The case of a signed measure

X

k

ˆ

µ^q(U_k) : A_k⊂U_k open set )

=X

k

ˆ µ^q(A_k).

Let us summarize the results contained in Proposition 4.11, Remark 4.5, Proposition 4.14 and Corollary 4.15:

Theorem 4.16. Let Ω⊂Rⁿbe an open set, letµ be a positive Borel measure inΩand let µˆ^q defined as in Definition 4.8 starting from the pre-measure q defined as in (4.8). Then, the following holds:

1. µˆ^q is a metric outer measure, coinciding with the measureµ˜^q.

2. there exists a dimensional constant C_n>1 such that for any Borel set A⊂Ω, 1

C_nµ(A)6µˆ^q(A)6inf{µ(U)|A⊂U open set }.

3. if moreover µ is outer regular, for instance if µ is finite on bounded sets (i.e., if µ is a Radon measure), then µ and the positive Borel measure associated with the outer measure µˆ^q (still denoted as µˆ^q) are equivalent, that is, 1

C_nµ6µˆ^q6µ.

Remark 4.8. We stress that µ is not generally assumed to be finite on open sets, unless explicitly mentioned.

4.2.3 The case of a signed measure

Our aim is to prove that the packing-type reconstruction applied to a signed measure µ with pre-measures q±(Br(x)) = ¹_rRr

s=0µ(Bs(x))ds

± produces a signed measure ˆµ^p whose positive and negative parts are comparable with those of µ.

Theorem 4.17. Let Ω ⊂ Rⁿ be an open set, let µ = µ⁺−µ⁻ be a signed Borel measure in Ω, finite on bounded sets. Let C ={closed balls Br(x)⊂X} and take µˆ^q+ and µˆ^q− as in Definition 4.8, corresponding to the pre-measures q±:C →R+∪ {+∞}defined by

q±(Br(x)) = 1

r Z r

s=0

µ(Bs(x))ds

±

. Then the following holds.

(i) µˆ^q+, µˆ^q− are metric outer measures, finite on bounded sets.

Proof. Showing the countable sub-additivity of ˆµ^q+ and ˆµ^q− under the assumptionP

kµ(A_k)<+∞

(see Proposition 4.11) does not require the special form of the pre-measureq but only the fact that, for any closed ballB,

q(B)6µ(B). (4.21) thus (4.21) is satisfied. This is sufficient to get the sub-additivity, under the assumptionP

kµ(Ak)<

+∞. It is also sufficient for concluding that, for any open set U ⊂Ω, ˆ

µ^q+(U)6µ⁺(U). This gives the proof of (i).

Let now A ⊂ Ω be a Borel set. If µ⁺(A) < +∞, let A ⊂ U open set, let δ > 0 and apply

By outer regularity ofµ⁻ (which is Borel and finite on bounded sets) there exists a sequence of open sets (U_k²)_k such that, for all k, we getA⊂U_k² and

µ⁻(U_k²)−−−→

k→∞ µ⁻(A).

For all k, letU_k =U_k¹∩U_k², thenU_k is an open set, A⊂U_k and, by monotonicity, ˆ

µ^q+(A)6µˆ^q+(U_k)6µˆ^q+(U_k¹), µ⁻(A)6µ⁻(Uk)6µ⁻(U_k²), therefore

ˆ

µ^q+(U_k)−−−→

k→∞ µˆ^q+(A) and µ⁻(U_k)−−−→

k→∞ µ⁻(A).

Evaluating (4.23) on the sequence (U_k)_k and letting kgo to +∞, we eventually get ˆ

µ^q+(A)>C_nµ⁺(A)−µ⁻(A). (4.24) Owing to Hahn decomposition of signed measures, we consider a Borel set P ⊂ Ω such that for all Borel A⊂Ω it holds

µ⁺(A) =µ⁺(A∩P) =µ(A∩P) and µ⁻(A) =µ(A∩(Ω−P)). Finally, let A⊂Ω be a Borel set, then by (4.24) applied to A∩P we find

ˆ

µ^q+(A)>µˆ^q+(A∩P)>Cnµ⁺(A∩P)−µ⁻(A∩P)

=C_nµ⁺(A)

It remains to show that if µ⁺(A) = +∞, then ˆµ^q+(A) = +∞. Let A ⊂Ω be a Borel set such that µ⁺(A) = +∞, and recall that, by definition of signed measure, µ⁻ must be finite. Let U ⊂Ω be an open set containing A. The next argument is exactly the same as in the proof of Proposition 4.14 (positive case). Given δ >0, we apply Corollary 4.13 toµ⁺ with

F_δ^U ∩ {B|B ⊂U} .

This gives ζ_n countable families G_δ¹, . . . ,G_δ^ζn of balls inF_δ∩ {B|B ⊂U} such that

U ⊂U0∪

ζn

[

j=1

G

B∈G_δ^j

B withµ⁺(U0) = 0,

hence

ζn

X

j=1

µ⁺





 G

B∈G_δ^j

B





>µ⁺(U) = +∞.

Consequently there exists j₀ ∈ {1, . . . , ζ_n}such that µ⁺ F

B∈G_δ^j0B

= +∞ and ˆ

µ^q_δ⁺(U)> X

B∈G_δ^j0

q₊(B) =X

l

1 rl

Z rl

r=0

µ(B_r(x_l))dr

+

>X

l

1 rl

Z rl

r=0

µ(Br(xl))dr=X

l

1 rl

Z rl

r=0

µ⁺(Br(xl))dr−X

l

1 rl

Z rl

r=0

µ⁻(Br(xl))dr

>X

l

1

2µ⁺(Brl

2(xl))−µ⁻





 G

B∈G^j0

δ

B







> 1 2

X

l

1

2ⁿ⁺¹µ⁺(B_rl(x_l))−µ⁻(U) =C_nµ⁺





 G

B∈G_δ^j0

B





−µ⁻(U)

= +∞.

Finally, letting δ→0, we obtain that ˆµ^q(U) = +∞for all open set U ⊃A, hence that ˆµ^q(A) = +∞.

This completes the proof of (ii) and thus of the theorem.

CHAPTER 5

R´ egularisation de la variation premi` ere

Introduction

Recall that our purpose is to study surface representation and discretization, and then geometric energies defined on surfaces, in the unified setting provided by varifolds. In Chapter 2 we introduced discrete volumetric varifolds, which are volumetric approximations associated with a family of meshes of the space, and point cloud varifolds. We stated an approximation result (Theorem 2.1) ensuring that rectifiable varifolds can be approximated by discrete varifolds in the sense of weak–∗convergence, and under some additional assumptions, in the sense of flat distance, with an estimate on the convergence rate depending on the size of the mesh and on the H¨older regularity of the tangent plane of the approximated varifold. The result raised the following question: given a sequence of approximating d–varifolds (V_i)_i weakly–∗ converging to some d–varifold V, is it possible to introduce a notion of approximate regularity for the approximating sequence (Vi) and to connect it with the regularity of the limitV? We have already studied this question from the point of view of rectifiability (Question 1.2) and proposed an answer in Chapter 3 (Theorem 3.4). We then asked the question from the point of view of curvature, let us recall it:

Question. 1.3 What conditions on a weakly–∗ converging sequence of varifolds (not necessarily rectifiable) ensure that the limit varifold has bounded first variation?

In Chapter 4, we tried to answer this question by observing that when a d–varifold has bounded first variation, some averaged quantity (involving the first variation of balls centered at a same point) can be written in a way that makes sense for any varifold:

1 R

Z R r=0

δV(B_r(x))dr=−1 R

Z

BR(x)×G_d,n

Π_S(y−x)

|y−x| dV(y, S). (5.1) We then used Carath´eodory and packing–type constructions to recover the vector–valued measureδV using only the values (5.1) for any ball B_R(x). We also noticed that (5.1) is simply the convolution ofδV with a suitable kernelTR: that is why the right hand side of (5.1) makes sense for any varifold.

Indeed, even when a varifold does not have bounded variation, the first variationδV is by definition a linear form on C¹_c or equivalently a distribution of order 1 which can thus always be convolved with a C¹ or Lipschitz function as is the kernel TR (see (5.9)). And it led us to the approach developed in this chapter. We obtain the following results, giving an answer to Question 1.3, which can be compared with Theorems 3.3 and 3.4 of Chapter 3.

Theorem. 5.4. Let Ω⊂Rⁿbe an open set and V be ad–varifold inΩwith finite mass kVk(Ω). Let Then V has bounded first variation and |δV|(Ω)is bounded by the previous supremum.

In the particular case when ρ_ε=T_ε is the tent kernel (5.9), the assumption (5.2) rewrites sup be a symmetric non negative function such thatR

Rⁿρ= 1andsuppρ⊂B1(0)and letρε(x) = _ε¹nρ ^x_ε . Assume that there exists a positive decreasing sequence (εi)i, tending to0, such that

sup Then there exists a subsequence (V_ϕ(i))_i weakly–∗ converging in Ω to a d–varifold V, V has bounded first variation and |δV|(Ω)is bounded by the previous supremum.

This convolution of the first variation actually provides a notion of approximate curvature which is convenient to define approximate Willmore energies. More precisely, with the above notations, we define the approximate Willmore energie associated with ρ andε >0 as

W_ε^p(V) =

Question 5.1. Do the approximate Willmore energies W_ε^p Γ–converge in the space of d–varifolds?

And if so, is the classical Willmore energy the Γ–limit?

LetW^p denote the classicalp–Willmore energy. We obtained the following Γ–convergence results (the detailed statements are given in Theorems 5.8 and 5.10). Notice that in the case p = 1, the Γ–limit is not the 1–Willmore energy but the total variation of the first variation.

W_ε^p −−−^Γ*

ε→0 W^p for 1< p <+∞

W_ε¹ −−−^Γ*

ε→0 the total variation of the first variation6=W¹.

However, this Γ–convergence result is not fully satisfactory. Indeed, in a practical way, for a fixed ε, we want to minimize W_ε^p, but not in the whole space of varifolds, we rather want to minimize in some subclass like the family of discrete volumetric varifolds. We thus need to make a link between the scale parameter ε of the energy W_ε^p and the scale of the discrete objects (with the size δi of the meshesK_i if we consider discrete volumetric varifoldsV_i∈ A_δi(K_i), defined in Chapter 2 (2.1)). More generally, in terms of Γ–convergence, this means that to each ε > 0 corresponds an approximation space A_ε and that the Γ–convergence must hold in these approximation spaces, denoted

W_ε^p −−−−−→^Γ

A_ε, ε→0 W^p. (5.4)

In other words:

– for any sequence (V_ε)_ε of d–varifolds such thatV_ε∈ A_ε and V_ε −−−^∗*

ε→0 V, W^p(V)6lim inf

ε→0 W_ε^p(V_ε).

Notice that as the Γ–lim inf holds for any sequence of d–varifolds (V_ε)_ε, it holds in particular when restricting the approximation spaces.

– for any d–varifold V, there exists a sequence (Vε)ε ofd–varifolds such thatVε∈ A_ε,Vε

−−−∗*

ε→0 V and

lim sup

ε→0

W_ε^p(Vε)6W^p(V).

Conversely to the Γ–lim inf property, the Γ–lim sup property must be reconsidered when re-stricting the approximation spaces.

So that we have to study the following question:

Question 5.2. Given

– a subset of d–rectifiable varifoldsA;

– a family (A_ε)_ε where eachA_ε is a prescribed subset of discrete volumetric varifolds.

For any rectifiable d–varifold V ∈ A, is there a sequence (V_ε)ε of discrete volumetric varifolds such that

– for any ε >0,V_ε∈ A_ε; – Vε

−−−∗*

ε→0 V, – and W_ε^p(Vε)−−−^∗*

ε→0 W^p(V)?

We study this question for A_ε of the form A_δ(K) (the space of discrete volumetric varifolds associated to the mesh K of size sup_K∈Kdiam(K)6δ defined in (2.1)): forW_ε¹, we obtain a partial result, and for W_ε^p, with p >1, the question stands. One difficulty is to make a connection between the parameter ε and the size of the mesh δ. This is linked to what we called the “accuracy of the approximation spacesA_δ(K)” defined in Remark 2.2 by

d^asym_H (V, W) := sup

V∈A

W∈Ainf_δ(K)∆^1,1(W, V), (5.5) where ∆^1,1 is the distance introduced in Definition 1.15 by:

∆(V, W) = sup

Z

ϕ dV − Z

ϕ dW

: ϕ∈Lip₁, kϕk_∞61

.

Recall that (5.5) is not the Hausdorff distancedH(A,A_δ(K)) since we care only of the approximation of Aby A_δ(K) and not the contrary:

dH(A,A_δ(K)) = max

d^asym_H (A,A_δ(K)), d^asym_H (A_δ(K),A) . In Remark 2.2, we state that for

A^β ={rectifiabled–varifolds satisfying aβ–H¨older condition on the tangent plane(2.5)} , we have the estimate

d^asym_H (A^β,A_δ(K))6

δ+ 2Cδ^β

kVk(Ω) ,

which can be controlled by the size of the mesh δ and the mass of the varifold kVk(Ω). Thanks to this estimate, we explicit a condition (5.6) linkingεandδ and ensuring the Γ–lim sup property in the space of discrete volumetric varifolds:

Theorem. 5.13. Let Ω⊂Rⁿ be an open set and letV =v(M, θ) be a rectifiabled–varifold inΩ. Let δ_i↓0 be a sequence of infinitesimals and (K_i)_i a sequence of meshes satisfying

sup

K∈Ki

diam(K)6δ_i −−−−→

i→+∞ 0.

Let the approximation spaces (A_δi(K_i))_i be defined as in (2.1) and let the kernel ρ ∈ W^2,∞ be as in (5.13). Assume that there exist 0< β <1 and C such that for kVk–almost every x, y∈Ω,

kT_xM−TyMk6C|x−y|^β.

Then, there exists a sequence of discrete volumetric varifolds (Vi)i such that (i) for all i, Vi∈ A_δi(K_i),

(ii) V_i−−−−^∗ *

i→+∞ V,

(iii) For any sequence of infinitesimals ε_i ↓0 satisfying δ_i^β

ε²_i −−−−→

i→+∞ 0. (5.6)

we have,

W_ε¹

i(Vi)−−−−→

i→+∞ W¹(V).

In order to understand better the approximate first variation provided by the regularizationδV∗ρ_ε, we ask the following question:

Question 5.3. – Given ad–varifoldV, is the regularizationδV ∗ρ_ε of the first variationδV, the first variation δ

Vbε

of some varifold Vbε?

– And if so, isVbε the regularization (in a sense to be defined) ofV? The construction can be done explicitly:

Theorem. 5.14. LetΩ⊂Rⁿ be an open set andV ad–varifold inΩwith finite masskVk(Ω)<+∞.

Let ε >0 and ρε as in (5.13). Define thed–varifold Vbε by: for everyψ∈C⁰_c(Ω×Gd,n), D

Vbε, ψ E

=hV,(y, S)7→ψ(·, S)∗ρε(y)i or equivalently,

Z

Ω×G_d,n

ψ(y, S)dVb_ε(y, S) = Z

(y,S)∈Ω×G_d,n

Z

x∈Rⁿ

ψ(x, S)ρ_ε(y−x)dLⁿ(x)dV(y, S). Then,

1. kVb_εk=kVk ∗ρ_ε, 2. δ

Vbε

=δV ∗ρε.

We observe that the mass kVb_εk is the convolution of kVk and in Proposition 5.15, we point out that the tangential part νb_x^εofVbε=kVbεk ⊗νb_x^εis generally not a Dirac mass nor a combination of Dirac masses.

Section 5.1 deals with answering to Question 1.3 thanks to a regularization of the first variation by convolution. In Section 5.2, we build approximate Willmore energiesW_ε^pand study the Γ–convergence to the p–Willmore energy in the space of varifolds (Question 5.1). In Section 5.3, we address the Question 5.2 of a different Γ–convergence result: we want that the Γ–lim sup–approximation property holds for sequences of a prescribed type of varifolds, for instance for discrete volumetric varifolds. In section 5.4, we answer Question 5.3, giving a construction of ad–varifoldVb_εsuch thatδ

Vb_ε

=δV∗ρ_ε.

5.1 Regularization of the first variation and quantitative conditions of rectifiability for sequences of varifolds

Given a sequence of approximating d–varifolds (V_i)_i weakly–∗ converging to some d–varifold, if we want to ensure that V has locally bounded first variation (i.e. δV is a Radon measure) we can impose that

sup

i

kδV_ik<+∞. (5.7)

But this condition implies in particular that for fixed i, kδV_ik is finite, that is Vi has bounded first variation. This is already a strong assumption for discrete volumetric varifolds, as we explained in Example 2.1. As for point cloud varifolds, they do not even have bounded first variation. Moreover, the computation of the first variation of discrete volumetric varifolds in Proposition 2.2 and Example 2.1 shows that condition (5.7) is generally not satisfied by sequences of weakly–∗ converging discrete volumetric varifolds, even if the limit varifold is smooth. Then, with no additional assumption onV_i, it is not possible to consider the first variationδVi ofVi as a measure. Nevertheless, it is a distribution of order 1 (Definition 1.11) so that we will rather ask for a control of a regularized form of the first variation ofV_i.

Dans le document Thèse de l Université de Lyon. Diplôme de Doctorat. Blanche Buet (Page 116-125)