that ωk=ℒk⁢(B1) is the volume of the k-dimensional unit ball B1⊆ℝk and k⁢ωk=ℋk-1⁢(𝕊k-1) is the surface area of the (k-1)-dimensional sphere

𝕊 k - 1 = ∂ ⁡ B 1 = { x ∈ ℝ k ∣ | x | = 1 } .
Since

| E ⁢ Δ ⁢ F | = 0 implies P ⁢ ( E , Ω ) = P ⁢ ( F , Ω ) and P s ⁢ ( E , Ω ) = P s ⁢ ( F , Ω ) ,

in Section 2 we implicitly identify sets up to sets of negligible Lebesgue measure. Moreover, whenever needed we can choose a particular representative for the class of χE in Lloc1⁢(ℝn), as in Remark 1.9. We will not make this assumption in Section 3 since the Minkowski content can be affected even by changes in sets of measure zero, that is, in general

| Γ 1 ⁢ Δ ⁢ Γ 2 | = 0 does not imply ℳ ¯ r ⁢ ( Γ 1 , Ω ) = ℳ ¯ r ⁢ ( Γ 2 , Ω )

(see Section 3 for a more detailed discussion).
We consider the open tubular ϱ-neighborhood of ∂⁡Ω (see Section B),

N ϱ ( ∂ Ω ) := { x ∈ ℝ n ∣ d ( x , ∂ Ω ) < ϱ } = { | d ¯ Ω | < ϱ } = Ω ϱ ∖ Ω - ϱ ¯ .

Remark 1.9.

Let E⊆ℝn. Modifying E on a set of measure zero, we can assume (see Section C) that

E 1 ⊆ E , E ∩ E 0 = ∅ , ∂ ⁡ E = ∂ - ⁡ E = { x ∈ ℝ n ∣ 0 < | E ∩ B r ⁢ ( x ) | ⁢ < ω n ⁢ r n ⁢ for all ⁢ r > ⁢ 0 } .

2 Asymptotics as s→1-

We say that an open set Ω⊆ℝn is an extension domain if there exists a constant C=C⁢(n,s,Ω)>0 such that for every u∈Ws,1⁢(Ω) there exists u~∈Ws,1⁢(ℝn) with u~|Ω=u and

∥ u ~ ∥ W s , 1 ⁢ ( ℝ n ) ≤ C ⁢ ∥ u ∥ W s , 1 ⁢ ( Ω ) .

Every open set with bounded Lipschitz boundary is an extension domain (see [8] for a proof). By definition, we consider ℝn itself as an extension domain.

We begin with the following embedding.

Proposition 2.1.

Let Ω⊆Rn be an extension domain. Then there exists a constant C=C⁢(n,s,Ω)≥1 such that for every u:Ω→R,

(2.1) ∥ u ∥ W s , 1 ⁢ ( Ω ) ≤ C ⁢ ∥ u ∥ BV ⁢ ( Ω ) .

In particular, we have the continuous embedding

BV ⁢ ( Ω ) ↪ W s , 1 ⁢ ( Ω ) .

Proof.

The claim is trivially satisfied if the right-hand side of (2.1) is infinite, so let u∈BV⁢(Ω). Suppose {uk}⊆C∞⁢(Ω)∩BV⁢(Ω) is an approximating sequence as in [16, Theorem 1.17], that is,

∥ u - u k ∥ L 1 ⁢ ( Ω ) → 0 and lim k → ∞ ⁡ ∫ Ω | ∇ ⁡ u k | ⁢ 𝑑 x = | D ⁢ u | ⁢ ( Ω ) .

We only need to check that the Ws,1-seminorm of u is bounded by its BV-norm. Since Ω is an extension domain, we know (see [8, Proposition 2.2]) that C⁢(n,s)≥1 such that

∥ v ∥ W s , 1 ⁢ ( Ω ) ≤ C ⁢ ∥ v ∥ W 1 , 1 ⁢ ( Ω ) .

Then

[ u k ] W s , 1 ⁢ ( Ω ) ≤ ∥ u k ∥ W s , 1 ⁢ ( Ω ) ≤ C ⁢ ∥ u k ∥ W 1 , 1 ⁢ ( Ω ) = C ⁢ ∥ u k ∥ BV ⁢ ( Ω ) ,

and hence, by using Fatou’s Lemma,

[ u ] W s , 1 ⁢ ( Ω ) ≤ lim inf k → ∞ ⁡ [ u k ] W s , 1 ⁢ ( Ω ) ≤ C ⁢ lim inf k → ∞ ⁡ ∥ u k ∥ BV ⁢ ( Ω ) = C ⁢ lim k → ∞ ⁡ ∥ u k ∥ BV ⁢ ( Ω ) = C ⁢ ∥ u ∥ BV ⁢ ( Ω ) ,

proving (2.1). ∎

Given a set E⊆ℝn and r∈ℝ, we denote

E r := { x ∈ ℝ n ∣ d ¯ E ⁢ ( x ) < r } ,

where d¯E is the signed distance function from E (see Section B).

Corollary 2.2.

If E ⊆ ℝ n has finite perimeter, i.e. χ E ∈ BV ⁢ ( ℝ n ) , then E has also finite s -perimeter for every s ∈ ( 0 , 1 ) .
Let Ω ⊆ ℝ n be a bounded open set with Lipschitz boundary. Then there exists r 0 > 0 such that

(2.2) sup | r | < r 0 ⁡ P s ⁢ ( Ω r ) < ∞ .
If Ω ⊆ ℝ n is a bounded open set with Lipschitz boundary, then

P s N ⁢ L ⁢ ( E , Ω ) ≤ 2 ⁢ P s ⁢ ( Ω ) < ∞

for every E ⊆ ℝ n .
Let Ω ⊆ ℝ n be a bounded open set with Lipschitz boundary. Then

P ⁢ ( E , Ω ) < ∞ 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 P s ⁢ ( E , Ω ) < ∞ for every ⁢ s ∈ ( 0 , 1 ) .

Proof.

(i) This follows from

P s ⁢ ( E ) = 1 2 ⁢ [ χ E ] W s , 1 ⁢ ( ℝ n )

and Proposition 2.1 with Ω=ℝn.

(ii) Let r0 be as in Proposition B.1 and notice that

P ( Ω r ) = ℋ n - 1 ( { d ¯ Ω = r } ) ,

so that

∥ χ Ω r ∥ BV ⁢ ( ℝ n ) = | Ω r | + ℋ n - 1 ( { d ¯ Ω = r } ) .

Thus,

sup | r | < r 0 P s ( Ω r ) ≤ C ( | Ω r 0 | + sup | r | < r 0 ℋ n - 1 ( { d ¯ Ω = r } ) ) < ∞ .

(iii) Notice that

ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ E ∖ Ω ) ≤ ℒ s ⁢ ( Ω , 𝒞 ⁢ Ω ) = P s ⁢ ( Ω ) ,

ℒ s ⁢ ( 𝒞 ⁢ E ∩ Ω , E ∖ Ω ) ≤ ℒ s ⁢ ( Ω , 𝒞 ⁢ Ω ) = P s ⁢ ( Ω ) ,

and use (2.2) (with Ω0=Ω).

(iv) The nonlocal part of the s-perimeter is finite thanks to (iii). As for the local part, recall that

P ⁢ ( E , Ω ) = | D ⁢ χ E | ⁢ ( Ω ) and P s L ⁢ ( E , Ω ) = 1 2 ⁢ [ χ E ] W s , 1 ⁢ ( Ω ) ,

then use Proposition 2.1. ∎

2.1 Asymptotics of the Local Part of the s-Perimeter

We recall the results of [4, 7], which straightforwardly give Theorem 1.4.

Theorem 2.3 ([4, Theorem 3’]).

Let Ω⊆Rn be a smooth bounded domain. Let u∈L1⁢(Ω). Then u∈BV⁢(Ω) if and only if

lim inf n → ∞ ⁡ ∫ Ω ∫ Ω | u ⁢ ( x ) - u ⁢ ( y ) | | x - y | ⁢ ϱ n ⁢ ( x - y ) ⁢ 𝑑 x ⁢ 𝑑 y < ∞ .

Then

C 1 ⁢ | D ⁢ u | ⁢ ( Ω ) ≤ lim inf n → ∞ ⁡ ∫ Ω ∫ Ω | u ⁢ ( x ) - u ⁢ ( y ) | | x - y | ⁢ ϱ n ⁢ ( x - y ) ⁢ 𝑑 x ⁢ 𝑑 y

≤ lim sup n → ∞ ⁡ ∫ Ω ∫ Ω | u ⁢ ( x ) - u ⁢ ( y ) | | x - y | ⁢ ϱ n ⁢ ( x - y ) ⁢ 𝑑 x ⁢ 𝑑 y ≤ C 2 ⁢ | D ⁢ u | ⁢ ( Ω )

for some constants C1, C2 depending only on Ω.

This result was refined by Dávila.

Theorem 2.4 ([7, Theorem 1]).

Let Ω⊆Rn be a bounded open set with Lipschitz boundary. Let u∈BV⁢(Ω). Then

lim k → ∞ ⁡ ∫ Ω ∫ Ω | u ⁢ ( x ) - u ⁢ ( y ) | | x - y | ⁢ ϱ k ⁢ ( x - y ) ⁢ 𝑑 x ⁢ 𝑑 y = K 1 , n ⁢ | D ⁢ u | ⁢ ( Ω ) ,

where

K 1 , n = 1 n ⁢ ω n ⁢ ∫ 𝕊 n - 1 | v ⋅ e | ⁢ 𝑑 σ ⁢ ( v ) ,

with e∈Rn being any unit vector.

In the above theorems, ϱk is any sequence of radial mollifiers, i.e. of functions satisfying

(2.3) ϱ k ⁢ ( x ) ≥ 0 , ϱ k ⁢ ( x ) = ϱ k ⁢ ( | x | ) , ∫ ℝ n ϱ k ⁢ ( x ) ⁢ 𝑑 x = 1

and

(2.4) lim k → ∞ ⁡ ∫ δ ∞ ϱ k ⁢ ( r ) ⁢ r n - 1 ⁢ 𝑑 r = 0 for all ⁢ δ > 0 .

In particular, for R big enough with R>diam⁡(Ω), we can consider

ϱ ⁢ ( x ) := χ [ 0 , R ] ⁢ ( | x | ) ⁢ 1 | x | n - 1

and define, for any sequence {sk}⊆(0,1), sk↗1,

ϱ k ⁢ ( x ) := ( 1 - s k ) ⁢ ϱ ⁢ ( x ) ⁢ c s k ⁢ 1 | x | s k ,

where the csk are normalizing constants. Then

∫ ℝ n ϱ k ⁢ ( x ) ⁢ 𝑑 x = ( 1 - s k ) ⁢ c s k ⁢ n ⁢ ω n ⁢ ∫ 0 R 1 r n - 1 + s k ⁢ r n - 1 ⁢ 𝑑 r = ( 1 - s k ) ⁢ c s k ⁢ n ⁢ ω n ⁢ ∫ 0 R 1 r s k ⁢ 𝑑 r = c s k ⁢ n ⁢ ω n ⁢ R 1 - s k ,

and hence taking csk:=1n⁢ωn⁢Rsk-1 gives (2.3); notice that csk→1n⁢ωn. Also,

lim k → ∞ ⁡ ∫ δ ∞ ϱ k ⁢ ( r ) ⁢ r n - 1 ⁢ 𝑑 r = lim k → ∞ ⁡ ( 1 - s k ) ⁢ c s k ⁢ ∫ δ R 1 r s k ⁢ 𝑑 r = lim k → ∞ ⁡ c s k ⁢ ( R 1 - s k - δ 1 - s k ) = 0 ,

giving (2.4). With this choice we obtain

∫ Ω ∫ Ω | u ⁢ ( x ) - u ⁢ ( y ) | | x - y | ⁢ ϱ k ⁢ ( x - y ) ⁢ 𝑑 x ⁢ 𝑑 y = c s k ⁢ ( 1 - s k ) ⁢ [ u ] W s k , 1 ⁢ ( Ω ) .

Then, if u∈BV⁢(Ω), Dávila’s theorem gives

(2.5) lim s → 1 ⁡ ( 1 - s ) ⁢ [ u ] W s , 1 ⁢ ( Ω ) = lim s → 1 ⁡ 1 c s ⁢ ( c s ⁢ ( 1 - s ) ⁢ [ u ] W s , 1 ⁢ ( Ω ) ) = n ⁢ ω n ⁢ K 1 , n ⁢ | D ⁢ u | ⁢ ( Ω ) .

2.2 Proof of Theorem 1.8

We split the proof of Theorem 1.8 into several steps, which we believe are interesting on their own.

2.2.1 The Constant ωn-1

We need to compute the constant K1,n. Notice that we can choose e in such a way that v⋅e=vn. Then, using spheric coordinates for 𝕊n-1, we obtain |v⋅e|=|cos⁡θn-1| and

d ⁢ σ = sin ⁡ θ 2 ⁢ ( sin ⁡ θ 3 ) 2 ⁢ ⋯ ⁢ ( sin ⁡ θ n - 1 ) n - 2 ⁢ d ⁢ θ 1 ⁢ ⋯ ⁢ d ⁢ θ n - 1 ,

with θ1∈[0,2⁢π) and θj∈[0,π) for j=2,…,n-1. Notice that

ℋ k ⁢ ( 𝕊 k ) = ∫ 0 2 ⁢ π 𝑑 θ 1 ⁢ ∫ 0 π sin ⁡ θ 2 ⁢ d ⁢ θ 2 ⁢ ⋯ ⁢ ∫ 0 π ( sin ⁡ θ k - 1 ) k - 2 ⁢ 𝑑 θ k - 1 = ℋ k - 1 ⁢ ( 𝕊 k - 1 ) ⁢ ∫ 0 π ( sin ⁡ t ) k - 2 ⁢ 𝑑 t .

Then we get

∫ 𝕊 n - 1 | v ⋅ e | ⁢ 𝑑 σ ⁢ ( v ) = ℋ n - 2 ⁢ ( 𝕊 n - 2 ) ⁢ ∫ 0 π ( sin ⁡ t ) n - 2 ⁢ | cos ⁡ t | ⁢ 𝑑 t

= ℋ n - 2 ⁢ ( 𝕊 n - 2 ) ⁢ ( ∫ 0 π 2 ( sin ⁡ t ) n - 2 ⁢ cos ⁡ t ⁢ d ⁢ t - ∫ π 2 π ( sin ⁡ t ) n - 2 ⁢ cos ⁡ t ⁢ d ⁢ t )

= ℋ n - 2 ⁢ ( 𝕊 n - 2 ) n - 1 ⁢ ( ∫ 0 π 2 d d ⁢ t ⁢ ( sin ⁡ t ) n - 1 ⁢ 𝑑 t - ∫ π 2 π d d ⁢ t ⁢ ( sin ⁡ t ) n - 1 ⁢ 𝑑 t )

= 2 ⁢ ℋ n - 2 ⁢ ( 𝕊 n - 2 ) n - 1 .

Therefore,

n ⁢ ω n ⁢ K 1 , n = 2 ⁢ ℋ n - 2 ⁢ ( 𝕊 n - 2 ) n - 1 = 2 ⁢ ℒ n - 1 ⁢ ( B 1 ⁢ ( 0 ) ) = 2 ⁢ ω n - 1 ,

and hence (2.5) becomes

lim s → 1 ⁡ ( 1 - s ) ⁢ [ u ] W s , 1 ⁢ ( Ω ) = 2 ⁢ ω n - 1 ⁢ | D ⁢ u | ⁢ ( Ω )

for any u∈BV⁢(Ω).

2.2.2 Estimating the Nonlocal Part of the s-Perimeter

The aim of this subsection consists in proving that if Ω⊆ℝn is a bounded open set with Lipschitz boundary and E⊆ℝn has finite perimeter in Ωβ for some β∈(0,r0) and r0 as in Proposition B.1, then

(2.6) lim sup s → 1 ⁡ ( 1 - s ) ⁢ P s N ⁢ L ⁢ ( E , Ω ) ≤ 2 ⁢ ω n - 1 ⁢ lim ϱ → 0 + ⁡ P ⁢ ( E , N ϱ ⁢ ( ∂ ⁡ Ω ) ) .

Actually, we prove something slightly more general than (2.6), namely that to estimate the nonlocal part of the s-perimeter we do not necessarily need to use the sets Ωϱ: any “regular” approximation of Ω will do.

More precisely, let Ak,Dk⊆ℝn be two sequences of bounded open sets with Lipschitz boundary strictly approximating Ω respectively from the inside and from the outside, that is,

A k ⊆ A k + 1 ⋐ Ω and Ak↗Ω, i.e. ⋃kAk=Ω,
Ω ⋐ D k + 1 ⊆ D k and Dk↘Ω¯, i.e. ⋂kDk=Ω¯.

We define for every k,

Ω k + := D k ∖ Ω ¯ , Ω k - := Ω ∖ A k ¯ ⁢ T k := Ω k + ∪ ∂ ⁡ Ω ∪ Ω k - ,

d k := min ⁡ { d ⁢ ( A k , ∂ ⁡ Ω ) , d ⁢ ( D k , ∂ ⁡ Ω ) } > 0 .

In particular, we observe that we can consider Ωϱ with ϱ<0 in place of Ak, and with ϱ>0 in place of Dk. Then Tk would be Nϱ⁢(∂⁡Ω) and dk=ϱ.

Proposition 2.5.

Let Ω⊆Rn be a bounded open set with Lipschitz boundary and let E⊆Rn be a set having finite perimeter in D1. Then

lim sup s → 1 ⁡ ( 1 - s ) ⁢ P s N ⁢ L ⁢ ( E , Ω ) ≤ 2 ⁢ ω n - 1 ⁢ lim k → ∞ ⁡ P ⁢ ( E , T k ) .

In particular, if P⁢(E,∂⁡Ω)=0, then

lim s → 1 ⁡ ( 1 - s ) ⁢ P s ⁢ ( E , Ω ) = ω n - 1 ⁢ P ⁢ ( E , Ω ) .

Proof.

Since Ω is regular and P⁢(E,Ω)<∞, we already know that

lim s → 1 ⁡ ( 1 - s ) ⁢ P s L ⁢ ( E , Ω ) = ω n - 1 ⁢ P ⁢ ( E , Ω ) .

Notice that, since |D⁢χE| is a finite Radon measure on D1 and Tk↘∂⁡Ω as k↗∞, we have that

lim k → ∞ ⁡ P ⁢ ( E , T k ) = P ⁢ ( E , ∂ ⁡ Ω ) .

Consider the nonlocal part of the fractional perimeter,

P s N ⁢ L ⁢ ( E , Ω ) = ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ E ∖ Ω ) + ℒ s ⁢ ( 𝒞 ⁢ E ∩ Ω , E ∖ Ω ) ,

and take any k. Then

ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ E ∖ Ω ) = ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ E ∩ Ω k + ) + ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ E ∩ ( 𝒞 ⁢ Ω ∖ D k ) )

≤ ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ E ∩ Ω k + ) + n ⁢ ω n s ⁢ | Ω | ⁢ 1 d k s

≤ ℒ s ⁢ ( E ∩ Ω k - , 𝒞 ⁢ E ∩ Ω k + ) + 2 ⁢ n ⁢ ω n s ⁢ | Ω | ⁢ 1 d k s

≤ ℒ s ⁢ ( E ∩ ( Ω k - ∪ Ω k + ) , 𝒞 ⁢ E ∩ ( Ω k - ∪ Ω k + ) ) + 2 ⁢ n ⁢ ω n s ⁢ | Ω | ⁢ 1 d k s

= P s L ⁢ ( E , T k ) + 2 ⁢ n ⁢ ω n s ⁢ | Ω | ⁢ 1 d k s .

Since we can bound the other term in the same way, we get

P s N ⁢ L ⁢ ( E , Ω ) ≤ 2 ⁢ P s L ⁢ ( E , T k ) + 4 ⁢ n ⁢ ω n s ⁢ | Ω | ⁢ 1 d k s .

By hypothesis, we know that Tk is a bounded open set with Lipschitz boundary

∂ ⁡ T k = ∂ ⁡ A k ∪ ∂ ⁡ D k .

Therefore, using (1.8), we have

lim s → 1 ⁡ ( 1 - s ) ⁢ P s L ⁢ ( E , T k ) = ω n - 1 ⁢ P ⁢ ( E , T k ) ,

and hence

lim sup s → 1 ⁡ ( 1 - s ) ⁢ P s N ⁢ L ⁢ ( E , Ω ) ≤ 2 ⁢ ω n - 1 ⁢ P ⁢ ( E , T k ) .

Since this holds true for any k, we get the claim. ∎

2.2.3 Convergence in Almost Every Ωϱ

Having a “continuous” approximating sequence (the Ωϱ) rather than numerable ones allows us to improve Proposition 2.5.

We first recall that if E has finite perimeter, then De Giorgi’s structure theorem (see, e.g., [17, Theorem 15.9]) guarantees in particular that

| D ⁢ χ E | = ℋ n - 1 ⁢ ⌞ ⁢ ∂ * ⁡ E ,

and hence

P ⁢ ( E , B ) = ℋ n - 1 ⁢ ( ∂ * ⁡ E ∩ B ) for every Borel set ⁢ B ⊆ ℝ n ,

where ∂*⁡E is the reduced boundary of E.

Corollary 2.6.

Let Ω⊆Rn be a bounded open set with Lipschitz boundary and let r0 be as in Proposition B.1. Let E⊆Rn be a set having finite perimeter in Ωβ for some β∈(0,r0), and define

S := { δ ∈ ( - r 0 , β ) ∣ P ⁢ ( E , ∂ ⁡ Ω δ ) > 0 } .

Then the set S is at most countable. Moreover,

(2.7) lim s → 1 ⁡ ( 1 - s ) ⁢ P s ⁢ ( E , Ω δ ) = ω n - 1 ⁢ P ⁢ ( E , Ω δ )

for every δ∈(-r0,β)∖S.

Proof.

We observe that

P ( E , ∂ Ω δ ) = ℋ n - 1 ( ∂ * E ∩ { d ¯ Ω = δ } )

for every δ∈(-r0,β), and

(2.8) M := ℋ n - 1 ⁢ ( ∂ * ⁡ E ∩ ( Ω β ∖ Ω - r 0 ¯ ) ) ≤ P ⁢ ( E , Ω β ) < ∞ .

Then we define the sets

S k := { δ ∈ ( - r 0 , β ) | ℋ n - 1 ( ∂ * E ∩ { d ¯ Ω = δ } ) > 1 k }

for every k∈ℕ and we remark that

S = ⋃ k ∈ ℕ S k .

Since by (2.8) we have

ℋ n - 1 ( ⋃ - r 0 < δ < β ( ∂ * E ∩ { d ¯ Ω = δ } ) ) = M ,

the number of elements in each Sk is at most

♯ ⁢ S k ≤ M ⁢ k .

As a consequence, the set S is at most countable, as claimed.

Finally, since Ωδ is a bounded open set with Lipschitz boundary for every δ∈(-r0,r0) (see Proposition B.1), we obtain (2.7) by Proposition 2.5. ∎

2.2.4 Conclusion

We are now ready to prove Theorem 1.8.

Proof of Theorem 1.8.

We begin by observing that if E⊆ℝn and we have two open sets 𝒪1⊆𝒪2, then

P s ⁢ ( E , 𝒪 1 ) ≤ P s ⁢ ( E , 𝒪 2 ) .

More precisely, we have

P s ⁢ ( E , 𝒪 2 ) = P s ⁢ ( E , 𝒪 1 ) + ℒ s ⁢ ( E ∩ ( 𝒪 2 ∖ 𝒪 1 ) , 𝒞 ⁢ E ∩ ( 𝒪 2 ∖ 𝒪 1 ) )

(2.9) + ℒ s ⁢ ( E ∩ ( 𝒪 2 ∖ 𝒪 1 ) , 𝒞 ⁢ E ∖ 𝒪 2 ) + ℒ s ⁢ ( 𝒞 ⁢ E ∩ ( 𝒪 2 ∖ 𝒪 1 ) , E ∖ 𝒪 2 ) .

Moreover, we also have

P s L ⁢ ( E , 𝒪 1 ) ≤ P s ⁢ ( E , 𝒪 2 ) and P ⁢ ( E , 𝒪 1 ) ≤ P ⁢ ( E , 𝒪 2 ) .

Now suppose that E has locally finite perimeter in Ω and let Ω′⋐Ω. Notice that we can find a bounded open set 𝒪 with Lipschitz boundary such that

Ω ′ ⋐ 𝒪 ⋐ Ω .

Since E has finite perimeter in 𝒪, by Corollary 2.2 (iv) we know that E has finite s-perimeter in 𝒪 (and hence also in Ω′⋐𝒪) for every s∈(0,1). Moreover, by Theorem 1.4 we obtain

lim inf s → 1 ⁡ ( 1 - s ) ⁢ P s L ⁢ ( E , Ω ′ ) ≤ lim inf s → 1 ⁡ ( 1 - s ) ⁢ P s L ⁢ ( E , 𝒪 ) < ∞ .

The converse implication is proved similarly.

Now suppose that E has locally finite perimeter in Ω and let 𝒪⋐Ω have Lipschitz boundary. Let r0=r0⁢(𝒪)>0 be as in Proposition B.1. Since 𝒪⋐Ω, we can find β∈(0,r0) small enough such that 𝒪β⋐Ω. Moreover, E has finite perimeter in 𝒪β since E has locally finite perimeter in Ω.

Then, by Corollary 2.6, we can find δ∈(0,β) such that P⁢(E,∂⁡𝒪δ)=0 and we have

(2.10) lim s → 1 ⁡ ( 1 - s ) ⁢ P s ⁢ ( E , 𝒪 δ ) = ω n - 1 ⁢ P ⁢ ( E , 𝒪 δ ) .

We also remark that, since |∂⁡𝒪|=0, we can rewrite (2.9) as

(2.11) P s ⁢ ( E , 𝒪 δ ) = P s ⁢ ( E , 𝒪 ) + P s L ⁢ ( E , 𝒪 δ ∖ 𝒪 ¯ ) + ℒ s ⁢ ( E ∩ ( 𝒪 δ ∖ 𝒪 ¯ ) , 𝒞 ⁢ E ∖ 𝒪 δ ) + ℒ s ⁢ ( 𝒞 ⁢ E ∩ ( 𝒪 δ ∖ 𝒪 ¯ ) , E ∖ 𝒪 δ ) .

Let

I s := ℒ s ⁢ ( E ∩ ( 𝒪 δ ∖ 𝒪 ¯ ) , 𝒞 ⁢ E ∖ 𝒪 δ ) + ℒ s ⁢ ( 𝒞 ⁢ E ∩ ( 𝒪 δ ∖ 𝒪 ¯ ) , E ∖ 𝒪 δ )

and notice that

(2.12) I s ≤ P s N ⁢ L ⁢ ( E , 𝒪 δ ) .

Hence, since P⁢(E,∂⁡𝒪δ)=0, by (2.12) and Proposition 2.5 we obtain

(2.13) lim s → 1 ⁡ ( 1 - s ) ⁢ I s = 0 .

Furthermore, since E has finite perimeter in 𝒪δ∖𝒪¯, which is a bounded open set with Lipschitz boundary, by (1.8) of Theorem 1.4 we find

(2.14) lim s → 1 ⁡ ( 1 - s ) ⁢ P s L ⁢ ( E , 𝒪 δ ∖ 𝒪 ¯ ) = ω n - 1 ⁢ P ⁢ ( E , 𝒪 δ ∖ 𝒪 ¯ ) .

Therefore, by (2.11), (2.10), (2.13) and (2.14), and exploiting the fact that P⁢(E,⋅) is a measure, we get

(2.15) lim s → 1 ⁡ ( 1 - s ) ⁢ P ⁢ ( E , 𝒪 ) = ω n - 1 ⁢ ( P ⁢ ( E , 𝒪 δ ) - P ⁢ ( E , 𝒪 δ ∖ 𝒪 ¯ ) ) = ω n - 1 ⁢ P ⁢ ( E , 𝒪 ¯ ) .

Finally, since by (1.8) we know that

(2.16) lim s → 1 ⁡ ( 1 - s ) ⁢ P s L ⁢ ( E , 𝒪 ) = ω n - 1 ⁢ P ⁢ ( E , 𝒪 ) ,

by (2.15) and (2.16) we obtain

lim s → 1 ⁡ ( 1 - s ) ⁢ P s N ⁢ L ⁢ ( E , 𝒪 ) = ω n - 1 ⁢ P ⁢ ( E , ∂ ⁡ 𝒪 ) ,

concluding the proof of the theorem. ∎

3 Irregularity of the Boundary

3.1 The Measure Theoretic Boundary as “Support” of the Local Part of the s-Perimeter

First of all we show that the (local part of the) s-perimeter does indeed measure a quantity related to the measure theoretic boundary.

Lemma 3.1.

Let E⊆Rn be a set of locally finite s-perimeter. Then

∂ - ⁡ E = { x ∈ ℝ n ∣ P s L ⁢ ( E , B r ⁢ ( x ) ) > 0 ⁢ for every ⁢ r > 0 } .

Proof.

The claim follows from the following observation: let A,B⊆ℝn such that A∩B=∅; then

ℒ s ⁢ ( A , B ) = 0 if and only if | A | = 0 or | B | = 0 .

Therefore, x∈∂-⁡E is equivalent to

| E ∩ B r ⁢ ( x ) | > 0 ⁢ and ⁢ | 𝒞 ⁢ E ∩ B r ⁢ ( x ) | > 0 for all ⁢ r > 0 ,

which is equivalent to

ℒ s ⁢ ( E ∩ B r ⁢ ( x ) , 𝒞 ⁢ E ∩ B r ⁢ ( x ) ) > 0 for all ⁢ r > 0 ,

concluding the proof ∎

This characterization of ∂-⁡E can be thought of as a fractional analogue of (C.7). However, we can not really think of ∂-⁡E as the support of

P s L ⁢ ( E , ⋅ ) : Ω ↦ P s L ⁢ ( E , Ω ) ,

in the sense that in general

∂ - ⁡ E ∩ Ω = ∅ does not imply P s L ⁢ ( E , Ω ) = 0 .

For example, consider E:={xn≤0}⊆ℝn and notice that ∂-E={xn=0}. Let Ω:=B1⁢(2⁢en)∪B1⁢(-2⁢en). Then ∂-⁡E∩Ω=∅, but

P s L ⁢ ( E , Ω ) = ℒ s ⁢ ( B 1 ⁢ ( 2 ⁢ e n ) , B 1 ⁢ ( - 2 ⁢ e n ) ) > 0 .

On the other hand, the only obstacle is the non-connectedness of the set Ω, and indeed we obtain the following proposition.

Proposition 3.2.

Let E⊆Rn be a set of locally finite s-perimeter and let Ω⊆Rn be an open set. Then

∂ - ⁡ E ∩ Ω ≠ ∅ 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 P s L ⁢ ( E , Ω ) > 0 .

Moreover, if Ω is connected, then

∂ - ⁡ E ∩ Ω = ∅ 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 P s L ⁢ ( E , Ω ) = 0 .

Therefore, if O^⁢(Rn) denotes the family of bounded and connected open sets, then ∂-⁡E can be considered as the “support” of

P s L ⁢ ( E , ⋅ ) : 𝒪 ^ ⁢ ( ℝ n ) → [ 0 , ∞ ) , Ω ↦ P s L ⁢ ( E , Ω ) ,

in the sense that if Ω∈O^⁢(Rn), then

P s L ⁢ ( E , Ω ) > 0 if and only if ∂ - ⁡ E ∩ Ω ≠ ∅ .

Proof.

Let x∈∂-⁡E∩Ω. Since Ω is open, we have Br⁢(x)⊆Ω for some r>0, and hence

P s L ⁢ ( E , Ω ) ≥ P s L ⁢ ( E , B r ⁢ ( x ) ) > 0 .

Let Ω be connected and suppose ∂-⁡E∩Ω=∅. We have the partition of ℝn as ℝn=E0∪∂-⁡E∪E1 (see Section C). Thus we can write Ω as the disjoint union

Ω = ( E 0 ∩ Ω ) ∪ ( E 1 ∩ Ω ) .

However, since Ω is connected and both E0 and E1 are open, we must have E0∩Ω=∅ or E1∩Ω=∅. Now, if E0∩Ω=∅ (the other case is analogous), then Ω⊆E1, and hence |𝒞⁢E∩Ω|=0. Thus,

P s L ⁢ ( E , Ω ) = ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ E ∩ Ω ) = 0 ,

concluding the proof. ∎

3.2 A Notion of Fractal Dimension

Let Ω⊆ℝn be an open set. Then (see, e.g., [8, Proposition 2.1])

t > s implies W t , 1 ⁢ ( Ω ) ↪ W s , 1 ⁢ ( Ω ) .

As a consequence, for every u:Ω→ℝ there exists a unique R⁢(u)∈[0,1] such that

[ u ] W s , 1 ⁢ ( Ω ) ⁢ { < ∞ for all ⁢ s ∈ ( 0 , R ⁢ ( u ) ) , = ∞ for all ⁢ s ∈ ( R ⁢ ( u ) , 1 ) ,

that is,

(3.1) R ⁢ ( u ) = sup ⁡ { s ∈ ( 0 , 1 ) ∣ [ u ] W s , 1 ⁢ ( Ω ) < ∞ } = inf ⁡ { s ∈ ( 0 , 1 ) ∣ [ u ] W s , 1 ⁢ ( Ω ) = ∞ } .

In particular, exploiting this result for characteristic functions, Visintin [22] suggested the following definition of fractal dimension.

Definition 3.3.

Let Ω⊆ℝn be an open set and let E⊆ℝn. If ∂-⁡E∩Ω≠∅, we define

Dim F ⁡ ( ∂ - ⁡ E , Ω ) := n - R ⁢ ( χ E ) ,

the fractal dimension of ∂-⁡E in Ω, relative to the fractional perimeter. If Ω=ℝn, we drop it in the formulas.

Notice that in the case of sets (3.1) becomes

(3.2) R ⁢ ( χ E ) = sup ⁡ { s ∈ ( 0 , 1 ) ∣ P s L ⁢ ( E , Ω ) < ∞ } = inf ⁡ { s ∈ ( 0 , 1 ) ∣ P s L ⁢ ( E , Ω ) = ∞ } .

In particular, we can take Ω to be the whole of ℝn, or a bounded open set with Lipschitz boundary. In the first case, the local part of the fractional perimeter coincides with the whole fractional perimeter, while in the second case we know that we can bound the nonlocal part with 2⁢Ps⁢(Ω)<∞ for every s∈(0,1). Therefore, in both cases in (3.2) we can as well take the whole fractional perimeter Ps⁢(E,Ω) instead of just the local part.

Now we recall the definition of Minkowski dimension, given in terms of the Minkowski contents. For equivalent definitions of the Minkowski dimension and for the main properties, we refer to [18, 13] and the references cited therein.

For simplicity, given Γ⊆ℝn, we set

N ¯ ϱ Ω ⁢ ( Γ ) := N ϱ ⁢ ( Γ ) ¯ ∩ Ω = { x ∈ Ω ∣ d ⁢ ( x , Γ ) ≤ ϱ }

for any ϱ>0.

Definition 3.4.

Let Ω⊆ℝn be an open set. For any Γ⊆ℝn and r∈[0,n] we define the inferior and superior r-dimensional Minkowski contents of Γ relative to the set Ω respectively by

ℳ ¯ r ⁢ ( Γ , Ω ) := lim inf ϱ → 0 ⁡ | N ¯ ϱ Ω ⁢ ( Γ ) | ϱ n - r , ℳ ¯ r ⁢ ( Γ , Ω ) := lim sup ϱ → 0 ⁡ | N ¯ ϱ Ω ⁢ ( Γ ) | ϱ n - r .

Then we define the lower and upper Minkowski dimensions of Γ in Ω by

Dim ¯ ℳ ⁢ ( Γ , Ω ) := inf ⁡ { r ∈ [ 0 , n ] ∣ ℳ ¯ r ⁢ ( Γ , Ω ) = 0 } = n - sup ⁡ { r ∈ [ 0 , n ] ∣ ℳ ¯ n - r ⁢ ( Γ , Ω ) = 0 } ,

Dim ¯ ℳ ⁢ ( Γ , Ω ) := sup ⁡ { r ∈ [ 0 , n ] ∣ ℳ ¯ r ⁢ ( Γ , Ω ) = ∞ } = n - inf ⁡ { r ∈ [ 0 , n ] ∣ ℳ ¯ n - r ⁢ ( Γ , Ω ) = ∞ } .

If they agree, we write

Dim ℳ ⁡ ( Γ , Ω )

for the common value and call it the Minkowski dimension of Γ in Ω. If Ω=ℝn or Γ⋐Ω, we drop the Ω in the formulas.

Remark 3.5.

Let Dimℋ denote the Hausdorff dimension. In general, one has

Dim ℋ ⁡ ( Γ ) ≤ Dim ¯ ℳ ⁢ ( Γ ) ≤ Dim ¯ ℳ ⁢ ( Γ ) ,

and all the inequalities might be strict (for some examples, see, e.g., [18, Section 5.3]). However, for some sets, like self-similar sets which satisfy appropriate symmetric and regularity conditions, they are all equal (see, e.g., [18, Corollary 5.8]).

Now we give a proof of relation (1.2) (obtained in [22]). For related results, see also [21, 14].

Proposition 3.6.

Let Ω⊆Rn be a bounded open set. Then for every E⊆Rn such that ∂-⁡E∩Ω≠∅ and Dim¯M⁢(∂-⁡E,Ω)≥n-1, we have

Dim F ⁡ ( ∂ - ⁡ E , Ω ) ≤ Dim ¯ ℳ ⁢ ( ∂ - ⁡ E , Ω ) .

Proof.

By hypothesis, we have

Dim ¯ ℳ ⁢ ( ∂ - ⁡ E , Ω ) = n - inf ⁡ { r ∈ ( 0 , 1 ) ∣ ℳ ¯ n - r ⁢ ( ∂ - ⁡ E , Ω ) = ∞ } ,

and we need to show that

inf ⁡ { r ∈ ( 0 , 1 ) ∣ ℳ ¯ n - r ⁢ ( ∂ - ⁡ E , Ω ) = ∞ } ≤ sup ⁡ { s ∈ ( 0 , 1 ) ∣ P s L ⁢ ( E , Ω ) < ∞ } .

Modifying E on a set of Lebesgue measure zero, we can suppose that ∂⁡E=∂-⁡E, as in Remark 1.9. Notice that this does not affect the s-perimeter.

Now for any s∈(0,1),

2 ⁢ P s L ⁢ ( E , Ω ) = ∫ Ω 𝑑 x ⁢ ∫ Ω | χ E ⁢ ( x ) - χ E ⁢ ( y ) | | x - y | n + s ⁢ 𝑑 y

= ∫ Ω 𝑑 x ⁢ ∫ 0 ∞ 𝑑 ϱ ⁢ ∫ ∂ ⁡ B ϱ ⁢ ( x ) ∩ Ω | χ E ⁢ ( x ) - χ E ⁢ ( y ) | | x - y | n + s ⁢ 𝑑 ℋ n - 1 ⁢ ( y )

= ∫ Ω 𝑑 x ⁢ ∫ 0 ∞ d ⁢ ϱ ϱ n + s ⁢ ∫ ∂ ⁡ B ϱ ⁢ ( x ) ∩ Ω | χ E ⁢ ( x ) - χ E ⁢ ( y ) | ⁢ 𝑑 ℋ n - 1 ⁢ ( y ) .

Notice that

d ⁢ ( x , ∂ ⁡ E ) > ϱ implies χ E ⁢ ( y ) = χ E ⁢ ( x ) for all ⁢ y ∈ B ϱ ⁢ ( x ) ¯ ,

and hence

∫ ∂ ⁡ B ϱ ⁢ ( x ) ∩ Ω | χ E ⁢ ( x ) - χ E ⁢ ( y ) | ⁢ 𝑑 ℋ n - 1 ⁢ ( y ) ≤ ∫ ∂ ⁡ B ϱ ⁢ ( x ) ∩ Ω χ N ¯ ϱ ⁢ ( ∂ ⁡ E ) ⁢ ( x ) ⁢ 𝑑 ℋ n - 1 ⁢ ( y ) ≤ n ⁢ ω n ⁢ ϱ n - 1 ⁢ χ N ¯ ϱ ⁢ ( ∂ ⁡ E ) ⁢ ( x ) .

Therefore,

2 ⁢ P s L ⁢ ( E , Ω ) ≤ n ⁢ ω n ⁢ ∫ 0 ∞ d ⁢ ϱ ϱ 1 + s ⁢ ∫ Ω χ N ¯ ϱ ⁢ ( ∂ ⁡ E ) ⁢ ( x ) = n ⁢ ω n ⁢ ∫ 0 ∞ | N ¯ ϱ Ω ⁢ ( ∂ ⁡ E ) | ϱ 1 + s ⁢ 𝑑 ϱ .

We claim that

(3.3) ℳ ¯ n - r ⁢ ( ∂ ⁡ E , Ω ) < ∞ implies P s L ⁢ ( E , Ω ) < ∞ for all ⁢ s ∈ ( 0 , r ) .

Indeed,

lim sup ϱ → 0 ⁡ | N ¯ ϱ Ω ⁢ ( ∂ ⁡ E ) | ϱ r < ∞

implies that there exist C,M>0 such that

sup ϱ ∈ ( 0 , C ] ⁡ | N ¯ ϱ Ω ⁢ ( ∂ ⁡ E ) | ϱ r ≤ M < ∞ .

Hence,

2 ⁢ P s L ⁢ ( E , Ω ) ≤ n ⁢ ω n ⁢ { ∫ 0 C | N ¯ ϱ Ω ⁢ ( ∂ ⁡ E ) | ϱ 1 - ( r - s ) + r ⁢ 𝑑 ϱ + ∫ C ∞ | N ¯ ϱ Ω ⁢ ( ∂ ⁡ E ) | ϱ 1 + s ⁢ 𝑑 ϱ }

≤ n ⁢ ω n ⁢ { M ⁢ ∫ 0 C 1 ϱ 1 - ( r - s ) ⁢ 𝑑 ϱ + | Ω | ⁢ ∫ C ∞ 1 ϱ 1 + s ⁢ 𝑑 ϱ }

= n ⁢ ω n ⁢ { M r - s ⁢ C r - s + | Ω | s ⁢ C s } < ∞ ,

proving (3.3).

This implies that

r ≤ sup ⁡ { s ∈ ( 0 , 1 ) ∣ P s L ⁢ ( E , Ω ) < ∞ }

for every r∈(0,1) such that ℳ¯n-r⁢(∂⁡E,Ω)<∞. Thus for ϵ>0 very small, we have

inf ⁡ { r ∈ ( 0 , 1 ) ∣ ℳ ¯ n - r ⁢ ( ∂ - ⁡ E , Ω ) = ∞ } - ϵ ≤ sup ⁡ { s ∈ ( 0 , 1 ) ∣ P s L ⁢ ( E , Ω ) < ∞ } .

Letting ϵ tend to zero, we conclude the proof. ∎

In particular, if Ω has Lipschitz boundary, we obtain the following corollary.

Corollary 3.7.

Let Ω⊆Rn be a bounded open set with Lipschitz boundary. Let E⊆Rn such that ∂-⁡E∩Ω≠∅ and Dim¯M⁢(∂-⁡E,Ω)∈[n-1,n). Then

P s ⁢ ( E , Ω ) < ∞ for every ⁢ s ∈ ( 0 , n - Dim ¯ ℳ ⁢ ( ∂ - ⁡ E , Ω ) ) .

Remark 3.8.

Actually, Proposition 3.6 and Corollary 3.7 still remain true when Ω=ℝn, provided the set E we are considering is bounded. Indeed, if E is bounded, we can apply the previous results with Ω=BR such that E⊆Ω. Moreover, since Ω has a regular boundary, as remarked above we can take the whole s-perimeter in (3.2), instead of just the local part. But then, since Ps⁢(E,Ω)=Ps⁢(E), we see that

Dim F ⁡ ( ∂ - ⁡ E , Ω ) = Dim F ⁡ ( ∂ - ⁡ E , ℝ n ) .

3.2.1 Remarks About the Minkowski Content of ∂-⁡E

In the beginning of the proof of Proposition 3.6 we chose a particular representative for the class of E in order to have ∂⁡E=∂-⁡E. This can be done since it does not affect the s-perimeter and we are already considering the Minkowski dimension of ∂-⁡E.

On the other hand, if we consider a set F such that |E⁢Δ⁢F|=0, we can use the same proof to obtain the inequality

Dim F ⁡ ( ∂ - ⁡ E , Ω ) ≤ Dim ¯ ℳ ⁢ ( ∂ ⁡ F , Ω ) .

It is then natural to ask whether we can find a “better” representative F, whose (topological) boundary ∂⁡F has Minkowski dimension strictly smaller than that of ∂-⁡E.

First of all, we remark that the Minkowski content can be influenced by changes in sets of measure zero. Roughly speaking, this is because the Minkowski content is not a purely measure theoretic notion, but rather a combination of metric and measure.

For example, let Γ⊆ℝn and define Γ′:=Γ∪ℚn. Then |Γ⁢Δ⁢Γ′|=0, but Nδ⁢(Γ′)=ℝn for every δ>0.

In particular, considering different representatives for E, we will get different topological boundaries and hence different Minkowski dimensions.

However, since the measure theoretic boundary minimizes the size of the topological boundary, that is,

∂ - ⁡ E = ⋂ | F ⁢ Δ ⁢ E | = 0 ∂ ⁡ F

(see Section C), it minimizes also the Minkowski dimension. Indeed, for every F such that |F⁢Δ⁢E|=0, we have that ∂-⁡E⊆∂⁡F implies

N ¯ ϱ Ω ⁢ ( ∂ - ⁡ E ) ⊆ N ¯ ϱ Ω ⁢ ( ∂ ⁡ F ) ,

from which

ℳ ¯ r ⁢ ( ∂ - ⁡ E , Ω ) ≤ ℳ ¯ r ⁢ ( ∂ ⁡ F , Ω )

follows, which further implies

Dim ¯ ℳ ⁢ ( ∂ - ⁡ E , Ω ) ≤ Dim ¯ ℳ ⁢ ( ∂ ⁡ F , Ω ) .

3.3 Fractal Dimension of the von Koch Snowflake

The von Koch snowflake S⊆ℝ2 is an example of a bounded open set with fractal boundary for which the Minkowski dimension and the fractal dimension introduced above coincide.

Moreover, its boundary is “nowhere rectifiable”, in the sense that ∂⁡S∩Br⁢(p) is not (n-1)-rectifiable for any r>0 and p∈∂⁡S.

First of all, we recall how to construct the von Koch curve. Then the snowflake is made of three von Koch curves.

Let Γ0 be a line segment of unit length. The set Γ1 consists of the four segments obtained by removing the middle third of Γ0 and replacing it by the other two sides of the equilateral triangle based on the removed segment. We construct Γ2 by applying the same procedure to each of the segments in Γ1 and so on. Thus, Γk comes from replacing the middle third of each straight line segment of Γk-1 by the other two sides of an equilateral triangle.

Figure 2

The first three steps of the construction of the von Koch snowflake.

As k tends to infinity, the sequence of polygonal curves Γk approaches a limiting curve Γ, called the von Koch curve. If we start with an equilateral triangle with unit length side and perform the same construction on all three sides, we obtain the von Koch snowflake Σ (see Figure 2). Let S be the bounded region enclosed by Σ so that S is open and ∂⁡S=Σ. We still call S the von Koch snowflake.

It can be shown (see, e.g., [13]) that the Hausdorff dimension of the von Koch snowflake is equal to its Minkowski dimension and

Dim ℋ ⁡ ( Σ ) = Dim ℳ ⁡ ( Σ ) = log ⁡ 4 log ⁡ 3 .

Now we explain how to construct S in a recursive way and we observe that

∂ - ⁡ S = ∂ ⁡ S = Σ .

As starting point for the snowflake take the equilateral triangle T of side 1 with barycenter in the origin and a vertex P=(0,t) on the y-axis with t>0. Then T1 is made of three triangles of side 1/3, T2 of 3⋅4 triangles of side 1/32 and so on. In general Tk is made of 3⋅4k-1 triangles of side 1/3k, call them Tk1,…,Tk3⋅4k-1. Let xki be the barycenter of Tki and let Pki be the vertex which does not touch Tk-1.

Then S=T∪⋃Tk. Also notice that Tk and Tk-1 touch only on a set of measure zero.

For each triangle Tki there exists a rotation ℛki∈SO⁢(n) such that

T k i = F k i ⁢ ( T ) := ℛ k i ⁢ ( 1 3 k ⁢ T ) + x k i .

We choose the rotations so that Fki⁢(P)=Pki.

Notice that for each triangle Tki we can find a small ball which is contained in the complementary of the snowflake, Bki⊆𝒞⁢S, and touches the triangle in the vertex Pki. Actually these balls can be obtained as the images of the affine transformations Fki of a fixed ball B.

To be more precise, fix a small ball contained in the complementary of T which has the center on the y-axis and touches T in the vertex P, say B:=B1/1000⁢(0,t+1/1000). Then

(3.4) B k i := F k i ⁢ ( B ) ⊆ 𝒞 ⁢ S

for every i, k. To see this, imagine constructing the snowflake S using the same affine transformations Fki, but starting with T∪B in place of T.

We know that ∂-⁡S⊆∂⁡S (see Section C). On the other hand, let p∈∂⁡S. Then every ball Bδ⁢(p) contains at least a triangle Tki⊆S and its corresponding ball Bki⊆𝒞⁢S (and actually infinitely many). Therefore, 0<|Bδ⁢(p)∩S|<ωn⁢δn for every δ>0, and hence p∈∂-⁡S.

Proof of Theorem 1.1.

Since S is bounded, its boundary is ∂-⁡S=Σ and Dimℳ⁡(Σ)=log⁡4log⁡3, we obtain (1.3) from Corollary 3.7 and Remark 3.8.

Exploiting (3.4) and the construction of S given above, we prove (1.4). We have

P s ⁢ ( S ) = ℒ s ⁢ ( S , 𝒞 ⁢ S ) = ℒ s ⁢ ( T , 𝒞 ⁢ S ) + ∑ k = 1 ∞ ℒ s ⁢ ( T k , 𝒞 ⁢ S )

= ℒ s ⁢ ( T , 𝒞 ⁢ S ) + ∑ k = 1 ∞ ∑ i = 1 3 ⋅ 4 k - 1 ℒ s ⁢ ( T k i , 𝒞 ⁢ S )

≥ ∑ k = 1 ∞ ∑ i = 1 3 ⋅ 4 k - 1 ℒ s ⁢ ( T k i , 𝒞 ⁢ S )

≥ ∑ k = 1 ∞ ∑ i = 1 3 ⋅ 4 k - 1 ℒ s ⁢ ( T k i , B k i ) ⁢ (by (3.4))

= ∑ k = 1 ∞ ∑ i = 1 3 ⋅ 4 k - 1 ℒ s ⁢ ( F k i ⁢ ( T ) , F k i ⁢ ( B ) )

= ∑ k = 1 ∞ ∑ i = 1 3 ⋅ 4 k - 1 ( 1 3 k ) 2 - s ⁢ ℒ s ⁢ ( T , B ) ⁢ (by Proposition 3.12)

= 3 3 2 - s ⁢ ℒ s ⁢ ( T , B ) ⁢ ∑ k = 0 ∞ ( 4 3 2 - s ) k .

We remark that ℒs⁢(T,B)≤ℒs⁢(T,𝒞⁢T)=Ps⁢(T)<∞ for every s∈(0,1). To conclude, notice that the last series is divergent if s≥2-log⁡4log⁡3. ∎

Exploiting the self-similarity of the von Koch curve, we show that the fractal dimension of S is the same in every open set which contains a point of ∂⁡S.

Corollary 3.9.

Let S⊆R2 be the von Koch snowflake. Then

Dim F ⁡ ( ∂ ⁡ S , Ω ) = log ⁡ 4 log ⁡ 3

for every open set Ω such that ∂⁡S∩Ω≠∅.

Proof.

Since Ps⁢(S,Ω)≤Ps⁢(S), we have

P s ⁢ ( S , Ω ) < ∞ for all ⁢ s ∈ ( 0 , 2 - log ⁡ 4 log ⁡ 3 ) .

On the other hand, if p∈∂⁡S∩Ω, then Br⁢(p)⊆Ω for some r>0. Now notice that Br⁢(p) contains a rescaled version of the von Koch curve, including all the triangles Tki which constitute it and the relative balls Bki. We can thus repeat the argument above to obtain

P s ⁢ ( S , Ω ) ≥ P s ⁢ ( S , B r ⁢ ( p ) ) = ∞ for all ⁢ s ∈ [ 2 - log ⁡ 4 log ⁡ 3 , 1 ) ,

concluding the proof. ∎

3.4 Self-similar Fractal Boundaries

Proof of Theorem 1.2.

Arguing as we did with the von Koch snowflake, we show that Ps⁢(T) is bounded both from above and from below by the series

∑ k = 0 ∞ ( b λ n - s ) k ,

which converges if and only if s<n-log⁡blog⁡λ.

Indeed,

P s ⁢ ( T ) = ℒ s ⁢ ( T , 𝒞 ⁢ T ) = ∑ k = 1 ∞ ∑ i = 1 a ⁢ b k - 1 ℒ s ⁢ ( T k i , 𝒞 ⁢ T )

≤ ∑ k = 1 ∞ ∑ i = 1 a ⁢ b k - 1 ℒ s ⁢ ( T k i , 𝒞 ⁢ T k i )

= ∑ k = 1 ∞ ∑ i = 1 a ⁢ b k - 1 ℒ s ⁢ ( F k i ⁢ ( T 0 ) , F k i ⁢ ( 𝒞 ⁢ T 0 ) )

= a λ n - s ⁢ ℒ s ⁢ ( T 0 , 𝒞 ⁢ T 0 ) ⁢ ∑ k = 0 ∞ ( b λ n - s ) k ,

P s ⁢ ( T ) = ℒ s ⁢ ( T , 𝒞 ⁢ T ) = ∑ k = 1 ∞ ∑ i = 1 a ⁢ b k - 1 ℒ s ⁢ ( T k i , 𝒞 ⁢ T )

≥ ∑ k = 1 ∞ ∑ i = 1 a ⁢ b k - 1 ℒ s ⁢ ( T k i , S k i )

= ∑ k = 1 ∞ ∑ i = 1 a ⁢ b k - 1 ℒ s ⁢ ( F k i ⁢ ( T 0 ) , F k i ⁢ ( S 0 ) )

= a λ n - s ⁢ ℒ s ⁢ ( T 0 , S 0 ) ⁢ ∑ k = 0 ∞ ( b λ n - s ) k .

Also notice that, since P⁢(T0)<∞, we have

ℒ s ⁢ ( T 0 , S 0 ) ≤ ℒ s ⁢ ( T 0 , 𝒞 ⁢ T 0 ) = P s ⁢ ( T 0 ) < ∞

for every s∈(0,1). ∎

Now suppose that T does not satisfy (1.6). Then we can obtain a set T′ which does, simply by removing a portion S0 from the building block T0. To be more precise, let S0⊆T0 be such that

| S 0 | > 0 , | T 0 ∖ S 0 | > 0 , P ⁢ ( T 0 ∖ S 0 ) < ∞ .

Then define a new building block T0′:=T0∖S0 and the set

T ′ := ⋃ k = 1 ∞ ⋃ i = 1 a ⁢ b k - 1 F k i ⁢ ( T 0 ′ ) .

This new set has exactly the same structure of T since we are using the same collection {Fki} of affine maps.

Notice that

S 0 ⊆ T 0 implies F k i ⁢ ( S 0 ) ⊆ F k i ⁢ ( T 0 ) ,

and

F k i ⁢ ( T 0 ′ ) = F k i ⁢ ( T 0 ) ∖ F k i ⁢ ( S 0 )

for every k, i. Thus,

T ′ = T ∖ ( ⋃ k = 1 ∞ ⋃ i = 1 a ⁢ b k - 1 F k i ⁢ ( S 0 ) )

satisfies (1.6).

Remark 3.10.

Roughly speaking, what matters in order to obtain a set which satisfies the hypothesis of Theorem 1.2 is that there exists a bounded open set T0 such that

| F k i ⁢ ( T 0 ) ∩ F h j ⁢ ( T 0 ) | = 0 if ⁢ i ≠ j ⁢ or ⁢ k ≠ h .

This can be thought of as a compatibility criterion for the family of affine maps {Fki}. Moreover, we also need to require that the ratio of the logarithms of the growth factor and the scaling factor is log⁡blog⁡λ∈(n-1,n). Then we are free to choose as building block any set T0′⊆T0 such that

| T 0 ′ | > 0 , | T 0 ∖ T 0 ′ | > 0 , P ⁢ ( T 0 ′ ) < ∞ ,

and the set

T ′ := ⋃ k = 1 ∞ ⋃ i = 1 a ⁢ b k - 1 F k i ⁢ ( T 0 ′ )

satisfies the hypothesis of Theorem 1.2.

Therefore, even if the Sierpinski triangle and the Menger sponge do not satisfy (1.6), we can exploit their structure to construct new sets which do.

However, we remark that the new boundary ∂-⁡T′ will look very different from the original fractal. Actually, in general it will be a mix of unrectifiable pieces and smooth pieces. In particular, we can not hope to get an analogue of Corollary 3.9. Still, the following remark shows that the new (measure theoretic) boundary retains at least some of the “fractal nature” of the original set.

Remark 3.11.

If the set T of Theorem 1.2 is bounded, exploiting Proposition 3.6 and Remark 3.8, we obtain

Dim ¯ ℳ ⁢ ( ∂ - ⁡ T ) ≥ log ⁡ b log ⁡ λ > n - 1 .

Moreover, notice that if Ω is a bounded open set with Lipschitz boundary, then

P ⁢ ( E , Ω ) < ∞ implies Dim F ⁡ ( E , Ω ) = n - 1 .

Therefore, if T⋐BR, then

P ⁢ ( T ) = P ⁢ ( T , B R ) = ∞ ,

even if T is bounded (and hence ∂-⁡T is compact).

3.4.1 Sponge-like Sets

The simplest way to construct the set T′ consists in simply removing a small ball S0:=B⋐T0 from T0.

In particular, suppose that |T0⁢Δ⁢T|=0, as with the Sierpinski triangle. Define

S := ⋃ k = 1 ∞ ⋃ i = 1 a ⁢ b k - 1 F k i ⁢ ( B ) and T ′ := ⋃ k = 1 ∞ ⋃ i = 1 a ⁢ b k - 1 F k i ⁢ ( T 0 ∖ B ) = T ∖ S .

Then

(3.5) | T 0 ⁢ Δ ⁢ T | = 0 implies | T ′ ⁢ Δ ⁢ ( T 0 ∖ S ) | = 0 .

Now the set E:=T0∖S looks like a sponge, in the sense that it is a bounded open set with an infinite number of holes (each one at a positive, but non-fixed distance from the others).

From (3.5) we get Ps⁢(E)=Ps⁢(T′). Thus, since T′ satisfies the hypothesis of Theorem 1.2, we obtain

Dim F ⁡ ( ∂ - ⁡ E ) = log ⁡ b log ⁡ λ .

3.4.2 Dendrite-like Sets

Depending on the form of the set T0 and on the affine maps {Fki}, we can define more intricate sets T′.

As an example, we consider the Sierpinski triangle E⊆ℝ2. It is of the form E=T0∖T, where the building block T0 is an equilateral triangle, say with side length one, a vertex on the y-axis and barycenter in 0. The pieces Tki are obtained with a scaling factor λ=2 and the growth factor is b=3 (see, e.g., [13] for the construction). As usual, we consider the set

T = ⋃ k = 1 ∞ ⋃ i = 1 3 k - 1 T k i .

However, as remarked above, we have |T⁢Δ⁢T0|=0.

Starting from k=2, each triangle Tki touches with (at least) a vertex (at least) another triangle Thj. Moreover, each triangle Tki gets touched in the middle point of each side (and actually it gets touched in infinitely many points).

Exploiting this situation, we can remove from T0 six smaller triangles, so that the new building block T0′ is a star polygon centered in 0 with six vertices, one in each vertex of T0 and one in each middle point of the sides of T0; see Figure 3.

The resulting set

T ′ = ⋃ k = 1 ∞ ⋃ i = 1 3 k - 1 F k i ⁢ ( T 0 ′ )

will have an infinite number of ramifications; see Figure 4.

Since T′ satisfies the hypothesis of Theorem 1.2, we obtain

Dim F ⁡ ( ∂ - ⁡ T ′ ) = log ⁡ 3 log ⁡ 2 .

$Figure 3 Removing the six triangles (in green) to obtain the new “building block” T0′{T^{\prime}_{0}} (on the right).$

Figure 3

Removing the six triangles (in green) to obtain the new “building block” T0′ (on the right).

$Figure 4 The third and fourth steps of the iterative construction of the set T′{T^{\prime}}.$

Figure 4

The third and fourth steps of the iterative construction of the set T′.

3.4.3 “Exploded” Fractals

In all the previous examples, the sets Tki are accumulated in a bounded region.

On the other hand, imagine making a fractal like the von Koch snowflake or the Sierpinski triangle “explode” and then rearrange the pieces Tki in such a way that d⁢(Tki,Thj)≥d for some fixed d>0.

Since the shape of the building block is not important, we can consider T0:=B1/4⁢(0)⊆ℝn with n≥2. Moreover, since the parameter a does not influence the dimension, we can fix a=1.

Then we rearrange the pieces, obtaining

(3.6) E := ⋃ k = 1 ∞ ⋃ i = 1 b k - 1 B 1 4 ⁢ λ k ⁢ ( k , 0 , … , 0 , i ) .

Define for simplicity

B k i := B 1 4 ⁢ λ k ⁢ ( k , 0 , … , 0 , i ) and x k i := k ⁢ e 1 + i ⁢ e n ,

and notice that

B k i = λ - k ⁢ B 1 4 ⁢ ( 0 ) + x k i .

Since for every k, h and i≠j we have

d ⁢ ( B k i , B h j ) ≥ 1 2 ,

the boundary of the set E is the disjoint union of (n-1)-dimensional spheres

∂ - ⁡ E = ∂ ⁡ E = ⋃ k = 1 ∞ ⋃ i = 1 b k - 1 ∂ ⁡ B k i ,

and in particular is smooth.

The (global) perimeter of E is

P ⁢ ( E ) = ∑ k = 1 ∞ ∑ i = 1 b k - 1 P ⁢ ( B k i ) = 1 λ ⁢ P ⁢ ( B 1 / 4 ⁢ ( 0 ) ) ⁢ ∑ k = 0 ∞ ( b λ n - 1 ) k = + ∞

since log⁡blog⁡λ>n-1.

However, E has locally finite perimeter, since its boundary is smooth and every ball BR intersects only finitely many Bki:

P ⁢ ( E , B R ) < ∞ for all ⁢ R > 0 .

Therefore, it also has locally finite s-perimeter for every s∈(0,1):

P s ⁢ ( E , B R ) < ∞ for all ⁢ R > 0 ⁢ and all ⁢ s ∈ ( 0 , 1 ) .

What is interesting is that the set E satisfies the hypothesis of Theorem 1.2, and hence it also has finite global s-perimeter for every s<σ0:=n-log⁡blog⁡λ:

P s ⁢ ( E ) < ∞ for all ⁢ s ∈ ( 0 , σ 0 ) and P s ⁢ ( E ) = ∞ for all ⁢ s ∈ [ σ 0 , 1 ) .

Thus we obtain Proposition 1.3.

Proof of Proposition 1.3.

It is enough to choose a natural number b≥2 and take λ:=b1/(n-σ). Notice that λ>1 and

log ⁡ b log ⁡ λ = n - σ ∈ ( n - 1 , n ) .

Then we can define E as in (3.6) and we are done. ∎

3.5 Elementary Properties of the s-Perimeter

In the following proposition, we collect some elementary but useful properties of the fractional perimeter which we have exploited throughout the paper.

Proposition 3.12.

Let Ω⊆Rn be an open set.

(Subadditivity:) Let E , F ⊆ ℝ n be such that | E ∩ F | = 0 . Then

P s ⁢ ( E ∪ F , Ω ) ≤ P s ⁢ ( E , Ω ) + P s ⁢ ( F , Ω ) .
(Translation invariance:) Let E ⊆ ℝ n and x ∈ ℝ n . Then

P s ⁢ ( E + x , Ω + x ) = P s ⁢ ( E , Ω ) .
(Rotation invariance:) Let E ⊆ ℝ n and let ℛ ∈ SO ⁢ ( n ) be a rotation. Then

P s ⁢ ( ℛ ⁢ E , ℛ ⁢ Ω ) = P s ⁢ ( E , Ω ) .
(Scaling:) Let E ⊆ ℝ n and λ > 0 . Then

P s ⁢ ( λ ⁢ E , λ ⁢ Ω ) = λ n - s ⁢ P s ⁢ ( E , Ω ) .

Proof.

Claim (i) follows from the following observations: Let A1,A2,B⊆ℝn. If |A1∩A2|=0, then

ℒ s ⁢ ( A 1 ∪ A 2 , B ) = ℒ s ⁢ ( A 1 , B ) + ℒ s ⁢ ( A 2 , B ) .

Moreover,

A 1 ⊆ A 2 implies ℒ s ⁢ ( A 1 , B ) ≤ ℒ s ⁢ ( A 2 , B ) ,

and

ℒ s ⁢ ( A , B ) = ℒ s ⁢ ( B , A ) .

Therefore,

P s ⁢ ( E ∪ F , Ω ) = ℒ s ⁢ ( ( E ∪ F ) ∩ Ω , 𝒞 ⁢ ( E ∪ F ) ) + ℒ s ⁢ ( ( E ∪ F ) ∖ Ω , 𝒞 ⁢ ( E ∪ F ) ∩ Ω )

= ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ ( E ∪ F ) ) + ℒ s ⁢ ( F ∩ Ω , 𝒞 ⁢ ( E ∪ F ) ) + ℒ s ⁢ ( E ∖ Ω , 𝒞 ⁢ ( E ∪ F ) ∩ Ω )

+ ℒ s ⁢ ( F ∖ Ω , 𝒞 ⁢ ( E ∪ F ) ∩ Ω )

≤ ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ E ) + ℒ s ⁢ ( F ∩ Ω , 𝒞 ⁢ F ) + ℒ s ⁢ ( E ∖ Ω , 𝒞 ⁢ E ∩ Ω ) + ℒ s ⁢ ( F ∖ Ω , 𝒞 ⁢ F ∩ Ω )

= P s ⁢ ( E , Ω ) + P s ⁢ ( F , Ω ) .

Claims (ii), (iii) and (iv) follow simply by changing variables in ℒs and by the following observations:

( x + A 1 ) ∩ ( x + A 2 ) = x + A 1 ∩ A 2 , ⁢ x + 𝒞 ⁢ A = 𝒞 ⁢ ( x + A ) ,

ℛ ⁢ A 1 ∩ ℛ ⁢ A 2 = ℛ ⁢ ( A 1 ∩ A 2 ) , ⁢ ℛ ⁢ ( 𝒞 ⁢ A ) = 𝒞 ⁢ ( ℛ ⁢ A ) ,

( λ ⁢ A 1 ) ∩ ( λ ⁢ A 2 ) = λ ⁢ ( A 1 ∩ A 2 ) , ⁢ λ ⁢ ( 𝒞 ⁢ A ) = 𝒞 ⁢ ( λ ⁢ A ) .

For example, for claim (iv) we have

ℒ s ⁢ ( λ ⁢ A , λ ⁢ B ) = ∫ λ ⁢ A ∫ λ ⁢ B d ⁢ x ⁢ d ⁢ y | x - y | n + s = ∫ A λ n ⁢ 𝑑 x ⁢ ∫ B λ n ⁢ d ⁢ y λ n + s ⁢ | x - y | n + s = λ n - s ⁢ ℒ s ⁢ ( A , B ) .

Then

P s ⁢ ( λ ⁢ E , λ ⁢ Ω ) = ℒ s ⁢ ( λ ⁢ E ∩ λ ⁢ Ω , 𝒞 ⁢ ( λ ⁢ E ) ) + ℒ s ⁢ ( λ ⁢ E ∩ 𝒞 ⁢ ( λ ⁢ Ω ) , 𝒞 ⁢ ( λ ⁢ E ) ∩ λ ⁢ Ω )

= ℒ s ⁢ ( λ ⁢ ( E ∩ Ω ) , λ ⁢ 𝒞 ⁢ E ) + ℒ s ⁢ ( λ ⁢ ( E ∖ Ω ) , λ ⁢ ( 𝒞 ⁢ E ∩ Ω ) )

= λ n - s ⁢ ( ℒ s ⁢ ( E ∩ Ω , 𝒞 ⁢ E ) + ℒ s ⁢ ( E ∖ Ω , 𝒞 ⁢ E ∩ Ω ) )

= λ n - s ⁢ P s ⁢ ( E , Ω ) .

This concludes the proof of the proposition. ∎

Communicated by Enrico Valdinoci

A Proof of Example 1.6

Note that E⊆(0,a2]. Let Ω:=(-1,1)⊆ℝ. Then E⋐Ω and dist(E,∂Ω)=1-a2=:d>0. Now,

P s ⁢ ( E ) = ∫ E ∫ 𝒞 ⁢ E ∩ Ω d ⁢ x ⁢ d ⁢ y | x - y | 1 + s + ∫ E ∫ 𝒞 ⁢ Ω d ⁢ x ⁢ d ⁢ y | x - y | 1 + s .

As for the second term, we have

∫ E ∫ 𝒞 ⁢ Ω d ⁢ x ⁢ d ⁢ y | x - y | 1 + s ≤ 2 ⁢ | E | s ⁢ d s < ∞ .

We split the first term into three pieces:

∫ E ∫ 𝒞 ⁢ E ∩ Ω d ⁢ x ⁢ d ⁢ y | x - y | 1 + s = ∫ E ∫ - 1 0 d ⁢ x ⁢ d ⁢ y | x - y | 1 + s + ∫ E ∫ 𝒞 ⁢ E ∩ ( 0 , a ) d ⁢ x ⁢ d ⁢ y | x - y | 1 + s + ∫ E ∫ a 1 d ⁢ x ⁢ d ⁢ y | x - y | 1 + s = ℐ 1 + ℐ 2 + ℐ 3 .

Note that

𝒞 ⁢ E ∩ ( 0 , a ) = ⋃ k ∈ ℕ I 2 ⁢ k - 1 = ⋃ k ∈ ℕ ( a 2 ⁢ k , a 2 ⁢ k - 1 ) .

A simple calculation shows that if a<b≤c<d, then

(A.1) ∫ a b ∫ c d d ⁢ x ⁢ d ⁢ y | x - y | 1 + s = 1 s ⁢ ( 1 - s ) ⁢ [ ( c - a ) 1 - s + ( d - b ) 1 - s - ( c - b ) 1 - s - ( d - a ) 1 - s ] .

Also note that if n>m≥1, then

( 1 - a n ) 1 - s - ( 1 - a m ) 1 - s = ∫ m n d d ⁢ t ⁢ ( 1 - a t ) 1 - s ⁢ 𝑑 t

= ( s - 1 ) ⁢ log ⁡ a ⁢ ∫ m n a t ( 1 - a t ) s ⁢ 𝑑 t

≤ a m ⁢ ( s - 1 ) ⁢ log ⁡ a ⁢ ∫ m n 1 ( 1 - a t ) s ⁢ 𝑑 t

(A.2) ≤ ( n - m ) ⁢ a m ⁢ ( s - 1 ) ⁢ log ⁡ a ( 1 - a ) s .

Now consider the first term

ℐ 1 = ∑ k = 1 ∞ ∫ a 2 ⁢ k + 1 a 2 ⁢ k ∫ - 1 0 d ⁢ x ⁢ d ⁢ y | x - y | 1 + s .

Use (A.1) and notice that (c-a)1-s-(d-a)1-s≤0 to get

∫ - 1 0 ∫ a 2 ⁢ k + 1 a 2 ⁢ k d ⁢ x ⁢ d ⁢ y | x - y | 1 + s ≤ 1 s ⁢ ( 1 - s ) ⁢ [ ( a 2 ⁢ k ) 1 - s - ( a 2 ⁢ k + 1 ) 1 - s ] ≤ 1 s ⁢ ( 1 - s ) ⁢ ( a 2 ⁢ ( 1 - s ) ) k .

Then, as a2⁢(1-s)<1, we get

ℐ 1 ≤ 1 s ⁢ ( 1 - s ) ⁢ ∑ k = 1 ∞ ( a 2 ⁢ ( 1 - s ) ) k < ∞ .

As for the last term

ℐ 3 = ∑ k = 1 ∞ ∫ a 2 ⁢ k + 1 a 2 ⁢ k ∫ a 1 d ⁢ x ⁢ d ⁢ y | x - y | 1 + s ,

use (A.1) and notice that (d-b)1-s-(d-a)1-s≤0 to get

∫ a 2 ⁢ k + 1 a 2 ⁢ k ∫ a 1 d ⁢ x ⁢ d ⁢ y | x - y | 1 + s ≤ 1 s ⁢ ( 1 - s ) ⁢ [ ( 1 - a 2 ⁢ k + 1 ) 1 - s - ( 1 - a 2 ⁢ k ) 1 - s ] ≤ - log ⁡ a s ⁢ ( 1 - a ) s ⁢ a 2 ⁢ k (by (A.2)) .

Thus,

ℐ 3 ≤ - log ⁡ a s ⁢ ( 1 - a ) s ⁢ ∑ k = 1 ∞ ( a 2 ) k < ∞ .

Finally, we split the second term

ℐ 2 = ∑ k = 1 ∞ ∑ j = 1 ∞ ∫ a 2 ⁢ k + 1 a 2 ⁢ k ∫ a 2 ⁢ j a 2 ⁢ j - 1 d ⁢ x ⁢ d ⁢ y | x - y | 1 + s

into three pieces according to the cases j>k, j=k and j<k.

If j=k, using (A.1), we get

∫ a 2 ⁢ k + 1 a 2 ⁢ k ∫ a 2 ⁢ k a 2 ⁢ k - 1 d ⁢ x ⁢ d ⁢ y | x - y | 1 + s = 1 s ⁢ ( 1 - s ) ⁢ [ ( a 2 ⁢ k - a 2 ⁢ k + 1 ) 1 - s + ( a 2 ⁢ k - 1 - a 2 ⁢ k ) 1 - s - ( a 2 ⁢ k - 1 - a 2 ⁢ k + 1 ) 1 - s ]

= 1 s ⁢ ( 1 - s ) ⁢ [ a 2 ⁢ k ⁢ ( 1 - s ) ⁢ ( 1 - a ) 1 - s + a ( 2 ⁢ k - 1 ) ⁢ ( 1 - s ) ⁢ ( 1 - a ) 1 - s - a ( 2 ⁢ k - 1 ) ⁢ ( 1 - s ) ⁢ ( 1 - a 2 ) 1 - s ]

= 1 s ⁢ ( 1 - s ) ⁢ ( a 2 ⁢ ( 1 - s ) ) k ⁢ [ ( 1 - a ) 1 - s + ( 1 - a ) 1 - s a 1 - s - ( 1 - a 2 ) 1 - s a 1 - s ] .

Summing over k∈ℕ, we get

∑ k = 1 ∞ ∫ a 2 ⁢ k + 1 a 2 ⁢ k ∫ a 2 ⁢ k a 2 ⁢ k - 1 d ⁢ x ⁢ d ⁢ y | x - y | 1 + s = 1 s ⁢ ( 1 - s ) ⁢ a 2 ⁢ ( 1 - s ) 1 - a 2 ⁢ ( 1 - s ) ⁢ [ ( 1 - a ) 1 - s + ( 1 - a ) 1 - s a 1 - s - ( 1 - a 2 ) 1 - s a 1 - s ] < ∞ .

In particular, note that

( 1 - s ) ⁢ P s ⁢ ( E ) ≥ ( 1 - s ) ⁢ ℐ 2 ≥ 1 s ⁢ ( 1 - a 2 ⁢ ( 1 - s ) ) ⁢ [ a 2 ⁢ ( 1 - s ) ⁢ ( 1 - a ) 1 - s + a 1 - s ⁢ ( 1 - a ) 1 - s - a 1 - s ⁢ ( 1 - a 2 ) 1 - s ] ,

which tends to +∞ when s→1. This shows that E cannot have finite perimeter.

To conclude let j>k, the case j<k being similar, and consider

∑ k = 1 ∞ ∑ j = k + 1 ∞ ∫ a 2 ⁢ j a 2 ⁢ j - 1 ∫ a 2 ⁢ k + 1 a 2 ⁢ k d ⁢ x ⁢ d ⁢ y | x - y | 1 + s .

Again, using (A.1) and (d-b)1-s-(d-a)1-s≤0, we get

∫ a 2 ⁢ j a 2 ⁢ j - 1 ∫ a 2 ⁢ k + 1 a 2 ⁢ k d ⁢ x ⁢ d ⁢ y | x - y | 1 + s ≤ 1 s ⁢ ( 1 - s ) ⁢ [ ( a 2 ⁢ k + 1 - a 2 ⁢ j ) 1 - s - ( a 2 ⁢ k + 1 - a 2 ⁢ j - 1 ) 1 - s ]

= a 1 - s s ⁢ ( 1 - s ) ⁢ ( a 2 ⁢ ( 1 - s ) ) k ⁢ [ ( 1 - a 2 ⁢ ( j - k ) - 1 ) 1 - s - ( 1 - a 2 ⁢ ( j - k ) - 2 ) 1 - s ]

≤ a 1 - s s ⁢ ( 1 - s ) ⁢ ( a 2 ⁢ ( 1 - s ) ) k ⁢ ( s - 1 ) ⁢ log ⁡ a ( 1 - a ) s ⁢ a 2 ⁢ ( j - k ) - 2 ⁢ (by (A.2))

= - log ⁡ a s ⁢ ( 1 - a s ) ⁢ a s + 1 ⁢ ( a 2 ⁢ ( 1 - s ) ) k ⁢ ( a 2 ) j - k

for j≥k+2. Then

∑ k = 1 ∞ ∑ j = k + 2 ∞ ∫ a 2 ⁢ j a 2 ⁢ j - 1 ∫ a 2 ⁢ k + 1 a 2 ⁢ k d ⁢ x ⁢ d ⁢ y | x - y | 1 + s ≤ - log ⁡ a s ⁢ ( 1 - a s ) ⁢ a s + 1 ⁢ ∑ k = 1 ∞ ( a 2 ⁢ ( 1 - s ) ) k ⁢ ∑ h = 2 ∞ ( a 2 ) h < ∞ .

If j=k+1, we get

∑ k = 1 ∞ ∫ a 2 ⁢ k + 2 a 2 ⁢ k + 1 ∫ a 2 ⁢ k + 1 a 2 ⁢ k d ⁢ x ⁢ d ⁢ y | x - y | 1 + s ≤ 1 s ⁢ ( 1 - s ) ⁢ ∑ k = 1 ∞ ( a 2 ⁢ k + 1 - a 2 ⁢ k + 2 ) 1 - s = a 1 - s ⁢ ( 1 - a ) 1 - s s ⁢ ( 1 - s ) ⁢ ∑ k = 1 ∞ ( a 2 ⁢ ( 1 - s ) ) k < ∞ .

This shows that also ℐ2<∞, so that Ps⁢(E)<∞ for every s∈(0,1), as claimed.

B Signed Distance Function

Given ∅≠E⊆ℝn, the distance function from E is defined by

d E ⁢ ( x ) = d ⁢ ( x , E ) := inf y ∈ E ⁡ | x - y | for ⁢ x ∈ ℝ n .

The signed distance function from ∂⁡E, negative inside E, is then defined by

d ¯ E ⁢ ( x ) = d ¯ ⁢ ( x , E ) := d ⁢ ( x , E ) - d ⁢ ( x , 𝒞 ⁢ E ) .

For the details of the main properties we refer, e.g., to [1, 3].

We also define the sets

E r := { x ∈ ℝ n ∣ d ¯ E ⁢ ( x ) < r } .

Let Ω⊆ℝn be a bounded open set with Lipschitz boundary. By definition, we can locally describe Ω near its boundary as the subgraph of appropriate Lipschitz functions. To be more precise, we can find a finite open covering {Cϱi}i=1m of ∂⁡Ω made of cylinders, and Lipschitz functions φi:Bϱi′→ℝ such that Ω∩Cϱi is the subgraph of φi. That is, up to rotations and translations,

C ϱ i = { ( x ′ , x n ) ∈ ℝ n ∣ | x ′ | < ϱ i , | x n | < ϱ i }

and

Ω ∩ C ϱ i = { ( x ′ , x n ) ∈ ℝ n ∣ x ′ ∈ B ϱ i ′ , - ϱ i < x n < φ i ⁢ ( x ′ ) } ,

∂ ⁡ Ω ∩ C ϱ i = { ( x ′ , φ i ⁢ ( x ′ ) ) ∈ ℝ n ∣ x ′ ∈ B ϱ i ′ } .

Let L be the supremum of the Lipschitz constants of the functions φi.

Now, [12, Theorem 4.1] guarantees that also the bounded open sets Ωr have Lipschitz boundary when r is small enough, say |r|<r0. Moreover, these sets Ωr can locally be described, in the same cylinders Cϱi used for Ω, as subgraphs of Lipschitz functions φir which approximate φi (see [12] for the precise statement) and whose Lipschitz constants are less than or equal to L. Notice that

∂ Ω r = { d ¯ Ω = r } .

Now, since in Cϱi the set Ωr coincides with the subgraph of φir, we have

ℋ n - 1 ⁢ ( ∂ ⁡ Ω r ∩ C ϱ i ) = ∫ B ϱ i ′ 1 + | ∇ ⁡ φ i r | 2 ⁢ 𝑑 x ′ ≤ M i ,

with Mi depending on ϱi and L, but not on r. Therefore,

ℋ n - 1 ( { d ¯ Ω = r } ) ≤ ∑ i = 1 m ℋ n - 1 ( ∂ Ω r ∩ C ϱ i ) ≤ ∑ i = 1 m M i

independently on r, proving the following proposition.

Proposition B.1.

Let Ω⊆Rn be a bounded open set with Lipschitz boundary. Then there exists r0=r0⁢(Ω)>0 such that Ωr is a bounded open set with Lipschitz boundary for every r∈(-r0,r0) and

sup | r | < r 0 ℋ n - 1 ( { d ¯ Ω = r } ) < ∞ .

C Measure Theoretic Boundary

Since

(C.1) | E ⁢ Δ ⁢ F | = 0 implies P ⁢ ( E , Ω ) = P ⁢ ( F , Ω ) and P s ⁢ ( E , Ω ) = P s ⁢ ( F , Ω ) ,

we can modify a set making its topological boundary as big as we want without changing its (fractional) perimeter. For example, let E⊆ℝn be a bounded open set with Lipschitz boundary. Then, if we set

F := ( E ∖ ℚ n ) ∪ ( ℚ n ∖ E ) ,

we have |E⁢Δ⁢F|=0, and hence we get (C.1). However, ∂⁡F=ℝn.

For this reason, one considers measure theoretic notions of interior, exterior and boundary, which solely depend on the class of χE in Lloc1⁢(ℝn). In some sense, by considering the measure theoretic boundary ∂-⁡E defined below, we can also minimize the size of the topological boundary (see (C.6)). Moreover, this measure theoretic boundary is actually the topological boundary of a set which is equivalent to E. Thus we obtain a “good” representative for the class of E.

We refer to [22, Section 3.2] (see also [16, Proposition 3.1]). For some details about the good representative of an s-minimal set, see [10, Appendix].

Definition C.1.

Let E⊆ℝn. For every t∈[0,1] define the set

(C.2) E ( t ) := { x ∈ ℝ n | ∃ lim r → 0 ⁡ | E ∩ B r ⁢ ( x ) | ω n ⁢ r n = t }

of points density t of E. The sets E(0) and E(1) are respectively the measure theoretic exterior and interior of the set E. The set

∂ e ⁡ E := ℝ n ∖ ( E ( 0 ) ∪ E ( 1 ) )

is the essential boundary of E.

Using the Lebesgue points theorem for the characteristic function χE, we see that the limit in (C.2) exists for a.e. x∈ℝn, and

lim r → 0 ⁡ | E ∩ B r ⁢ ( x ) | ω n ⁢ r n = { 1 for a.e. ⁢ x ∈ E , 0 for a.e. ⁢ x ∈ 𝒞 ⁢ E .

So,

| E ⁢ Δ ⁢ E ( 1 ) | = 0 , | 𝒞 ⁢ E ⁢ Δ ⁢ E ( 0 ) | = 0 , | ∂ e ⁡ E | = 0 .

In particular, every set E is equivalent to its measure theoretic interior.

However, notice that E(1) in general is not open.

We have another natural way to define a measure theoretic boundary.

Definition C.2.

Let E⊆ℝn and define the sets

E 1 := { x ∈ ℝ n ∣ there exists ⁢ r > 0 ⁢ such that ⁢ | E ∩ B r ⁢ ( x ) | = ω n ⁢ r n } ,

E 0 := { x ∈ ℝ n ∣ there exists ⁢ r > 0 ⁢ such that ⁢ | E ∩ B r ⁢ ( x ) | = 0 } .

Then we define

∂ - ⁡ E := ℝ n ∖ ( E 0 ∪ E 1 ) = { x ∈ ℝ n ∣ 0 < | E ∩ B r ⁢ ( x ) | ⁢ < ω n ⁢ r n ⁢ for every ⁢ r > ⁢ 0 } .

Notice that E0 and E1 are open sets, and hence ∂-⁡E is closed. Moreover, since

(C.3) E 0 ⊆ E ( 0 ) and E 1 ⊆ E ( 1 ) ,

we get

∂ e ⁡ E ⊆ ∂ - ⁡ E .

We have that

(C.4) if ⁢ F ⊆ ℝ n ⁢ is such that ⁢ | E ⁢ Δ ⁢ F | = 0 , then ⁢ ∂ - ⁡ E ⊆ ∂ ⁡ F .

0 < | F ∩ B r ⁢ ( x ) | < ω n ⁢ r n ,

which implies

F ∩ B r ⁢ ( x ) ≠ ∅ and 𝒞 ⁢ F ∩ B r ⁢ ( x ) ≠ ∅ for every ⁢ r > 0 ,

and hence x∈∂⁡F.

In particular, ∂-⁡E⊆∂⁡E.

Moreover,

(C.5) ∂ - ⁡ E = ∂ ⁡ E ( 1 ) .

Indeed, since |E⁢Δ⁢E(1)|=0, we already know that ∂-⁡E⊆∂⁡E(1). The converse inclusion follows from (C.3) and the fact that both E0 and E1 are open. From (C.4) and (C.5) we obtain

(C.6) ∂ - ⁡ E = ⋂ F ∼ E ∂ ⁡ F ,

where the intersection is taken over all sets F⊆ℝn such that |E⁢Δ⁢F|=0, so we can think of ∂-⁡E as a way to minimize the size of the topological boundary of E. In particular,

if ⁢ F ⊆ ℝ n ⁢ is such that ⁢ | E ⁢ Δ ⁢ F | = 0 , then ⁢ ∂ - ⁡ F = ∂ - ⁡ E .

From (C.3) and (C.5) we see that we can take E(1) as a “good” representative for E, obtaining Remark 1.9.

Recall that the support of a Radon measure μ on ℝn is defined as the set

supp ⁡ μ := { x ∈ ℝ n ∣ μ ⁢ ( B r ⁢ ( x ) ) > 0 ⁢ for every ⁢ r > 0 } .

Notice that, being the complementary of the union of all open sets of measure zero, it is a closed set. In particular, if E is a Caccioppoli set, we have

(C.7) supp ⁡ | D ⁢ χ E | = { x ∈ ℝ n ∣ P ⁢ ( E , B r ⁢ ( x ) ) > 0 ⁢ for every ⁢ r > 0 } ,

and it is easy to verify that

∂ - ⁡ E = supp ⁡ | D ⁢ χ E | = ∂ * ⁡ E ¯ ,

where ∂*⁡E denotes the reduced boundary (see, e.g., [17, Chapter 15]). Moreover, ∂*⁡E⊆∂e⁡E, and by Federer’s theorem (see, e.g., [17, Theorem 16.2]) we have

ℋ n - 1 ⁢ ( ∂ e ⁡ E ∖ ∂ * ⁡ E ) = 0 .

We remark that in general the inclusions

∂ * ⁡ E ⊆ ∂ e ⁡ E ⊆ ∂ - ⁡ E ⊆ ∂ ⁡ E

are all strict; see, e.g., Figure 5. Indeed, we have already observed in the previous discussion that in general ∂-⁡E is much smaller than the topological boundary ∂⁡E. In order to have an example of a point p∈∂-⁡E∖∂e⁡E it is enough to consider sublinear cusps. For example, if E:={(x,y)∈ℝ2∣y<-|x|1/2} and p:=(0,0), then it is easy to verify that p∈E(0) and hence p∉∂e⁡E. On the other hand, p∈∂-⁡E. Finally, the vertex of an angle is an example of a point p∈∂e⁡E∖∂*⁡E (see, e.g., [17, Example 15.4]).

$Figure 5 The point A belongs to ∂-⁡E{\partial^{-}E} but A∉∂e⁡E{A\not\in\partial_{e}E}. The point B belongs to ∂e⁡F{\partial_{e}F} but B∉∂*⁡F{B\not\in\partial^{*}F}.$

Figure 5

The point A belongs to ∂-⁡E but A∉∂e⁡E. The point B belongs to ∂e⁡F but B∉∂*⁡F.

Acknowledgements

I would like to express my gratitude to Enrico Valdinoci for his valuable advice and never lacking support.

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Received: 2018-01-06

Revised: 2018-04-10

Accepted: 2018-04-12

Published Online: 2018-06-13

Published in Print: 2019-02-01

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https://doi.org/10.1515/ans-2018-2016

Keywords for this article

28A80; 35R11; 49Q05

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