The Γ − lim inf inequality for the dimension-reduced problem . 69

Z

Σ∩∂Ω

min{α⁰m, α₁m + β₁/2, . . . , α_Nm + β_N/2} + α⁰|σ^⊥|(Ω) , where fluxes along the boundary are cheaper and thus preferred.

Remark 5 (Divergence measure vector fields and flat chains). Any divergence measure vector field can be identified with a flat 1-chain or a locally normal 1-current (see for instance [ˇS07, Sec. 5] or [BW17, Rem. 2.29(3)]; comprehensive references for flat chains and currents are [Whi57, Fed69]). Furthermore, for a sequence σ^j, j ∈ N, of divergence measure vector fields with uniformly boundedk∇·σ^jk^M, weak-∗ convergence is equivalent to convergence of the corresponding flat 1-chains with respect to the flat norm [BW17, Rem. 2.29(4)]. Analogously, scalar Radon measures of finite total variation and bounded support can be identified with flat 0-chains or locally normal 0-currents [Whi99b, Thm. 2.2], and for a bounded sequence of compactly supported scalar measures, weak-∗ convergence is equivalent to convergence with respect to the flat norm of the corresponding flat 0-chains.

From the above it follows that in Theorems 4.1 and 4.2 we may replace weak-∗ con-vergence by concon-vergence with respect to the flat norm. Indeed, for both results it suf-fices to consider sequences (σ^ε, ϕ^ε₁, . . . , ϕ^ε_N) with uniformly bounded costFε. For those we have uniformly bounded kσ^εk^M (by Theorem 4.2) as well as uniformly bounded k∇ · σ^εk^M=kµ^ε+− µ^ε−k^M so that weak-∗ and flat norm convergence are equivalent.

4.3 The Γ-limit of the phase field functional

In this section we prove the Γ-convergence result. As it is canonical, we begin with the lim inf-inequality, after which we prove the lim sup-inequality as well as equicoercivity.

4.3.1 The Γ −lim inf inequality for the dimension-reduced prob-lem

Here we consider the energy reduced to codimension-1 slices of the domain Ω. In our particular case of a two-dimensional domain, each slice is just one-dimensional, which will simplify notation a little (the procedure would be the same for higher codimensions, though). The reduced functional depends on the (scalar) normal flux ϑ through the slice as well as the scalar phase fields ϕ1, . . . , ϕN restricted to the slice. First observe that any measure ϑ∈ M(I) can be decomposed into its atoms and a remainder, namely

ϑ = m_ϑH⁰

x

^{Sϑ+ ϑ⊥,}

where S_ϑ ⊂ I is the set of atoms of ϑ, m^ϑ : S_ϑ → R is H⁰

x

^Sϑ-measurable, and ϑ^⊥ contains no atoms. Analogously to the functional introduced above we define the

reduced cost functional Gh(·; I) : M(I) → [0, ∞),

with ω_ε and γ_ε from equation (4.1). Eventually we extend the above functional on M(I) × L¹(I)^N by setting Gε(ϑ, ϕ₁, . . . , ϕ_N; I) = +∞ if (ϑ, ϕ¹, . . . , ϕ_N) 6∈ L²(I)× W^1,2(I)^N.

For notational convenience, we next introduce the sets K_i^ε on which the pointwise minimum inside Gε (or also Fε) is realized by the i^th element. we use the same notation (which case is referred to will be clear from the context).

We now show the following lower bound on the energy, from which the Γ− lim inf inequality for the dimension-reduced situation will automatically follow.

Proposition 4.2 (Lower bound on reduced phase field functional). Let I = (a, b)⊂ R and 0 ≤ δ ≤ η ≤ 1. Let Iη ⊂ {x ∈ I | ϕ1(x), . . . , ϕ_N(x) ≥ η}, and denote the collec-tion of connected components of I \ I^η by C^η. Furthermore define the subcollection of connected components Cη^≥ ={C ∈ C^η| inf^Cϕ₁, . . . , inf_Cϕ_N ≥ δ} and C^≥ =S

C∈Cη^≥C.

Finally assume ϕ_i(a), ϕ_i(b)≥ η for all i = 1, . . . , N.

1. If α₀ <∞ we have

Proof. 1. (α₀ <∞) We first show that without loss of generality we may assume

|ϑ| ≥ α₀ε 2γ_ε

1

1− (η − δ)² on K₀^ε. (4.3)

The motivation is that there may be regions in which a phase field ϕ_i is (still) small, but in which we actually have to pay α0|ϑ|. Thus, in those regions we would like ω_ε(α₀,^γε(x)_ε ,|ϑ(x)|) to approximate α⁰|ϑ(x)| sufficiently well, and the above condition We achieve (4.3) by modifying ϑ while keeping the cost as well as R

Iη|ϑ| dx and R

C|ϑ| dx for all C ∈ C^η the same so that the overall estimate of the proposition is not affected. The modification mimics the relaxation from Remark 2: the modified ϑ oscillates between small and very large values. Indeed, for fixed C ∈ C^η ∪ {I^η} and x∈ C set

where t_C is chosen such that R

C| ˆϑ| dx =R

C|ϑ| dx (this is possible, since for tC =∞ we have| ˆϑ| ≥ |ϑ| and for tC =−∞ we have | ˆϑ| ≤ |ϑ| everywhere on C). The cost did

not change by this modification since where we have employed (4.4) and Jensen’s inequality. Upon minimizing in m₀, which yields the optimal value α₀εH¹((I_η∪ C^≥)\ K0^ε)/δ² for m₀, we thus obtain Similarly, using Jensen’s inequality we have

Z

where in the last step we optimized forH¹(C_i^<). Finally, if inf_Cϕ_i ≤ δ we have (using Summa-rizing the previous estimates we obtain

Gε(ϑ, ϕ1, . . . , ϕN; I\ (I^η ∪ C^≥)) Finally, we obtain the desired estimate,

Gε(ϑ, ϕ₁, . . . , ϕ_N; I) =Gε(ϑ, ϕ₁, . . . , ϕ_N; I_η∪ C^≥) +Gε(ϑ, ϕ₁, . . . , ϕ_N; I\ (I^η∪ C^≥))

With Jensen’s inequality we thus compute Furthermore, the same calculation as in the case α₀ <∞ yields

Gε( ˜ϑ, ϕ1, . . . , ϕN; I\ (I^η∪ C^≥))≥ so that we obtain the desired estimate

Gε(ϑ, ϕ₁, . . . , ϕ_N; I) = Gε(ϑ, ϕ₁, . . . , ϕ_N; I_η∪ C^≥) +Gε(ϑ, ϕ₁, . . . , ϕ_N; I\ (I^η∪ C^≥))

Corollary 4.1 (Γ− lim inf inequality for reduced functionals). Let J ⊂ R be open and bounded, ϑ∈ M(J), and ϕ¹, . . . , ϕ_N ∈ L¹(Ω). Then

Γ− lim inf

ε→0 Gε/ϑ, ϕ₁, . . . , ϕ_N; J )≥ G (ϑ, ϕ1, . . . , ϕ_N; J )

with respect to weak-∗ convergence in M(J) and strong convergence in L¹(J )^N. Proof. Let (ϑ^ε, ϕ^ε₁, . . . , ϕ^ε_N) be an arbitrary sequence converging to (ϑ, ϕ₁, . . . , ϕ_N) in the considered topology as ε → 0, and assume without loss of generality that the limit inferior of Gε[ϑ^ε, ϕ^ε₁, . . . , ϕ^ε_N; J ] is actually a limit and is finite (else there is nothing to show).

It suffices to show the lim inf-inequality for each connected component ˜I = (a, b) of J separately. Due to Gε[ϑ^ε, ϕ^ε₁, . . . , ϕ^ε_N; ˜I]≥ _2ε^βikϕ^εi − 1k²L² for i = 1, . . . , N we must have ϕ^ε_i → 1 in L²( ˜I) and thus also in L¹( ˜I) so that ϕ_i = 1. Even more, after passing to another subsequence, by Egorov’s Theorem all ϕ^ε_i converge uniformly to 1 outside a set of arbitrarily small measure. In particular, for any ξ > 0 we can find an open interval (a + ξ, b− ξ) ⊂ I ⊂ ˜I such that ϕ^ε_i → 1 uniformly on ∂I, and for any η < 1 there is an open set I_η ⊂ I with H¹(I\ I^η)≤ 1 − η such that ϕ^εi ≥ η on I^η∪ ∂I for all i = 1, . . . , N and all ε small enough.

We now choose δ = ε^1/3and η = 1−ε and denote by Cη(ε) andCη^≥(ε) the collections of connected components of I\I^η from Theorem 4.2 (which now depend on ε). Further we abbreviate Cη^<(ε) = C^η(ε)\ Cη^≥(ε). The bound of Theorem 4.2 implies that the number of elements in Cη^<(ε) is bounded uniformly in ε and η. Passing to another

subsequence we may assumeCη^<(ε) to contain K sets C₁(ε), . . . , C_K(ε) whose midpoints where in the second inequality we used

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for all measurable A, B, C ⊂ I (due to the subadditivity of h) and in the third inequality we used h(m)≤ α0m as well as the lower semi-continuity of the mass on an open set.

Letting now ζ→ 0 (so that by the σ-continuity of ϑ we have |ϑ|(I\B^ζ({x¹, . . . , xK})) →

where R

4.3.2 The Γ − lim inf inequality

We now prove the desired lim inf-inequality, which will be obtained by slicing.

Proposition 4.3 (Γ− lim inf of phase field functional). Let µ⁺, µ− ∈ P(Ω). We have Γ− lim inf_ε→0 Fε =E

with respect to weak-∗ convergence in M(Ω; R²) and strong convergence in L¹(Ω)^N. Proof. Let (σ^ε, ϕ^ε₁, . . . , ϕ^ε_N) converge to (σ, ϕ₁, . . . , ϕ_N) in the considered topology. We first extend σ^ε and σ to R²\ Ω by zero and ϕ^εi and ϕ_i to R²\ Ω by 1 for i = 1, . . . , N.

The phase field cost functional and the cost functional are extended to R² in the obvious way (their values do not change). Without loss of generality (potentially after extracting a subsequence) we may assume lim_ε→0Fε(σ^ε, ϕ^ε₁, . . . , ϕ^ε_N) to exist and to be finite (else there is nothing to show). As a consequence we have ∇ · σ^ε = µ^ε₊− µ^ε− as well as ∇ · σ = µ⁺− µ⁻ and ϕ₁ ≡ . . . ≡ ϕ^N ≡ 1 (since the phase field cost functional is bounded below by PN

i=1 βi

2εkϕ^εi − 1k²_L²).

Now let A ⊂ R² open and bounded; the restriction of the phase field cost

where Fε is the dimension-reduced phase field energy and for simplicity we identified the domain Aξ,t of the sliced functions with an open subset of the real line. Fatou’s lemma thus implies By assumption, the left-hand side is finite so that the right-hand side integrand is finite for almost all t ∈ R as well. Pick any such t and pass to a subsequence such that lim inf turns into lim. Indeed σ_ξ,t^ε * σ^∗ _ξ,t for every ξ and almost all t, as σ^ε * σ^∗ and Theorem C.1. Thus, Theorem 4.1 on the reduced dimension problem implies

lim inf

For notational convenience let us now define the auxiliary function κ, defined for open subsets A⊂ R², as

κ(A) = lim inf

ε→0 Fε(σ^ε, ϕ^ε₁, . . . , ϕ^ε_N; A) . Furthermore, introduce the nonnegative Borel measure

λ(A) = α₀|σ^⊥|(A) + Z

Sσ∩A

h(m_σ) dH¹ as well as the |σ|-measureable Borel functions

ψ_j : R² → R, ψ_j =

Since σ is a divergence measure vector field, we have

for all j ∈ N where we used Remark 9 in the last equality. By [Bra98, Prop. 1.16] the above inequality implies

for any open A⊂ R². In particular, choosing A as the 1-neighbourhood of Ω we obtain lim inf

Alaoglu theorem implies existence of a weakly-* converging subsequence (still denoted σ^ε) with limit σ ∈ M(Ω; R²). It is now a standard property of Γ-convergence that, due to the above equicoercivity, any sequence of minimizers of Fε contains a subsequence converging to a minimizer of E .

To finish the proof we show uniform boundedness of σ^εinM(Ω; R²). Indeed, using ωε(α0,^γε_ε^(x),|σ(x)|) ≥ ^α₂⁰|σ(x)| for x ∈ K0^ε (remember that K₀^ε = ∅ for α⁰ = ∞) we

(the first term is interpreted as zero for α0 =∞). Furthermore, by H¨older’s inequality we have

Nel documento Approximations par champs de phases pour des problèmes de transport branché (pagine 77-86)