Γ-liminf inequality

In this section we prove the Γ− lim inf inequality stated in Theorem 2.2.

Proof of Theorem 2.2. With no loss of generality we assume that lim inf_ε↓0Fε,β(σ_ε, ϕ_ε) <

+∞ otherwise the inequality is trivial. For a Borel set A ⊂ Ω, we define H(A) := lim inf

ε↓0 Fε,β(σ_ε, ϕ_ε; A),

so that H is a subadditive set function. By assumption, the limit measure σ is 1-rectifiable; we write σ = m τH¹

x

Σ. Furthermore we can assume σ to be compactly supported in Ω. Consider a convex open set Ω₀such that supp(∇·σ) = S ⊂⊂ Ω0 ⊂⊂ Ω

and let f := [0, 1]× Rⁿ→ Rⁿ be a smooth homotopy of the indentity map on Rⁿ onto a contraction of Ω into Ω₀ such that f (t,·) restricted to Ω0 is the identity map, for any t ∈ [0, 1]. Let σ^t = f (t,·)]σ, indeed lim inf^t↓0F (σt, 1) ≥ F (σ, 1) as σ^t * σ. Further^∗

∇ · σt=∇ · σ since h(t, ·) is the identity on S . Now we claim that

lim inf

r↓0

H

B(x, r)

2r ≥ hβ(m(x)) for H¹-almost every x∈ Σ. (2.22) Let us fix λ≥ 1 and let us note hβ,λ(t) := min(h_β(t), λ). We then introduce the Radon measure

H_λ⁰(A) :=

Z

Σ∩A

h_β,λ(m) dH¹.

Now, let δ ∈ (0, 1). Assuming that (2.22) holds true, there exists Σ⁰ ⊂ Σ with H¹(Σ\Σ⁰) = 0 such that for every x∈ Σ⁰, there exists r₀(x) > 0 with

(1 + δ)H



B(x, r)

 ≥ 2rh^β,λ(m(x)) for every r∈ (0, r⁰(x)).

By the Besicovitch differentiation Theorem, there exists Σ₁ ⊂ Σ with H¹(Σ\Σ1) = 0 such that for every x∈ Σ¹, there exists r₁(x) > 0 with

(1 + δ)2rh_β(m(x)) ≥ Hλ⁰



B(x, r)

for every r ∈ (0, r1(x)).

We consider the familly B of closed balls B(x, r) with x ∈ Σ0 ∩ Σ1 and 0 < r <

min(r₀(x), r₁(x)) and we apply the Vitali-Besicovitch covering theorem [AFP00, The-orem 2.19] to the family B and to the Radon measure Hλ⁰. We obtain a disjoint family of closed balls B⁰ ⊂ B such that

H_λ⁰(Ω) = H_λ⁰(Σ) = X

B(x,r)∈B⁰

H_λ⁰ 

B(x, r)

≤ (1 + δ)² X

B(x,r)∈B⁰

H

B(x, r)

≤ (1 + δ)²H(Ω).

Sending λ to infinity and then δ to 0, we get the lower bound H(Ω) ≥ R

Σh_β(m) dH¹ which proves the theorem.

Let us now establish the claim (2.22). Since σ is a rectifiable measure, we have for H¹-almost every x∈ Σ and for every ϕ ∈ C^c(Rⁿ),

1 2r

Z

ϕ(x + ry) d|σ|(y) −→ m(x)^r↓0 Z

R

ϕ(tτ (x)) dt (2.23)

and 1

2r Z

B(x,r)∩Σ|τ(y) − τ(x)| d|σ|(y) −→ 0.^r↓0 (2.24) Let x ∈ Σ \ S be such a point. Without loss of generality, we assume x = 0, τ(0) = e¹ and m := m(0) > 0. Let δ ∈ (0, 1). Our goal is to establish a precise lower bound for Fε,β(σ_ε, ϕ_ε; C) where C is a cylinder of the form

C_r^δ := {x ∈ Rⁿ : |x1| < δr, |x⁰| < r} .

For this we proceed as in the proof of Lemma 2.3, here, the rectifiability of σ simplifies We also introduce the mean value

gε^δ,r := 1 2r

Z r

−r

g_ε^δ,r(s) ds.

From (2.23), we have for r > 0 small enough, g₀^δ,r := 1

Using (2.25), we conclude that for r > 0 small enough and then for ε > 0 small enough, we have

g^δ,r_ε (t) ≥ (1 − 3δ)m, for a.e. t∈ (−r, r).

By definition of the reduced dimension problem, we conclude that Fε,β(σ_ε, ϕ_ε; C_r^δ) ≥ 2rhⁿ⁻¹ε,β ((1− 3δ)m) .

Sending ε↓ 0, we obtain

H(C_r^δ) ≥ 2rhⁿ⁻¹β ((1− 3δ)m) .

We notice that H(B^√_1+δ2r) ≥ H(Cr^δ). Recall that for the case n− 1 we omit the superscript in the definition of h, thus dividing by 2√

1 + δ²r and taking the liminf as r ↓ 0, we get

lim inf

r↓0

H(B^√_1+δ2r) 2√

1 + δ²r ≥ h_β((1− 3δ)m)

√1 + δ² . Sending δ to 0, we get (2.22) by lower semi-continuity of h_β.

2.5 Γ-limsup inequality

Proof of Theorem 2.3.

Let us suppose F (σ, ϕ; Ω) < +∞, so that in particular ϕ ≡ 1. From Xia [Xia04], we can assume σ to be supported on a finite union of compact segments and to have constant multiplicity on each of them, namely polyhedral vector measures are dense in energy. We first construct a recovery sequence for a measure σ concentrated on a segment with constant multiplicity. Then we show how to deal with the case of a polyhedral vector measures.

Step 1. (σ concentrated on a segment.) Assume that σ is supported on the segment I = [0, L]× {0} and writes as m e¹H¹

x

^I. Consider m constant so that∇ · σ = m(δ_(0,0)− δ(L,0)) and

Eβ(σ, 1; Ω) = h_β(m)H¹(I) = L h_β(m).

For δ > 0 fixed, we consider the profiles

ϕ_ε(t) :=











η, for 0≤ t ≤ r^∗ε, v_δ t

ε



, for r∗ε≤ t ≤ r,

1 for r≤ t,

and ϑ_ε = m χ_Br∗ε⁰ (x⁰) ω_n−1 (εr∗)ⁿ⁻¹

with r∗ and v_δ, defined in Proposition 2.2 with d = n− 1. Assume r^∗ ≥ 1 and let d(x, I) be the distance function from the segment I and introduce the sets

Ir∗ε :={x ∈ Ω : d(x, I) ≤ r^∗ε} , and Ir:={x ∈ Ω : d(x, I) ≤ r} . Set ϕ_ε(x) = ϕ_ε(d(x, I)) and σ¹_ε = (mH¹

x

^{I)∗ ρε, where ρε} is the mollifier of equa-tion (2.1). We first construct the vector measures

σ_ε¹ = σ¹_εe₁ and σ²_ε(x₁, x⁰) = ϑ_ε(|x⁰|) e¹.

Alternatively, σ_ε² = σ∗ ˜ρ^ε for the choice ˜ρ_ε(x₁, x⁰) = χ_B⁰_r∗ε(x⁰)/ ω_n−1(εr_∗)ⁿ⁻¹. Let us highlight some properties of σ_ε¹ and σ²_ε. Both vector measures are radial in x⁰, with an abuse of notation we denote σ¹_ε(x₁, s) = σ¹_ε(x₁,|x⁰|). Since, both σε¹ and σ_ε² are obtained

trough convolution it holds supp(σ¹_ε)∪ supp(σε²) ⊂ I^r∗ε and they are oriented by the vector e₁ therefore |σε¹| = σ¹ε and |σε²| = ϑε. Furthermore for any x₁, it holds

Z

{x₁}×B_r∗ε⁰

σ¹_ε(x1, x⁰)− ϑ^ε(x⁰)

dx⁰ = 0. (2.26)

We construct σ_ε by interpolating between σ_ε¹ and σ²_ε. To this aim consider a cutoff function ζ_ε : R→ R⁺ satisfying

ζ_ε(t) = 1 for t≤ r^∗ε or t≥ L − r^∗ε,

ζ_ε(t) = 0 for 2 r∗ε≤ t ≤ L − 2 r^∗ε, and |ζε⁰| ≤ 1 r∗ε and define σ_ε component-wise as







σ³_ε· e1 = 0,

σ³_ε· eⁱ(x₁, x⁰) = −ζε⁰(x₁) x_i

|x⁰|ⁿ⁻¹ Z |x⁰|

0

sⁿ⁻²σ¹_ε(x₁, s)− ϑ^ε(s)

ds, for i = 2, . . . , n.

The integral corresponds to the difference of the fluxes of σ_ε¹and σ_ε² through the (n −1)-dimensional disk {x1} × B⁰. For σ_ε³ we have the following

∇ · σε³ =−ζε⁰(x₁)

n

X

i=2

"

 1

|x⁰|ⁿ⁻¹ −(n− 1)x²i

|x⁰|ⁿ⁺¹

 Z |x⁰| 0

sⁿ⁻²σ¹_ε(x₁, s)− ϑ^ε(s) ds

+ x²_i

|x⁰|² σ¹_ε(x₁,|x⁰|) − ϑε(|x⁰|)



=−ζε⁰(x₁)σ¹_ε(x₁,|x⁰|) − ϑε(|x⁰|) . (2.27) Let

σ_ε= ζ_εσ_ε¹+ (1− ζε) σ_ε²+ σ³_ε.

In force of equation (2.27) and from the construction of σ¹_ε, σ_ε² and ζ_ε we have

∇ · σ^ε=∇ · (ζ^εσ¹_ε) +∇ · (1 − ζ^ε)σ_ε²+∇ · σ³ε

= ζ_ε∇ · σε¹+ ζ_ε⁰(σ¹_ε− ϑ^ε) +∇ · σε³

= ζε∇ · σε¹ =∇ · (σ ∗ ρ^ε).

In addition for any (x₁, x⁰) such that |x⁰| ≥ r^∗ε from (2.26) we derive σ_ε³· eⁱ(x₁, x⁰) = −ζε⁰(x₁) x_i

|x⁰|ⁿ⁻² Z |x⁰|

0

sⁿ⁻¹σ¹_ε(x₁, s)− ϑ^ε(s)

ds = 0 which justifies supp(σ_ε)⊂ I^r∗ε. Let us now prove

lim sup

ε↓0 Fε,β(σ_ε, ϕ_ε; Ω) ≤ L h^β(m) + Cδ.

We split Ω as the union of Ω\ I^r, C_r,ε := I_r∩ [2 ε, L − 2 ε] × Rⁿ⁻¹ and D_ε and D_ε⁰, as show in figure 2.2, where D_ε ={x¹ ≤ 2 r^∗ε} ∩ I^r∗ε and D_ε⁰ ={x¹ ≥ L − 2 r^∗ε} ∩ I^r∗ε. On Ω\ Ir we notice that σ_ε = 0 and ϕ_ε = 1 therefore

Fε,β(σ_ε, ϕ_ε; Ω\ Ir) = 0.

Ω

O εr∗ L

2εr∗ L− 2εr∗

L− εr∗

Dε D⁰ε

Ir∗ε

Ir

Figure 2.2: Illustration of the interval I and both its r and (r∗ε)-enlargement for r∗ ≥ 1.

In grayscale we plot the levels of the function ζ_ε, whilst the striped region corresponds to the cylinder C_r,ε.

Observe that |Dε| = |Dε⁰| = Cεⁿ, then we have the upper bound Z

Dε

|σ^ε|² dx≤ 2 m² r_∗² εⁿ⁻²

Z

B1

ρ² dx + C

 .

Taking into consideration this estimate we obtain Fε,β(σ_ε, ϕ_ε; D_ε) = Fε,β(σ_ε, ϕ_ε; D_ε⁰)≤ (1− η)²

εⁿ⁻¹ Lⁿ(D_ε) + 2 m²r_∗² η

εⁿ⁻². (2.28) Finally on C_r,ε both σ_ε and ϕ_ε are independent of x₁ and are radial in x⁰ then by Fubini’s theorem and Proposition 2.2 we get

Fε,β(σ_ε, ϕ_ε; C_r,ε) =

Z L−2 εr∗

2 εr∗

Z

B⁰_r

Gε,β(ϑ_ε, ϕ_ε)≤ L (h^β(m) + C δ).

Adding all together gives the desired estimate. It remains to discuss the case r_∗ < 1.

From the point of view of the construction of σ_ε we need to replace the functions ζ_ε with ˜ζ_ε, satifying

ζ˜ε(t) = 1 for t≤ ε or t ≥ L − ε,

ζ˜_ε(t) = 0 for 2 ε≤ t ≤ L − 2 ε, and   

ζ˜_ε⁰    ≤

1 ε.

This choice ensures that σ_ε has all the properties previously obtained with r_∗ε replaced by ε accordingly. Define

w_ε(t) :=







η, for t ≤√

3ε, 1− η

r−√

3(t−√

3) + η, for √

3ε≤ t ≤ r, and set

ϕ_ε = min{ϕε(d(x, I)), w_ε(|x|), w^ε(|x − (L, 0)|)}.

With these choices for ϕ_εand σ_εthe estimates follow analogously with small differences in the constants.

O ε L

Figure 2.3: On the left the striped region corresponds to supp(σ_ε), remark that the balls of radius√

3ε centered respectively in (0; 0) and (L; 0) contain the modifications we have performed to satisfy the constraint. On the right we illustrate the level-lines of the cutoff function ˜ζ_ε in grayscale.

Step 2. (Case of a generic σ in polyhedral form.) Indeed, in force of the results quoted in Chapter 1 it is sufficient to show equation (2.3) for a polyhedral vector measure. Following the same notation introduced therein let

σ =

N

X

j=1

m_jH¹

x

^{Σj τj.}

With no loss of generality we can assume that the segments Σ_j intersect at most at their extremities. We consider measures σ satisfying constraint (2.4) so that if a point P belongs to Σ_j1, . . . , Σ_jP it must satisfy of Kirchhoff law,

where zj, is +1 if P is the ending point of the segment Σj with respect to its orientation, and −1 if it is the starting point. Let σε^j and ϕ^j_ε be the sequences constructed above for each segment Σ_k and define

σ_ε =

Let P_j and Q_j be respectively the initial and final point of the segment Σ_j and recall that, by the construction made above, for each j

∇ · σε^j = m_j δ_Pj − δ^Qj
∗ ρε

then by linearity of the divergence operator, it holds

∇ · σ^ε=

and the latter satisfies constraint (2.1) in force of equation (2.29). To conclude let us prove that

Indeed the following inequality holds true

And by inequality (2.28) follows

Fε,β(σ_ε, ϕ_ε; supp(σ^j_ε¹)∩ · · · ∩ supp(σε^jP))≤ C (1− η)² point. Since we are considering a polyhedral vector measure composed by N segments the worst case scenario is that we have 2N intersections in which at most N segments intersects. We conclude

which, passing to the limit, yields inequality (2.30).

The k-dimensional problem

3.1 Introduction

In this chapter we analyze how to address the problem of approximating the k-dimensional Plateau problem. In particular we aim at extending Theorems 2.1, 2.2 and 2.3 in the case where the 1-currents (vector measures) are replaced with k-currents. Let σ0 ∈ P^k(Ω) a polyhedral k-current with finite mass and let S := supp(∂σ0) be com-pactly contained in Ω. We want to minimize a functional of the type (10) where the set of candidates ranges among all currentsDk(Ω) such that

∂σ = ∂σ₀ in D^k(Rⁿ). (3.1)

Let us introduce a parameter η = η(ε) which satisfies

η(ε) = βε^n−k+1 for β ∈ R⁺ (3.2)

and let X_ε(Ω) be the set of pairs (σ_ε, ϕ_ε) where ϕ_ε∈ W^1,p(Ω, [η, 1]) and has trace 1 on

∂Ω and σ_ε is of finite mass with density absolutely continuous with respect toLⁿ. In this case we identify the current σ_ε with its L¹(Ω, Λ_k(Rⁿ)) density. Furthermore as in equation (2.1) given a convolution kernel ρ_ε we impose the constraint

∂σ_ε= (∂σ₀)∗ ρ^ε in D^k(Rⁿ).

For (σ_ε, ϕ_ε)∈ D^k(Ω)× L²(Ω) let

Fε,β^k (σ_ε, ϕ_ε; Ω) :=





 Z

Ω



ε^p−n+k|∇ϕε|^p+(1− ϕ^ε)²

ε^n−k +ϕε|σ^ε|² ε



dx, if (σ_ε, ϕ_ε)∈ Xε(Ω),

+∞, otherwise.

(3.3) Let us denote with X the set of pairs (σ, ϕ) such that σ is a k-rectifiable current satisfying (3.1) and ϕ ≡ 1. In this section we show that for any sequence ε ↓ 0 the Γ-limit of the family (Fε,β^k )_ε∈R+ is the functional

E_β^k(σ, ϕ; Ω) =





 Z

supp σ

h^n−k_β (m(x)) dH^k(x), if (σ, ϕ)∈ X,

+∞, otherwise inM(Ω, Rⁿ)× L²(Ω),

(3.4)

where the function h^n−k_β : R₊ → R⁺ is the function obtained in Appendix B for the choice d = n − k and is endowed with the same properties stated in Chapter 2. In particular under the assumption p > n− k we first prove a compactness theorem.

Theorem 3.1. Assume that β > 0. For any sequence ε↓ 0, (σ^ε, ϕ_ε)∈ D^k(Ω)× L²(Ω) such that

Fε,β^k (σ_ε, ϕ_ε; Ω)≤ F0 < +∞

then ϕ_ε → 1 in L²(Ω) and there exists a rectifiable k-current σ∈ D^k(Ω) such that, up to a subsequence, σε

* σ and (σ, 1)∗ ∈ X.

Then we show the Γ-convergence result, namely Theorem 3.2. Assume that β ≥ 0.

1. For any (σ, ϕ)∈ D^k(Ω)× L²(Ω) and any sequence (σε, ϕε)∈ D^k(Ω)× L²(Ω) such that (σ_ε, ϕ_ε)→ (σ, ϕ) it holds

lim inf

ε↓0 Fε,β^k (σ_ε, ϕ_ε; Ω)≥ Eβ^k(σ, ϕ; Ω).

2. For any couple (σ, ϕ)∈ D^k(Ω)×L²(Ω) there exists a sequence (σ_ε, ϕ_ε)∈ D^k(Ω)× L²(Ω) such that (σ_ε, ϕ_ε)→ (σ, ϕ) and

lim sup

ε↓0

Fε,β^k (σε, ϕε; Ω)≤ Eβ^k(σ, ϕ; Ω).

Nel documento Approximations par champs de phases pour des problèmes de transport branché (pagine 55-64)