Higher integrability of the gradient for the Thermal Insulation problem

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Higher integrability of the gradient for the Thermal Insulation problem

Camille Labourie, Emmanouil Milakis

To cite this version:

Camille Labourie, Emmanouil Milakis. Higher integrability of the gradient for the Thermal Insulation problem. 2021. �hal-03207514v3�

Higher integrability of the gradient for the Thermal Insulation problem

C. Labourie, E. Milakis

Abstract

We prove the higher integrability of the gradient for local mini- mizers of the thermal insulation problem, an analogue of De Giorgi’s conjecture for the Mumford-Shah functional. We deduce that the sin- gular part of the free boundary has Hausdorff dimension strictly less than n − 1.

AMS Subject Classifications: 35R35, 35J20, 49N60, 49Q20.

Keywords: Thermal Insulation, Higher Integrability, Free Boundary Problems.

Contents

1 Introduction 2

2 Generalities about minimizers 3

2.1 Definition . . . 3 2.2 Properties . . . 4

3 Porosity of the singular part 10

4 Higher integrability of the gradient 12

5 Dimension of the singular part 18

Appendices 20

A Generalities about BV functions 20

B A Robin problem 23

B.1 Statement . . . 23 B.2 Hölder continuity up to the boundary . . . 24 B.3 Gradient estimates . . . 29

C Extracts from David’s book 30

1 Introduction

We fix a bounded connected set Ω ⊂ Rⁿ. The thermal insulation problem consists in minimizing the functional

I(A, u) :=

ˆ

A

|∇u|²dLⁿ+ ˆ

∂A

|u^∗|²dHⁿ⁻¹+ Lⁿ(A) (1) among all pairs (A, u) where A ⊂ Rⁿ is an admissible domain and u ∈ W^1,2(A) is a function such that u = 1 for Lⁿ-a.e. on Ω. Here, u^∗ is the trace of u on ∂A.

The problem has been studied by Caffarelli–Kriventsov in [4], [12] and Bucur–Giacomini–Luckhaus in [2], [3]. The authors transpose the problem to a slightly different setting in order to apply the direct method of the calculus of variation. The authors represent a pair (A, u) by the function u1_A and relax the functional on SBV. The new problem consists in minimizing the functional

F (u) :=

ˆ

Rⁿ

|∇u|²dLⁿ+ ˆ

Ju

(u²+ u²) dHⁿ⁻¹+ Lⁿ({ u > 0 }) (2) among all functions u ∈ SBV(Rⁿ) such that u = 1 Lⁿ-a.e. on Ω. The definition of J_u and u, u are given in Appendix A. This new setting is more suited to a direct minimization since it enjoys the compactness and closure properties of SBV. In parenthesis, there always exist functions u ∈ SBV(Rⁿ) such that u = 1 Lⁿ-a.e. on Ω and F (u) < ∞. For example, u := 1_Bwhere B is an open ball containing Ω. In [4, Theorem 4.2], Caffarelli and Kriventsov prove that the SBV problem has a solution u. A key point property of solutions is that there exists 0 < δ < 1 (depending on n, Ω) such that spt(u) ⊂ B(0, δ⁻¹) and

u ∈ { 0 } ∪ [δ, 1] Lⁿ-a.e. on Rⁿ. (3) This property has also been proved in [2]. On another note, some minimality criteria have been proved by calibrations in [5].

The main goal of the present article is to prove that there exists p > 1 such that |∇u|² ∈ L^p_loc(Rⁿ\ Ω) (Theorem 4.1). A parallel property was conjectured by De Giorgi for minimizers of the Mumford-Shah functional and solved by De Lellis and Focardi in the planar case ([6]) and then De Philippis and Figalli ([9]) in the general case. Our proof is inspired by the technique of [9] and it relies on three key properties: the Ahlfors-regularity of the free boundary, the uniform rectifiability of the free boundary and the ε-regularity theorem. In particular, this implies a porosity property which means that the singular part Σ of the free boundary has many holes in a quantified way. In contrast to the Mumford-Shah situation, the ε-regularity theorem describes a regular part of the boundary as a pair of graphs rather

than just one graph. The minimizers satisfies an elliptic equation with a Robin boundary condition at the boundary rather than a Neumann boundary condition. We present the technique of [9] in a different way by singling out a higher integrability lemma and a covering lemma and by removing the need of [9, Lemma 3.2] (the existence of good radii). Once we establish the higher integrability of the gradient, we are also able to conclude that the dimension of Σ is strictly less than n − 1 (Theorem 5.1). The link between the higher integrability of the gradient and the dimension of the singular part has been first observed for the Mumford-Shah functional by Ambrosio, Fusco, Hutchinson in [8]. An open question of Caffarelli–Kriventsov hints that for all minimizers in the planar case, Σ is empty and the optimal exponent is p = ∞ (see also Remark 5.2).

Acknowledgement. We would like to thank Guido De Philippis for his helpful correspondence concerning estimates for elliptic equations.

2 Generalities about minimizers

2.1 Definition

Notations. Our ambient space is an open set X of Rⁿ. One can think of X as Rⁿ\ Ω. For x ∈ Rⁿ and r > 0, B(x, r) is the open ball centered in x and of radius r. If there is no ambiguity, it is simply denoted by B_r. Given an open ball B := B(x, r) and a scalar t > 0, the notation tB means B(x, tr). Given a set A ⊂ Rⁿ, the indicator function of A is denoted by 1_A. Given two sets A, B ⊂ Rⁿ, the notation A ⊂⊂ B means that there exists a compact set K ⊂ Rⁿ such that A ⊂ K ⊂ B.

Given u ∈ SBV_loc(X), we denote by K the support of the singular part of Du:

K := spt(|u − u|Hⁿ⁻¹ J_u) (4a)

:= spt(Hⁿ⁻¹ Ju). (4b)

For x ∈ K and r > 0 such that B(x, r) ⊂ X, we define ω2(x, r) := r⁻⁽ⁿ⁻¹⁾

ˆ

B(x,r)

|∇u|²dLⁿ, (5a)

β₂(x, r) := r⁻⁽ⁿ⁺¹⁾inf

V

ˆ

K∩B(x,r)

d(y, V )²dHⁿ⁻¹(y)

!¹₂

, (5b)

where V runs among (n − 1) planes V ⊂ Rⁿpassing through x. When there is an ambiguity, we will write β_K,2 instead of β₂. We have gathered some definitions and results from the theory of BV functions in the introduction of Appendix A.

For any open ball B such that B ⊂ X, we define a competitor of u in B as a function v ∈ SBV_loc(X) such that v = u Lⁿ-a.e. on X \ B. We fix a constant δ ∈ ]0, 1[ for all the paper.

Definition 2.1. We say that u ∈ SBV_loc(X) is a local minimizer if 1. for Lⁿ-a.e. x ∈ X, we have u ∈ { 0 } ∪ [δ, δ⁻¹];

2. for all open balls B such thatB ⊂ X, for all competitors v of u in B, ˆ

B

|∇u|²dLⁿ+ ˆ

Ju∩B

(u²+ u²) dHⁿ⁻¹+ Lⁿ({ u > 0 } ∩ B)

≤ ˆ

B

|∇v|²dLⁿ+ ˆ

Jv∩B

(v²+ v²) dHⁿ⁻¹+ Lⁿ({ v > 0 } ∩ B). (6)

As a first consequence, we have that u, u ∈ { 0 } ∪ [δ, δ⁻¹] everywhere in X. In particular, u ≥ δ everywhere on Su. For all open balls B such that B ⊂ X, we have

ˆ

B

|∇u|²dLⁿ+ ˆ

Ju∩B

(u²+ u²) dHⁿ⁻¹< ∞. (7)

This shows that |∇u|² ∈ L¹_loc(X) and that Su is Hⁿ⁻¹-locally finite in X.

In X \ S_u, the function u is W_loc^1,2 and locally minimizes its Dirichlet energy.

Therefore, u is harmonic (and thus continuous) in X \S_u. We conclude that in each connected component of X \ S_u, we have either u > δ everywhere or u = 0 everywhere.

2.2 Properties

The next results (Ahlfors-regularity, uniform rectifiability and ε-regularity theorem) also hold true for the almost-minimizers of [12, Definition 2.1]. We are going to cite [4, Corollary 3.3 and Theorem 5.1].

Proposition 2.2 (Ahlfors-regularity). Let u ∈ SBV_loc(X) be a local mini- mizer. There exists 0 < r₀ ≤ 1 and C ≥ 1 (both depending on n, δ) such that the following holds true.

1. For all x ∈ X, for all 0 ≤ r ≤ r₀ such that B(x, r) ⊂ X, ˆ

B(x,r)

|∇u|²dLⁿ+ Hⁿ⁻¹(K ∩ B(x, r)) ≤ Crⁿ⁻¹. (8)

2. For all x ∈ S_u, for all 0 ≤ r ≤ r₀ such that B(x, r) ⊂ X,

Hⁿ⁻¹(K ∩ B(x, r)) ≥ C⁻¹rⁿ⁻¹. (9)

Corollary 2.3. Let u ∈ SBV_loc(X) be a local minimizer.

(i) We have K = S_u = J_u and Hⁿ⁻¹(K \ J_u) = 0.

(ii) The set A_u := { u > 0 } \ K is open and ∂Au = K.

Proof. It is straighforward by definition that K ⊂ J_u ⊂ S_u. On the other hand, property (9) shows that Su ⊂ K. We justify that Hⁿ⁻¹(K \ Ju) = 0.

The jump set J_u is Borel and Hⁿ⁻¹ locally finite in X, so for Hⁿ⁻¹-a.e.

x ∈ X \ Ju,

r→0lim

Hⁿ⁻¹(Ju∩ B(x, r))

rⁿ⁻¹ = 0 (10)

(see [14, Theorem 6.2]). We draw our claim from the observation that this limit contradicts (9).

We study the set A_u. We recall that the function u is continuous in X \ K (since it coincides with u outside S_u) and u ∈ { 0 } ∪ [δ, 1] everywhere in X \ K. As a consequence, the sets

A_u := { u > 0 } \ K, (11)

B_u := { u = 0 } \ K (12)

are open subsets of X \ K and thus of X. The space X is the disjoint union

X = K ∪ Au∪ B_u, (13)

where A_u and B_u are open and K is relatively closed, so Au⊂ A_u∪ K.

We show that S_u ⊂ A_u. Let us suppose that there exists x ∈ S_uand r > 0 such that A_u∩ B(x, r) = ∅. Then B(x, r) \ K ⊂ { u = 0 } so we have u = 0 Lⁿ-a.e. on B(x, r) and thus x is a Lebesgue point of u (a contradiction). We conclude that S_u ⊂ A_u and in turn K ⊂ A_u so A_u = Au∪ K.

We are going to apply [7] to justify that K is locally contained in a uniformly rectifiable set. We underline that our local minimizers are not quasiminimizers as in [7, Definition 7.21]. In Appendix C, we have sum- marised the relevant results of [7] and how their proofs adapt to our case (Remark C.5).

Proposition 2.4 (Uniform Rectifiability). Let u ∈ SBV_loc(X) be a local minimizer. There exists 0 < r₀ ≤ 1 (depending on n, δ) such that the following holds true. For all x ∈ K and 0 < r ≤ r₀ such that B(x, r) ⊂ X, there is a closed, Ahlfors-regular, uniformly rectifiable set E of dimension (n − 1) such that K ∩¹₂B(x, r) ⊂ E. The constants for the Ahfors-regularity and uniform rectifiability depends on n, δ.

Proof. We want to show that (u, K) satisfy Definition C.1, or rather the alternative Definition given in Remark C.5. Then the Proposition will follow

from Theorem C.4. First, it is clear that (u, K) is an admissible pair. Let B be an open ball of radius r > 0 such that B ⊂ X. Let an admissible pair (v, L) be a competitor of (u, K) in B. As explained in Remark C.2, we can assume without loss of generality that L is Hⁿ⁻¹ locally finite. Therefore, v ∈ SBVloc(X) and Hⁿ⁻¹(Jv\ L) = 0. We have included more details about the construction of SBV functions in Appendix A. We can now apply the minimality inequality. We have

ˆ

B

|∇u|²dLⁿ+ ˆ

Ju∩B

(u²+ u²) dHⁿ⁻¹+ Lⁿ({ u > 0 } ∩ B)

≤ ˆ

B

|∇v|²dLⁿ+ ˆ

Jv∩B

(v²+ v²) dHⁿ⁻¹+ Lⁿ({ v > 0 } ∩ B) (14) so

ˆ

B

|∇u|²dLⁿ+ δ²Hⁿ⁻¹(Ju∩ B) + Lⁿ({ u > 0 } ∩ B)

≤ ˆ

B

|∇v|²dLⁿ+ δ⁻²Hⁿ⁻¹(Jv∩ B) + Lⁿ({ v > 0 } ∩ B). (15) We omit the term Lⁿ({ u > 0 } ∩ B) at the left and we bound the term Lⁿ({ v > 0 } ∩ B) at the right by ωnrⁿ where ω_n is the Lebesgue volume of the unit ball. We can replace J_u by K at the left since Hⁿ⁻¹(K \ J_u) = 0.

We can replace J_v by L at the right since Hⁿ⁻¹(J_v\ L) = 0. It follows that Hⁿ⁻¹(K ∩ B) ≤ δ⁻⁴Hⁿ⁻¹(L ∩ B) + δ⁻²∆E + δ⁻²ωnrⁿ (16) where

∆E :=

ˆ

B

|∇v|²− ˆ

B

|∇u|²dLⁿ. (17)

We are going to cite the ε-regularity theorem for our problem [12, The- orem 14.1]. Contrary to the ε-regularity theorem for the Mumford-Shah problem, it does not require ω₂(x, r) to be small. It says that when K is very close to a plane, K is given by a pair of smooth graphs. We describe this situation in the next definition.

Given a point x ∈ Rⁿ, a vector e_n∈ Sⁿ⁻¹, we can decompose each point y ∈ Rⁿ under the form y = x + (y⁰ + y_ne_n) where y⁰ ∈ e^⊥_n and y_n ∈ R.

Then for all function f : e^⊥_n → R, we define the graph of f in the coordinate system (x, e_n) as

Γ_(x,en)(f ) := { y ∈ Rⁿ| y_n= f (y⁰) } . (18) Definition 2.5. Let u ∈ SBV_loc(X) be a local minimizer. Let x ∈ K and R > 0 be such that B(x, R) ⊂ X. Let 0 < α ≤ 1. We say that K is C^1,α- regular in B := B(x, R) ⊂ X if it satisfies the three following conditions.

(i) There exists a vector e_n ∈ Sⁿ⁻¹ and two functions f_i: e^⊥_n → R (i = 1, 2) such that f1 ≤ f₂ and

K ∩ B =



 [

i=1,2

Γ_(x,en)(f_i)



∩ B. (19)

The functions f₁, f₂ are C^1,α and

R⁻¹|f_i|_∞+ |∇f_i|_∞+ R^α[∇f_i]_α≤ 1

4. (20)

(ii) There are two possible cases. The first case is

(u > 0 in { y ∈ B | y_n< f1(y⁰) or yn> f2(y⁰) }

u = 0 in { y ∈ B | f₁(y⁰) < y_n< f₂(y⁰) } (21) The second case is f₁ = f2 and

(u > 0 in { y ∈ B | y_n> f₁(y⁰) }

u = 0 in { y ∈ B | y_n< f1(y⁰) } (22) or inversely.

Theorem 2.6 (ε-regularity theorem). Let u ∈ SBV_loc(X) be a local mini- mizer and let x ∈ K.

(i) For all ε > 0, there exists ε₁> 0 (depending on n, δ, β) such that the following holds true. For r > 0 such that B(x, r) ⊂ X and β₂(x, r) + r ≤ ε₁, we have ω₂(x,^r₂) ≤ ε.

(ii) There exists ε > 0, C ≥ 1 and 0 < α < 1 (both depending on n, δ) such that the following holds true. For r > 0 such that B(x, r) ⊂ X and β₂(x, r) + r ≤ ε, the set K is C^1,α-regular in B(x, C⁻¹R).

This last result is specific to local minimizers and does not hold true for the general almost-minimizers of [12, Definition 2.1].

Proposition 2.7. Let u ∈ SBV_loc(X) be a local minimizer. Let x ∈ K and R > 0 be such that B(x, R) ⊂ X. We assume that K is regular in B := B(x, R) (Definition 2.5) and we denote by Γ_i the graph of f_i in B (i = 1, 2). Then for each i = 1, 2, u_|Ai solves the Robin problem

 ∆u = 0 in A_i

∂_νiu − u_i = 0 in Γ_i, (23) where

A1 := { y ∈ B | yn< f1(y⁰) } (24) A₂ := { y ∈ B | y_n> f₂(y⁰) } (25) and ν_i is the inner normal vector to A_i.

Proof. We only detail the case (21) of Definition 2.5 and we prove the Propo- sition for i = 2. We partition B in three sets (modulo Lⁿ)

A₁:= { z ∈ B | y_n< f₁(y⁰) } (26) A2:= { z ∈ B | yn> f2(y⁰) } (27) A3:= { z ∈ B | f1(y⁰) < xn< f2(y⁰) } . (28) The first paragraph is devoted to detail a few generalities about traces and upper/lower limits. We consider a general v ∈ L^∞(B) ∩ W_loc^1,2(B \ K) such that v = 0 in A₃. For each i = 1, 2, there exists v^∗_i ∈ L¹(Γi) such that for Hⁿ⁻¹-a.e. x ∈ Γ_i,

r→0limr⁻ⁿ ˆ

Ai∩B(x,r)

|v(y) − v_i^∗(x)| dLⁿ(y) = 0. (29) The boundary Γ_i is C¹ so for all x ∈ Γ_i, there a vector ν_i(x) ∈ Sⁿ⁻¹ such that

r→0limr⁻ⁿLⁿ((Ai∆H_i⁺(x)) ∩ B(x, r)) = 0 (30) where

H_i⁺(x) := { y ∈ Rⁿ| (y − x) · ν_i(x) > 0 } . (31) Therefore, (29) is equivalent to

r→0limr⁻ⁿ ˆ

H_i⁺(x)∩B(x,r)

|v(y) − v_i^∗(x)| dLⁿ(y) = 0. (32)

For Hⁿ⁻¹-a.e. x ∈ Γ₂, we detail the relationship between v(x)²+ v(x)² and v^∗_i(x). We fix x ∈ Γ2\ Γ₁ such that (32) is satisfied for i = 2. We have v = 0 on A₃ and B(x, r) is disjoint from A₁ for small r > 0 so

r→0limr⁻ⁿ ˆ

(X\A2)∩B(x,r)

|v| dLⁿ= 0 (33)

which is equivalent to

r→0limr⁻ⁿ ˆ

H₂⁻(x)∩B(x,r)

|v| dLⁿ= 0. (34)

Combining (32) for i = 2 and (34), we deduce

v(x)²+ v(x)²= v^∗₂(x)². (35) Next, we fix x ∈ Γ₁ ∩ Γ₂ such that (32) holds true for i = 1 and i = 2.

The surfaces Γ₁ and Γ₂ have necessary the same tangent plane at x and the vectors ν_i are opposed. Combining (32) for i = 1 and i = 2, we deduce

v²+ v²= (v^∗₁)²+ (v₂^∗)². (36)

We come back to our local minimizer u ∈ SBV_loc(X). We fix ϕ ∈ C_c¹(B).

For ε ∈ R, we define v : X → R by v :=

(u + εϕ in A₂

u in X \ A₂ (37)

It is clear that { v 6= u } ⊂⊂ B and that v is C¹ in X \ K. As K is Hⁿ⁻¹ locally finite in X, we have v ∈ SBV_loc(X) and S_v ⊂ K. Remember that u ≥ δ in A1 ∪ A₂, and u = 0 in A₃. We take ε small enough so that ε|ϕ|_∞ < δ. As a consequence v > 0 in A1 ∪ A₂ and v = 0 in A₃. Let us check the multiplicities on the discontinuity set. As we have seen before, Jv ∩ B ⊂ Γ₁ ∪ Γ₂. We observe that for x ∈ Γ₂ such that the trace u^∗₂(x) exists, we have

v₂^∗(x) = u^∗₂(x) + εϕ(x) (38) and for x ∈ Γ₁ such the trace u^∗₁(x) exists, we have

v₁^∗(x) = u^∗₁(x). (39)

Using the previous discussion, we deduce that for Hⁿ⁻¹-a.e. on Γ₂\ Γ₁,

v²+ v² = (u^∗₂+ εϕ)² (40)

= (u²+ u²) + 2εϕu^∗₂+ ε²|ϕ|² (41) that for Hⁿ⁻¹-a.e. on Γ₂∩ Γ₁,

v²+ v² = (u^∗₂+ εϕ)²+ (u^∗₁)² (42)

= (u²+ u²) + 2εϕu^∗₂+ ε²|ϕ|² (43) and that for Hⁿ⁻¹-a.e. on Γ₁\ Γ₂,

v²+ v²= (u^∗₁)² (44)

= u²+ u². (45)

Finally, it is clear that ˆ

B

|∇v|²dLⁿ= ˆ

B

|∇u|²dLⁿ+ 2ε ˆ

A2

h∇u, ∇ϕi dLⁿ + ε²

ˆ

A2

|∇ϕ|²dLⁿ. (46) We plug all these informations in the minimality inequality and we obtain that

0 ≤ 2ε ˆ

A2

h∇u, ∇ϕi dLⁿ+ 2ε ˆ

Γ2

ϕu^∗₂dHⁿ⁻¹+ C(ϕ)ε². (47) As this holds true for all small ε (positive or negative), we conclude that

ˆ

A2

h∇u, ∇ϕi dLⁿ+ ˆ

Γ2

ϕu^∗₂dHⁿ⁻¹= 0. (48)

3 Porosity of the singular part

The following result says that the part where K is not regular has many holes in a quantified way. It also holds true for the almost-minimizers of [12, Definition 2.1]. It is simpler to obtain than its Mumford-Shah counterpart (see [15]) because the ε-regularity theorem of [12] only requires to control the flatness.

Proposition 3.1 (Porosity). Let u ∈ SBV_loc(X) be a local minimizer. There exists 0 < r₀ ≤ 1, C ≥ 2 and 0 < α < 1 (all depending on n,δ) for which the following holds true. For all x ∈ K and all 0 < r ≤ r₀such that B(x, r) ⊂ X, there exists a smaller ball B(y, C⁻¹r) ⊂ B(x, r) in which K is C^1,α-regular.

Proof. The letter C is a constant ≥ 1 that depends on n, δ. The letter α is the constant of Theorem 2.6. For y ∈ K and t > 0 such that B(y, t) ⊂ X, we define the L^∞ flatness

β_K(y, t) := inf

V sup

z∈K∩B(y,t)

t⁻¹d(z, V ), (49)

where the infimum is taken over the affine hyperplanes V of Rⁿ passing through y. Note that in [7, (41.2)], the infimum is taken over all affine hyperplanes V of Rⁿ(not necessarily passing through y); this would decrease our β number but no more than a factor ¹₂. Indeed, if V is any hyperplane of Rⁿ and y⁰ is the orthogonal projection of y onto V , then the hyperplane V − (y⁰− y) is passing through y so we have we have

β_K(y, t) ≤ sup

z∈K∩B(y,t)

d(z, V − (y⁰− y)) (50)

≤ y⁰− y

+ sup

z∈K∩B(y,t)

d(z, V ) (51)

≤ 2 sup

z∈K∩B(y,t)

d(z, V ). (52)

We also observe that

βK,2(y, t)² ≤ t⁻⁽ⁿ⁻¹⁾Hⁿ⁻¹(K ∩ B(y, t))βK(y, t)² (53) so as soon as t is small enough for the Ahlfors-regularity to hold, we have β_K,2(y, t) ≤ Cβ_K(y, t).

Let r₀ be the minimum between the radius of Proposition 2.2 (Ahfors- regularity) and the radius of Proposition 2.4 (uniform rectifiability). We fix x ∈ K and 0 < r ≤ r₀ such that B(x, r) ⊂ X. According to Proposition 2.4, there exists an Ahlfors-regular and uniformly rectifiable set E such that K ∩ ¹₂B(x, r) ⊂ E. Moreover, the constants for the Ahfors-regularity and uniform rectifiability depends on n, δ. For y ∈ E and t > 0, we define as before

β_E(y, t) := inf

V sup

z∈E∩B(y,t)

t⁻¹d(z, V ), (54)

where the infimum is taken on the set of all affine hyperplanes V of Rⁿ passing through x. As E is Ahlfors-regular and uniformly rectifiable, the Weak Geometric Lemma [7, (73.13)] states that for all ε > 0, the set

{ (y, t) | y ∈ E, 0 < t < diam(E), β_E(y, t) > ε } (55) is a Carleson set. This means that for all ε > 0, there exists C₀(ε) ≥ 1 (depending on n, δ, ε) such that for all y ∈ E and all 0 < t < diam(E),

ˆ _t

0

ˆ

E∩B(y,t)

1{ β_E(z,s)>ε }(z) dHⁿ⁻¹(z)ds

s ≤ C₀(ε)tⁿ⁻¹. (56) We only apply this property with y := x. We observe that for all z ∈ K ∩ B(x,¹₄r) and for all 0 < s ≤ ¹₄r, we have K ∩ B(z, s) ⊂ E ∩ B(z, s) so β_K(z, s) ≤ β_E(z, s). Thus for all 0 < t < diam(K ∩ ¹₂B(x, r)) such that t ≤ ¹₄r, we have

ˆ _t

0

ˆ

K∩B(x,t)

1_{{ βK}(z,s)>ε }(z) dHⁿ⁻¹(z)ds

s ≤ C₀(ε)tⁿ⁻¹. (57) We only apply this property with t := ¹₄diam(K ∩ B(x, r)).

ˆ _t

0

ˆ

K∩B(x,t)

1{ β_K(z,s)>ε }(z) dHⁿ⁻¹(z)ds

s ≤ C₀(ε)tⁿ⁻¹. (58) Note that C⁻¹r ≤ t ≤ ¹₄r, where the first inequality comes from the Ahlfors- regularity of K. We are going to deduce from (58) that for all ε > 0, there exists C(ε) ≥ 1, a point z ∈ K ∩ B(x, t) and a radius s such that C(ε)⁻¹t ≤ s ≤ t and β_K(z, s) ≤ ε. We proceed by contradiction for some C(ε) to be precised. We have therefore

ˆ _t

0

ˆ

K∩B(x,t)

1{ β_K(z,s)>ε }(z) dHⁿ⁻¹(z)ds s

≥ Hⁿ⁻¹(K ∩ B(x, t)) ˆ _t

C(ε)⁻¹t

ds s

(59)

≥ Hⁿ⁻¹(K ∩ B(x, t)) ln(C(ε)) (60)

≥ C⁻¹tⁿ⁻¹ln(C(ε)) (61)

This contradicts (58) if C(ε) is too big compared to C₀(ε).

We fix ε > 0 (to be precised soon) and we assume that we have a cor- responding pair (z, s) as above. In particular, β_K,2(z, s) ≤ CβK(z, s) ≤ Cε.

According to the second statement of Theorem 2.6, we can fix ε (depending on n, δ) so that if r₀ ≤ ε, then K is C^1,α-regular in B(z, C⁻¹s).

4 Higher integrability of the gradient

Theorem 4.1. Let u ∈ SBV_loc(X) be minimal. There exists 0 < r0 ≤ 1, C ≥ 1 and p > 1 (depending on n, δ) such that the following holds true. For all x ∈ X, for all 0 ≤ r ≤ r₀ such that B(x, r) ⊂ X,

ˆ

1 2B(x,r)

|∇u|^2pdLⁿ≤ Cr^n−p. (62) The higher integrability is well known for weak solutions of elliptic sys- tems ([10, Theorem 2.1]). In this case, the proof consists in combining the Caccioppoli-Leray inequality and the Sobolev-Poincaré inequality to deduce that |∇u|ⁿ⁺²²ⁿ satisfies a reverse Hölder inequality. The higher integrability is then an immediate consequence of Gehring Lemma. In our case, u is still a weak solution of an elliptic system but we lack information about the regularity of K to carry out this method.

We draw inspiration from [9] but we simplify the proof by singling out an higher integrability lemma (Lemma 4.2 below) and a covering lemma (Lemma 4.3 below) and by removing the need of [9, Lemma 3.2] (the exis- tence of good radii).

Proof of Theorem 4.1. There exists 0 < r₀ ≤ 1, such that for all x ∈ X and all 0 ≤ R ≤ r₀ such that B(x, R) ⊂ X, one can applies Lemma 4.2 below in the ball B(x, R) to the function v := R|∇u|². The assumption (i) follows from the Ahlfors-regularity of K (Proposition 2.2). The assumption (ii) follows from the porosity (Proposition 3.1). The assumption (iii) follows from interior/boundary gradient estimates for the Robin problem and from the Ahlfors-regularity. In particular, the interior estimate can be derived from the subharmonicity of |∇u|² in X \ K and the boundary estimate is detailed in Lemma B.1 in Appendix B.

Lemma 4.2. We fix a radius R > 0 and an open ball B_Rof radius R. Let K be a closed subset of B_Rand v : B_R→ R⁺ be a non-negative Borel function.

We assume that there exists C₀ ≥ 1 and 0 < α ≤ 1 such that the following holds true.

(i) For all ball B(x, r) ⊂ B_R,

C₀rⁿ⁻¹≤ Hⁿ⁻¹(K ∩ B(x, r)) ≤ C₀rⁿ⁻¹. (63) (ii) For all ball B(x, r) ⊂ B_R centered in K, there exists a smaller ball

B(y, C₀⁻¹r) ⊂ B(x, r) in which K is C^1,α-regular (Definition 2.5).

(iii) For all ball B(x, r) ⊂ B_R such that K is disjoint from B(x, r) or K is C^1,α-regular in B(x, r) (Definition 2.5), we have

sup

1 2B(x,r)

v(x) ≤ C₀ R r



. (64)

Then there exists p > 1 and C ≥ 1 (depending on n, C₀) such that

1 2B_R

v^p ≤ C. (65)

The proof of Lemma 4.2 takes advantage of the following covering lemma.

We use the notation Γ_(x,en) defined at line (18). The assumption (ii) says that in each double ball 2B_k, the set E is an union of Lipschitz graphs which are close to an hyperplane.

Lemma 4.3 (Covering Lemma). Let E ⊂ Rⁿ be a bounded set. Let (B_k) be a family of open balls of center x_k∈ Rⁿand radius R_k> 0. We assume that

(i) for all k 6= l, 2B_k∩ B_l= ∅;

(ii) for all k, for all x ∈ E ∩ 2B_k, there exists a vector e_n ∈ Sⁿ⁻¹ and a

1

2-Lipschitz function f : e^⊥_n → R such that |f | ≤ ¹₂R_k and

x ∈ Γ_(xk,en)(f ) ∩ 2B_k⊂ E. (66) Let 0 < r ≤ inf_kR_k. There exists a sequence of open balls (D_i)_i∈I of radius r and centered in E \S

kBk such that E \[

k

Bk⊂[

i∈I

Di (67)

and the balls (20⁻¹Di)i∈I are pairwise disjoint and disjoint from S

kBk. Proof. Let 0 < r₀ ≤ inf_kR_k. We introduce the set

F := E \[

k

Bk. (68)

The goal is to cover F with a controlled number of balls of radius r₀. Let r be a radius 0 < r ≤ r0 which will be precised during the proof. As F is bounded, there exists a maximal sequence of points (x_i) ∈ F such that B(x_i, r) ⊂ Rⁿ\S

kB_k and |x_i− x_j| ≥ r. For i 6= j, we have |x_i− x_j| ≥ r so the balls (B(x_i,¹₂r))i are disjoint. Next, we show that

F ⊂[

i

B(xi, 10r). (69)

Let x ∈ F . If B(x, r) ⊂ Rⁿ\S

kB_k, then by maximality of (x_i), there exists i such that x ∈ B(x_i, r) ⊂ B(x_i, 10r). Now we focus on the case where there exists an index k₀ such that B(x, r) ∩ B_k0 6= ∅. The radius of B_k0 is denoted by R and we assume without loss of generality that its center is 0.

As x ∈ F = E \S

kB_k and B(x, r) ∩ B(0, R) 6= ∅, we have R < |x| < R + r.

We are going to build a point y ∈ E such that R + r < |y| < R + 7r and

|x − y| < 9r.

Since r ≤ R, we observe that x ∈ B(0, 2R). According to the assumptions of the lemma, there exists two scalars 0 < ε, L ≤ ¹₂, a vector e_n∈ Sⁿ⁻¹ and a L-Lipschitz f : e^⊥_n → R such that |f | ≤ εR and

x ∈ { y ∈ B(0, 2R) | y_n= f (y⁰) } ⊂ E. (70) Here, we have decomposed each point y ∈ Rⁿ under the form y = y⁰+ y_ne_n where y⁰ ∈ n^⊥ and y_n∈ R. The estimate R < |x| < R + r can be rewritten

R <

x⁰+ f (x⁰)e_n

< R + r. (71)

We consider t ≥ 1 such that |tx⁰+ f (x⁰)en| = R + 4r and we estimate how close tx⁰ is to x⁰. We have

 x⁰

≥ q

R²− |f (x⁰)|² (72)

 tx⁰

≤ q

(R + 4r)²− |f (x⁰)|² (73) so

tx⁰− x⁰ ≤

q

(R + 4r)²− |f (x⁰)|²− q

R²− |f (x⁰)|² (74)

≤ 4Rr + 8r² q

R²− |f (x⁰)|²

. (75)

We assume r ≤ ¹₈R and we recall that |f (x⁰)| ≤ εR with ε ≤ ¹₂ so this simplifies to

tx⁰− x⁰

≤ 5r

√

1 − ε² < 6r. (76)

Next, we define y := tx⁰+ f (tx⁰)e_n and we recall that f is L-Lipschitz with L ≤ ¹₂ to estimate

y − [tx⁰+ f (x⁰)e_n] =

f (tx⁰) − f (x⁰)

 (77)

< 3r. (78)

Since |tx⁰+ f (x⁰)en| = R + 4r, this yields

R + r < |y| < R + 7r. (79) We also estimate

|y − x| ≤

tx⁰− x⁰ +

f (tx⁰) − f (x⁰)

 (80)

< 9r. (81)

As r ≤ ¹₈R, the inequalities (79) imply y ∈ B(0, 2R) and thus y ∈ E. We are going to justify that B(y, r) ⊂ Rⁿ\S

kBk. We recall that B(0, 2R) is disjoint from all the other balls of the family (B_k). By (79), we observe that B(y, r) ⊂ B(0, R + 8r) \ B(0, R) (82)

⊂ B(0, 2R) \ B(0, R) (83)

and our claim follows. By maximality of the family (x_i), there exists i such that |y − x_i| < r and in turn by (81), |x − x_i| < 10r. We finally choose r := ₁₀¹r0. The balls (D_i) are given by Di:= B(xi, 10r) = B(xi, r0).

Proof of Lemma 4.2. We observe that any ball B ⊂ B_Rand for p ≥ 1, ˆ

B

v^pdLⁿ= ˆ ∞

0

Lⁿ(B ∩ { v^p > t }) dt (84)

= p ˆ ∞

0

s^p−1Lⁿ(B ∩ { v > s }) ds (85) and for M ≥ 1,

p ˆ _∞

1

s^p−1Lⁿ(B ∩ { v > s }) ds

≤ p

∞

X

h=0

ˆ _M^h+1

M^h

s^p−1Lⁿ(B ∩ { v > s }) ds

(86)

≤ p

∞

X

h=0

ˆ _M^h+1

M^h

s^p−1ds

!

Lⁿ(B ∩ { v > M^h}) (87)

≤ (M^p− 1)

∞

X

h=h

M^hpLⁿ(B ∩ { v > M^h}). (88)

Thus, it suffices to prove that there exists N > M ≥ 1, C ≥ 1 (depending on n, C₀) such that for all h ≥ 0,

Lⁿ(¹₂BR∩ { v > M^h}) ≤ CRⁿN^−h (89) and then take p > 1 such that M^pN⁻¹ < 1.

To simplify the notations, we change the constant C₀ so that (ii) yields that 40B(y, C₀⁻¹r) ⊂ B(x, r) and that K is C^1,α-regular in 4B(y, C₀⁻¹r).

Let M := max { 4C₀,¹₄C₀²} ≥ 4. We define for h ≥ 1,

A_h := { x ∈¹₂B_R\ K | v > M^h} . (90) The proof is based on the fact that A_h is at distance ∼ M^−hR from K and has many holes of size ∼ M^−hR near K. We justify more precisely these

observations. For the first one, let h ≥ 1, let x ∈ A_h and assume that B(x, C0M^−hR) is disjoint from K. Then we use property (ii) to estimate

v(x) ≤ M^h. (91)

This contradicts the definition of A_h. We deduce that there exists y ∈ K such that |x − y| < C₀M^−hR. For the second observation, let h ≥ 2, let x ∈ K ∩¹⁵₁₆BR and apply the porosity property to the ball B(x, M^−hR).

We obtain an open ball B ⊂ X centered in K, of radius C₀⁻¹M^−hR and such that K is C^1,α-regular in 4B. Then by point (iii) and since M ≥ ¹₄C₀²,

sup

2B

v ≤ ¹₄C₀²M^h≤ M^h+1 (92) In particular, 2B is disjoint from A_h+1.

We start the proof by defining for h ≥ 1,

r(h) := M^−hR (93)

R(h) := 3

4 + M^−h+1



R. (94)

The sequence (R(h)) is decreasing, lim_h→∞R(h) = ³₄R and R(h+1)+r(h) ≤ R(h). For each h ≥ 1, we build an index set I(h) and a family of balls (B_i)_i∈I(h) as follow. First we define I(1) := ∅ and (B_i)_i∈I(1) := ∅. Let h ≥ 2 be such that (B_i)_i∈I(1), . . . , (B_i)_i∈I(h−1)have been built. We assume that the index sets I(g), where g = 1, . . . , h−1, are pairwise disjoint. We assume that for all i ∈ I_g, the balls B_i have radius C₀⁻¹r(g) = C₀⁻¹M^−gR. We assume that for all indices i, j ∈Sh−1

g=1I(g) with i 6= j, we have that 2B_i ∩ B_j = ∅ and that K is C^1,α-regular in 2B_i. Then, we introduce the sets

K_h:= K ∩ B_R(h)\

h−1

[

g=1

[

i∈I(g)

B_i (95)

K_h^∗:= K ∩ B_R(h+1)\

h−1

[

g=1

[

i∈I(g)

Bi. (96)

According to Lemma 4.3, there exists a sequence of open balls (D_i)_i∈I(h) centered in K_h^∗ of radius r(h) = M^−hR such that

K_h^∗⊂ [

i∈I(h)

D_i, (97)

and such that the balls (20⁻¹D_i) are pairwise disjoint and disjoint from Sh−1

g=1

S

i∈I(g)Bi. We can assume that index set I(h) is disjoint from the sets

I(g), g = 1, . . . , h − 1. Since R(h + 1) + r(h) ≤ R(h), we observe that the balls (20⁻¹Di) are included in

B_R(h)\

h−1

[

g=1

[

i∈I(g)

B_i. (98)

Next, we apply the porosity to the balls (D_i). For each i ∈ I(h), there exists an open ball B_i centered in K, of radius C₀⁻¹M^−hR such that Bi ⊂ 40⁻¹Di, K is C^1,α-regular in 4 B_i and by (iii),

sup

2Bi

v ≤ ¹₄C₀²M^h≤ M^h+1 (99) We should not forget to mention that for all i ∈ I(h), we have 2B_i⊂ 20⁻¹D_i so 2B_i is disjoint from all the other balls we have built so far.

Now, we estimate Lⁿ(Ah) for h ≥ 1. We show first that the points of A_h cannot be too far from K_h^∗. Let x ∈ A_h. We have seen earlier that there exists y ∈ K such that |x − y| < C₀M^−hR. We are going to show that y ∈ K_h^∗. Since |x| ≤ ¹₂R and M ≥ 4C0, we have

|y| ≤ ¹₂R + C₀M^−hR ≤ ³₄R. (100) Let us assume that there exists g = 1, . . . , h − 1 and i ∈ I(g) such that y ∈ B_i. The radius of B_i is C₀⁻¹M^−(h−1)R and since |x − y| < C₀M^−hR, we have x ∈ 2B_i. However 2B_i is disjoint from A_h by construction. We have shown that y ∈ K_h^∗. As a consequence, there exists i ∈ I(h) such that y ∈ D_i. The radius of D_i is r(h) = M^−hR and |x − y| < C₀M^−hR so

Ah ⊂ [

i∈I(h)

(1 + C0)Di. (101)

This allows to estimate

Lⁿ(A_h) ≤ ωn(1 + C0)ⁿ|I(h)|r(h)ⁿ (102) where ω_n is the Lebesgue measure of the unit ball.

Next, we want to control |I(h)|. The balls (20⁻¹Di)_i∈I(h) are disjoint and included in the set B(R(h)) \Sh−1

g=2

S

i∈I(g)Bi so by Ahlfors-regularity, C₀⁻¹20⁻⁽ⁿ⁻¹⁾r(h)⁽ⁿ⁻¹⁾|I(h)| ≤ X

i∈I(h)

Hⁿ⁻¹(K ∩ 12⁻¹Di) (103)

≤ Hⁿ⁻¹(K_h). (104)

We are going to see that Hⁿ⁻¹(K_h) is bounded from above by a decreasing geometric sequence. We have

Hⁿ⁻¹(K_h^∗) ≤ X

i∈I(h)

Hⁿ⁻¹(K ∩ D_i) (105)

≤ C₀ X

i∈I(h)

r(h)ⁿ⁻¹ (106)

≤ C₀ⁿ⁺¹ X

i∈I(h)

(C₀⁻¹r(h))ⁿ⁻¹ (107)

≤ C₀ⁿ⁺¹ X

i∈I(h)

Hⁿ⁻¹(K ∩ Bi) (108)

≤ C₀ⁿ⁺¹Hⁿ⁻¹(Kh\ K_h+1). (109) We deduce

Hⁿ⁻¹(K_h) ≤ C₀ⁿ⁺¹Hⁿ⁻¹(K_h\ K_h+1) + Hⁿ⁻¹(K ∩ B_R(h)\ B_R(h+1)). (110) We rewrite this inequality as

Hⁿ⁻¹(Kh+1) ≤ λ⁻¹Hⁿ⁻¹(Kh) + C₀⁻⁽ⁿ⁺¹⁾Hⁿ⁻¹(K ∩ B_R(h)\ B_R(h+1)) (111) where λ := C₀ⁿ⁺¹(C₀ⁿ⁺¹− 1)⁻¹ > 1. Then, we multiply both sides of the inequality by λ^h+1:

λ^h+1Hⁿ⁻¹(K_h+1)

≤ λ^hHⁿ⁻¹(Kh) + C₀⁻⁽ⁿ⁺¹⁾λ^−hHⁿ⁻¹(K ∩ B_R(h)\ B_R(h+1))

(112)

≤ λ^hHⁿ⁻¹(K_h) + Hⁿ⁻¹(K ∩ B_R(h)\ B_R(h+1)). (113) Summing this telescopic inequality, we obtain that for all h ≥ 1,

λ^hHⁿ⁻¹(Kh) ≤ 2Hⁿ⁻¹(K ∩ BR) (114)

≤ 2C₀Rⁿ⁻¹. (115)

In summary, we have proved that for some constant C ≥ 1, λ > 1 (depending on n, C₀) and for h ≥ 1

Lⁿ(A_h) ≤ CRⁿ(λM )^−h. (116)

5 Dimension of the singular part

Notation. The Hausdorff dimension of a set A ⊂ Rⁿ is defined by

dimH(A) := inf { s ≥ 0 | H^s(A) = 0 } . (117)

We take the convention that for s < 0, the term H^s-a-e. means everywhere and the inequality dim_H(A) < 0 means A = ∅.

The goal of this section is to explain the link between the integrability exponent of the gradient and the dimension of the singular part. It has been first observed for the Mumford-Shah functional by Ambrosio, Fusco, Hutchinson in [8].

Theorem 5.1. Let u ∈ SBV_loc(X) be a local minimizer. We define

Σ := { x ∈ K | K is not regular at x } . (118) For p > 1 such that |∇u|² ∈ L^p_loc(X), we have

dimH(Σ) ≤ max { n − p, n − 8 } < n − 1. (119) Remark 5.2. In dimension n ≤ 7, Caffarelli–Kriventsov have shown that if a point x ∈ K is at the boundary of two local connected components where u > 0 or if it is a 0-density point of { u = 0 }, then x is a regular point ([4, Theorem 8.2]). In dimension n = 2, they show furthermore that if x is at the boundary of a connected component of { u = 0 }, then it is a regular point ([4, Corollary 9.2]). Thus in the planar case, a point of Σ must be an acculumation point of connected components of { u = 0 }. There is however no known example of such a situation.

Theorem 5.1 will be proved very easily with the help of [4, Theorem 8.2]

and the following well-known result.

Lemma 5.3. Let v ∈ L^p_loc(X) for some p ≥ 1 and let s < n. Then, for H^n−p(n−s)-a.e. x ∈ X,

r→0limr^−s ˆ

B(x,r)

v dLⁿ= 0. (120)

Proof. Without loss of generality, we assume v ≥ 0. We start with the case p = 1. We define µ as the measure vLⁿand we want to show that for H^s-a.e.

x ∈ X, we have

r→0limr^−sµ(B(x, r)) = 0. (121) If s < 0, the limit is indeed 0 for every x ∈ X. In the case 0 ≤ s < n, we fix a closed ballB ⊂ X, a scalar λ > 0 and a set

A := { x ∈ B | lim sup

r→0

r^−sµ(B(x, r)) > λ } . (122) According to [1, Theorem 2.56],

µ(A) ≥ λH^s(A). (123)

As A ⊂ B and µ is a Radon measure, we have µ(A) < ∞. Then (123) gives H^s(A) < ∞ and since s < n, Lⁿ(A) = 0. The measure µ is dominated by

Lⁿ so µ(A) = 0 and now (123) gives H^s(A) = 0. We can take a sequence of scalars λ_k→ 0 to deduce

H^s({ x ∈ B | lim sup

r→0

r^−sµ(B(x, r)) > 0 }) = 0. (124) We can then conclude that

H^s({ x ∈ X | lim sup

r→0

r^−sµ(B(x, r)) > 0 }) = 0. (125) by covering X with a a sequence of closed balls B_k⊂ X.

Now we come to the general case p ≥ 1. Let us fix t < n. For x ∈ X and for r > 0, the Hölder inequality shows that

r⁻

 n−ⁿ_p

ˆ

B(x,r)

v dLⁿ≤ ˆ

B(x,r)

v^pdLⁿ

!¹_p

(126) so

r⁻

 n+_p^t−ⁿ

p

ˆ

B(x,r)

v dLⁿ≤ r^−t ˆ

B(x,r)

v^pdLⁿ

!¹

p

. (127)

We apply the first part to see that for H^t-a.e. x ∈ X,

r→0limr⁻



n+_p^t−ⁿ_pˆ

B(x,r)

v dLⁿ= 0. (128)

The scalar t such that s = n +_p^t −ⁿ_p is t := n − p(n − s) < n.

Proof of Theorem 5.1. According to Lemma 5.3, we have for H^n−p-a.e. x ∈ X,

r→0limω2(x, r) = 0 (129)

and according to [4, Theorem 8.2], the set n

x ∈ X ∩ Σ    lim

r→0ω₂(x, r) = 0o

(130) has a Hausdorff dimension ≤ n − 8.

Appendices

A Generalities about BV functions

We recall a few definitions and results from the theory of BV functions ([1]).

We work in an open set X of the Euclidean space Rⁿ(n > 1). When a point x ∈ X is given, we abbreviate the open ball B(x, r) as B_r.

Let u ∈ L¹_loc(X). The upper and lower approximate limit of u at at a point x ∈ X are defined by

u(x) := inf { t ∈ R | lim

r→0r⁻ⁿ ˆ

{ u>t }∩B_r

(u − t) dLⁿ= 0 } , (131) u(x) := sup { t ∈ R | lim

r→0r⁻ⁿ ˆ

{ u<t }∩B_r

(t − u) dLⁿ= 0 } . (132) The functions u, u : X → R are Borel and satisfies u ≤ u. We have two examples in mind. We say that x is a Lebesgue point if there exists t ∈ R such that

r→0lim Br

|u − t| dLⁿ= 0. (133)

We then have u(x) = u(x) = t and we denote t by ˜u(x). The set of non- Lebesgue points x ∈ X is called the singular set S_u. Both the set S_u and the function X \ S_u → R, x 7→ ˜u(x) are Borel ([1, Proposition 3.64]). The Lebesgue differentiation theorem states that for Lⁿ-a.e. x ∈ X, we have x ∈ X \ S_u and u(x) = ˜u(x). We say that x is a jump point if there exist two real numbers s < t and a (unique) vector ν_u(x) ∈ Sⁿ⁻¹ such that

r→0lim H⁺∩Br

|u(y) − s| dLⁿ(y) = 0 (134a)

r→0lim H⁻∩Br

|u(y) − t| dLⁿ(y) = 0, (134b) where

H⁺ := { y ∈ Rⁿ| (y − x) · ν_u(x) > 0 } (135a) H⁻ := { y ∈ Rⁿ| (y − x) · ν_u(x) < 0 } . (135b) We then have u(x) = t and u(x) = s. The set of jump points x ∈ X is called the jump set J_u. Both the set J_u and the function J_u → Sⁿ⁻¹, x 7→ νu(x) are Borel ([1, Proposition 3.69]).

Here we summarize [1, Proposition 3.76, 3.78]. Let u ∈ BV(X). The singular set S_u is Hⁿ⁻¹ rectifiable and Hⁿ⁻¹(Su\ J_u) = 0. According to the Besicovitch derivation theorem, we can write

Du = D^au + D^su (136)

where D^au is the absolutely continuous part of Du with respect to Lⁿ and D^su is the singular part of Du with respect to Lⁿ. As a consequence, there exists a unique vector-valued map ∇u ∈ L¹(X; Rⁿ), the approximate gra- dient, such that D^au = ∇uLⁿ. The measures Lⁿ and kD^suk are mutually singular which means that there exists a Borel set S ⊂ X such that

Lⁿ(S) = kD^suk(Rⁿ\ S) = 0. (137)