Proofs of the main theorems

2.4 Proofs

2.4.1 Proofs of the main theorems

We first establish a few lemmas that will be used in the proof of Theorem 2.1.

Lemma 2.20. Let u, v >0 and q₀, q₁ be real. Then the function

F(t) :=tq0+ (1−t)q1+ut^1+s/d+v(1−t)^1+s/d, s≥d, (2.21) has a unique minimum on [0,1]. If there is some t∗ in (0,1) that satisfies

q₁−q₀

1 +s/d =ut^s/d∗ −v(1−t∗)^s/d, (2.22) then the minimum occurs att∗. Otherwise, the minimum occurs at t∗∈ {0,1} such that q1−t∗ = min{q₀, q₁}.

Proof. AsF is strictly convex on [0,1], it has a unique minimum in this interval. Differentiating yields F⁰(t) =q0−q1+ (1 +s/d)(ut^s/d−v(1−t)^s/d). If there is at∗ ∈(0,1) satisfying (2.22), thenF⁰(t∗) = 0 and the minimum ofF occurs at the valuet∗. Otherwise,F is strictly monotone in [0,1], and the minimum must occur at an endpoint. In fact, the minimum is at t∗ ∈ {0,1}

such thatq1−t∗ = min{q₀, q1}.

Lemma 2.21. Letµ be a finite Radon measure on the set Ω⊂R^p and q : Ω→Rbe measurable with respect to µ. Then for every ε >0 and µ-a.e. point x∈R^p there exists a positive number

R=R(x, ε) such that

µ[{z∈Ω∩B(x, r) :|q(z)−q(x)|< ε}]

µ[B(x, r)] >1−ε (2.23)

for allr < R.

Proof. Consider the following partition of set Ω:

Ω =

∞

[

m=1

{(m−1)ε≤q(x)< mε}=:

∞

[

m=1

Ω_m. (2.24)

By the Lebesgue-Besicovitch differentiation theorem (cf. [42, 1.7.1] or [81, Corollary 2.14]) (orLebesgue’s density theorem in this case), for µ-a.e. pointx∈A_m, m= 1,2, . . . ,

limr↓0

µ[A_m∩B(x, r)]

µ[B(x, r)] = 1. (2.25)

Therefore, (2.25) holds for µ-a.e. pointx∈Ω. In particular, fix such a pointx∈A_m. Because (A_m∩B(x, r)) ⊂ {z ∈ B(x, r) : |q(z)−q(x)| < ε}, equation (2.25) implies (2.23) for small

enoughR.

Lemma 2.22. LetΩ⊂R^p satisfy Ω =∪^M_m=1Am, where Am are sets from the class R^d_s. Let also the functionq be defined and lower semicontinuous onΩ. Assume that a sequence of configurations ω_n⊂Ω, n∈N, is such that

1. ω_n=SM

m=1ω_n^m and ω_n^m⊂A_m; 2. ω^k_n∩ω^l_n=∅ if k6=l;

3. limN3n→∞#ω_n^m/n=α_m, 1≤m≤M.

Then

lim inf

N3n→∞

E_s,d^q (ω_n, n) T(n) ≥

M

X

m=1

α^1+s/d_m C_s,d H_d(A_m)^s/d +

M

X

m=1

α_m min

x∈Am

q(x). (2.26) Proof. Observe that the minima on the RHS of (2.26) are attained due to the lower semicontinuity

ofq. For the LHS of (2.26) there holds

lim inf

N3n→∞

E_s,d^q (ωn, n)

T(n) = lim inf

N3n→∞

1 T(n)







 X

x6=y x,y∈ωn

kx−yk^−s+T(n) n

X

x∈ωn

q(x)









≥ lim inf

N3n→∞

1 T(n)

M

X

m=1

X

x6=y x,y∈ω^m_n

kx−yk^−s+ lim inf

n→∞

1 n

M

X

m=1

X

x∈ω_n^m

q(x)

≥ lim inf

N3n→∞

M

X

m=1

T(#ω^m_n) T(n)

E_s,d⁰ (ω_n^m)

T(#ω_n^m) + lim inf

n→∞

M

X

m=1

#ω_n^m

n min

x∈Am

q(x)

≥

M

X

m=1

α^1+s/d_m C_s,d H_d(Am)^s/d +

M

X

m=1

αm min

x∈Am

q(x),

where for the last inequality we used (2.20) and Theorem 1.3.

Remark 2.23. Observe that the only assertion about #ω_n we make is (3).

Corollary 2.24. Let the assumptions of Lemma 2.22 be satisfied and suppose qm, 1≤m≤M, are numbers such that the closure ofZ_m:={x∈A_m :q(x)< q_m} has H_d-measure zero. Then

lim inf

N3n→∞

E_s,d^q (ω_n, n) T(n) ≥

M

X

m=1

α^1+s/d_m Cs,d

H_d(Am)^s/d +

M

X

m=1

α_mq_m. (2.27) Proof. LetN⁰ ⊂Nbe such that

lim inf

N3n→∞

E_s,d^q (ωn, n)

T(n) = lim

N⁰3n→∞

E_s,d^q (ωn, n) T(n) . Then

N⁰3n→∞lim

E^q_s,d(ωn, n) T(n) ≥

M

X

m=1

α^1+s/d_m C_s,d

H_d(Am)^s/d + lim

N⁰3n→∞

1 n

M

X

m=1

X

x∈ω_n^m

q(x)

≥

M

X

m=1

α^1+s/d_m C_s,d

H_d(A_m)^s/d + lim

N⁰3n→∞

1 n

M

X

m=1

#(ω^m_n ∩(Ω\Z_m)) n

!

=

M

X

m=1

α^1+s/d_m C_s,d H_d(Am)^s/d +

M

X

m=1

αmqm.

Lemma 2.25. Let the set Ωbe such that the assumptions of Theorem 2.1 hold. Assume that a

sequence of N-point configurations {ω_N}_N≥2 in Ωsatisfies lim sup

N→∞

N∈N

E^q_s,d(ω_N)

T(N) <+∞ (2.28)

and 1

N X

x∈ω_N

δ_x * dµ,^∗ N3N → ∞, (2.29)

for some Borel probability measureµ on Ω. Then µ isH_d-absolutely continuous.

Proof. Indeed, otherwise let E⊂Ω be a Borel set such that H_d(E) = 0 andµ(E)>0. Sinceµ is inner regular as a Borel measure on a Radon space, [59, 434K(b)], without loss of generalityE is closed. For anε >0 pickr >0 such that Er:={x∈A: dist (x, E)≤r} satisfiesH_d(Er)< ε;

observe thatE_r is closed. By the definition of weak^∗ convergence, lim infN3N→∞ 1

N#{x∈ω_N : x∈E_r} ≥µ(E). Then according to Theorem 1.3 and the limit (2.20),

lim inf

N→∞N∈N

E_s,d⁰ (ω_N ∩E_r, N)

T(N) = lim inf

N→∞N∈N

E_s,d⁰ (ω_N∩E_r, N) T(#(ω_N∩E_r))

T(#(ωN∩Er)) T(N)

≥ Cs,d

H_d(Er)^s/dµ(E)^1+s/d≥ Cs,d

ε^s/dµ(E)^1+s/d.

Asεwas arbitrary, this contradicts (2.28). Thus µ must beH_d-absolutely continuous.

Lemma 2.26. Let the assumptions of Theorem 2.1 be satisfied. Let also the sequence of N-point configurations {ω_N}_N∈N be such that

Nlim→∞

N∈N

E_s,d^q (ω_N)

T(N) =g^q_s,d(Ω) (2.30)

and 1

N X

x∈ω_N

δx

* dµ,∗ N3N → ∞. (2.31)

Assume that{B_m}^M_m=1, M ≥1,is a collection of closed pairwise disjoint balls such thatH_d(Bm)>

0, H_d(∂B_m) = 0 and H_d({z∈B_m :q(z)≤q_m}) ≥ (1−δ)H_d(B_m), m = 1, . . . , M, for some positive δ <1.

Then

Nlim→∞

N∈N

E_s,d^q (ω_N ∩(∪_mB_m), N)

T(N) ≤min ( _M

X

m=1

C_s,dα^1+s/dm

((1−δ)H_d(B_m))^s/d +q_mα_m )

, (2.32)

where the minimum is taken over αm ≥0 such that P

αm =P

µ(Bm).

In particular, there exists a sequence {ω⁰_N}_N∈N for which (2.32) is an equality withω_N⁰ in place of ωN.

Proof. Fix anε >0 satisfyingε <1−δ. Consider the set{z∈Bm:q(z)≤qm}, m= 1, . . . , M. By the inner regularity of measureH_d, it has a closed subsetB_m⁰ contained in a ball concentric with Bi of smaller radius, for whichH_d(B_m⁰ ) >(1−δ−ε)H_d(Bm). Let a sequence ofN-point configurations {ω_N⁰ }_N∈N in Ω be such that ω_N⁰ ∩(Ω\ ∪_mBm) = ωN ∩(Ω\ ∪_mBm), and such that for 1 ≤m ≤ M, the collection ω_N⁰ ∩B_m is a minimizer of the (s, d,0)-energy in B⁰_m (in particular, is contained in it).

Equation (2.30) and Lemma 2.25 imply that µ(∂Bm) = 0, 1≤ m≤M. Hence the weak^∗ convergence in (2.31) implies lim #{x ∈ ω_N : x ∈ B_m}/N → µ(B_m) when N 3 N → ∞, [7, Theorem 2.1].

We will further assume that the following limits exist αm := limN3N→∞#(ω_N⁰ ∩Bm)/N, 1 ≤m ≤M. The assumptions on #(ω⁰_N ∩B_m) mean that P

mα_m = P

mµ(B_m). Finally, we observe that by the construction of the setsB_m⁰ , there exists a positiversuch that dist (∪_mB_m⁰ ,Ω\

∪_mB_m)≥r. Recall that

E_s,d^q (ω⁰_N, N) =E_s,d^q ω_N⁰ ∩(∪_mB_m), N

+E_s,d^q ω_N⁰ ∩(Ω\ ∪_mB_m), N

+ X

x,y∈ω⁰_N, x∈∪mBm, y∈Ω\∪mBm

kx−yk^−s. (2.33)

Because ω⁰_N ∩ ∪_mB_m =ω_N⁰ ∩ ∪_mB_m⁰ and because of the lower bound r for the distance between

∪_mB_m⁰ and Ω\ ∪_mBm, every term in the last sum is at most r^−s. Using the previous equation and the definition of g^q_s,d(Ω), we have:

0≤ lim

N→∞N∈N

E_s,d^q (ω_N⁰, N)

T(N) −g^q_s,d(Ω) = lim

N→∞

N∈N

E_s,d^q (ω⁰_N)

T(N) −E_s,d^q (ω_N) T(N)

!

≤ lim

N→∞N∈N

E_s,d^q (ω_N⁰ ∩(∪_mB_m), N)

T(N) − lim

N→∞N∈N

E_s,d^q (ω_N ∩(∪_mB_m), N) T(N)

+ lim

N→∞

N∈N

N² T(N)r^−s.

(2.34)

Since lim_N→∞

N∈N N²r^−s/T(N) = 0, there holds

Nlim→∞

N∈N

E_s,d^q (ωN ∩(∪_mBm), N)

T(N) ≤ lim

N→∞

N∈N

E_s,d^q (ω⁰_N∩(∪_mBm), N)

T(N) . (2.35)

From equation (2.30) and Lemma 2.25 follows thatµ(∂Bm) = 0, 1≤m≤M. Hence the weak^∗

convergence in (2.31) implies lim #{x ∈ ω_N : x ∈ B_m}/N → µ(B_m) when N 3 N → ∞, [7, Theorem 2.1]. The construction of the sequence {ω_N⁰}_N∈N and the limit (2.20) therefore imply

N→∞lim

N∈N

E_s,d^q (ω_N⁰ ∩(∪_mB_m), N)

T(N) ≤

M

X

m=1

C_s,dα^1+s/dm

((1−δ−ε)H_d(B_m))^s/d +q_mα_m

!

. (2.36)

We have so far only imposed the conditions that α1, . . . , αM are nonnegative and sum to P

mµ(B_m). Taking ε→0+ and minimizing over such α_m in (2.36) gives (2.32).

We first prove Theorem 2.1 for the case that q is a suitable simple function. The general case then follows by approximating an arbitrary lower semicontinuous q with such functions.

Lemma 2.27. LetΩ⊂R^p be a set from R^d_s, and B_m,1≤m≤M, be a collection of pairwise disjoint closed balls such thatH_d(Bm)>0 and H_d(Ω∩∂Bm) = 0, 1≤m≤M. Assume also q is a lower semicontinuous function and for D:= Ω\ ∪_mB_m,

q(x) =







q0, H_d-a.e.x∈D,

q_m, H_d-a.e.x∈B_m, 1≤m≤M,

(2.37)

for positive qm, 0≤m≤M.

Then equation (2.6) holds for the setΩ and function q.

Proof. For convenience, letA0:=D, Am:=Bm, 1≤m≤M, in this proof. We will first verify that for some positive{αˆ_m}^M_m=0 that add up to one,

N→∞lim

E_s,d^q (Ω, N) T_s,d(N) =

M

X

m=0

C_s,dαˆ^1+s/dm

H_d(A_m)^s/d +q_mαˆ_m

!

, (2.38)

where the values of ˆα_m, 0≤m≤M are such that ( ˆα₀, . . . ,αˆ_M) := arg min

αm≥0, Pαm=1

M

X

m=0

Cs,dα^1+s/dm

H_d(Am)^s/d +q_mα_m

!

. (2.39)

(i).Due to the weak^∗ compactness of the set Ω, Corollary 2.24 implies

g^q_s,d(Ω)≥ min

αm≥0, Pαm=1

M

X

m=0

C_s,dα^1+s/dm

H_d(Am)^s/d +qmαm

!

. (2.40)

Let a closedD⁰⊂Dsatisfyq(x)≡q0, x∈D⁰andH_d(D⁰)>(1−ε)H_d(D). By the same argument as in the proof of Lemma 2.26, for a fixedε >0 we construct a sequence ofN-element sets{ω_N⁰}_N∈N such that the subsetsω⁰_N∩Dandω⁰_N∩B_m, 1≤m≤M are (s, d,0)-energy minimizing inD⁰and

B_m⁰ respectively. Recall thatB_m⁰ are closed subsets of Ω∩B_m satisfyingH_d(B_m⁰ )>(1−ε)H_d(B_m) and dist (∪_mB_m⁰ , D)>0. As in Lemma 2.26, we construct{ω⁰_N}_N≥1 so that the following limits exist and are finite α₀ := limN3N→∞#(ω⁰_N ∩D⁰)/N and α_m := limN3N→∞#(ω⁰_N ∩B_m)/N, 1≤m≤M. Since E_s,d^q (Ω, N)≤E^q_s,d(ω⁰_N, N), equation (2.33) implies

g^q_s,d(Ω)≤ lim

N→∞

E_s,d^q (ω_N⁰, N) T(N) =

M

X

m=0

C_s,dα^1+s/dm

((1−ε)H_d(A_m))^s/d +q_mα_m

!

. (2.41)

This gives (2.38) after taking ε→0+.

(ii).Fix an Am with strictly positive ˆαm, say,A0 and assumeq0 ≤qm for definiteness. Pick any of the remaining setsA_k, 1≤k≤M and denoteβ =β(k) :=α₀+α_k. Consider the terms on the RHS of (2.39) that contain eitherα₀ orα_k:

F¯(α1, αk) := X

m=0,k

C_s,dα^1+s/dm

H_d(A_m)^s/d+qmαm

!

. (2.42)

Now choose the coefficients of the functionF(t) in Lemma 2.20 so that F(t) = t^1+s/d C_s,d

H_d(A₀)^s/d+t q₀

β^s/d + (1−t)^1+s/d C_s,d

H_d(A_k)^s/d + (1−t) q_k

β^s/d, (2.43) thenβ^1+s/dF(α0/β) = ¯F(α0, αk). Because of (2.39), it must be that ˆα0/β is the value ˆt∈(0,1]

for which the minimum of F(t) is attained. According to Lemma 2.20, either ˆt= 1, or qk−q0

C_s,d(1 +s/d) =

αˆ0

H_d(A₀) s/d

−

αˆk

H_d(A_k) s/d

. (2.44)

Equation (2.44) thus applies to any pair of sets inA0, . . . , AM provided both the corresponding ˆ

α_m’s is positive. Also, if ˆα_k >0, then q_k < q_l for every l such that ˆα_l= 0. To summarize, for someL₁ there holds

αˆm

H_d(A_m) s/d

=

L1−qm

C_s,d(1 +s/d)

+

, 0≤m≤M. (2.45)

It follows fromP

mαˆm = 1 that the first of equations (2.5) is satisfied for this L1. Finally, we can evaluate the RHS of (2.38):

N→∞lim

E_s,d^q (Ω, N) T_s,d(N) =

M

X

m=0

ˆ α_m

L₁−q_m 1 +s/d

+

+q_mαˆ_m

=

M

X

m=0

ˆ

α_mL₁+sq_m/d 1 +s/d ,

where in the last equality we used ˆα_m= 0 ⇐⇒ (L₁−q_m)₊= 0. This implies (2.6) because from

(2.45), ˆα₀ =µ^q(D) and ˆα_m =µ^q(A_m), 1≤m≤M, for the µ^q defined in (2.5).

Proof of Theorem 2.1. Note that as q is lower semicontinuous on the compact set Ω, it is bounded below there, so we may assume without loss of generality q is positive.

(i).Let Ω∈ R^d_s and let 0≤q < C for a positive constantC. We will further use that the restrictionH_d is a Radon measure onR^p. Namely, [81, Theorem 7.5] implies it is locally finite because Ω is the Lipschitz image of a compact set inR^d. It is also Borel regular as a restriction of Hausdorff measure, see [81, Theorem 4.2]. Then by [81, Corollary 1.11],H_dis a Radon measure.

Fix from now on a number 0< ε <1/4. Apply Lemma 2.21 to the measure H_d and function q, denote the set ofx∈Ω for which there exists anR(x, ε) as described in the Lemma by Ω⁰, and consider covering of Ω⁰ by the collection of closed balls{B(x, r) :x∈Ω⁰, 0< r < R(x, ε)}.

Choose for each x∈Ω⁰ a sequence of radiir_x,k →0, k→ ∞, k ∈N,for which H_d[{z∈B(x, rx,k) :|q(z)−q(x)|< ε}]

H_d[B(x, r_x,k)] >1−ε (2.46)

and also y ∈ Ω∩B(x, r_x,k) =⇒ q(y) > q(x)−ε. The latter is possible due to the lower semicontinuity of q.

Let {B(x, r_x,k)} be a Vitali cover of Ω⁰, so one can apply the version of Vitali’s covering theorem for Radon measures [81, Theorem 2.8] to produce a (at most) countable subcollection of pairwise disjoint{B_j :=B(xj, rj) :j ≥1} for whichH_d(Ω⁰ \ ∪_j≥1Bj) = 0. Using H_d(Ω)<∞, {B_j}_j≥1 can be chosen so that H_d(Ω∩∂B_j) = 0, j = 1,2, . . . (there are uncountable many options for the value ofrj, at most countably many of them positive). SinceH_d(Ω\Ω⁰) = 0, we can fix aJ ∈Nsuch that H_d

Ω\ ∪^J_j=1Bj

< ε. Let D:= Ω\ ∪_j=1^J Bj. AsH_d(∂Bj) = 0, 1≤j≤J, there holdsH_d(D)< ε.

Define the two simple functions q_ε, q_ε to be constant on each Bj, 1≤j≤J:

q_ε(x) :=







q(x_j) +ε, x∈B_j,

C, x∈D\S

jBj.

qε(x) :=







max(0, q(x_j)−ε), x∈B_j \D,

0, x∈D.

(2.47)

Such q_ε, q_ε are lower semicontinuous on Ω. Lemma 2.27 gives equation (2.6) applied toq_ε and q_ε on Ω. Let B_j⁰ :={z∈Ω∩B_j :|q(z)−q(x_j)|< ε}. Then,

q(xj)−ε≤q_ε(x)≤q(x)≤q_ε(X) =q(xj)ε, x∈B_j⁰. (2.48) In view of (2.46) forB_j⁰ andH_d(D)< ε, (2.48) implies that bothq

εandq_ε convergeH_d-a.e. toq as ε→0+. Since both are bounded by C+ 1, the dominated convergence theorem is applicable,

and

ε→0limS(q_ε,Ω) = lim

ε→0S(q_ε,Ω) =S(q,Ω). (2.49)

We now estimate

N→∞lim E_s,d^q (Ω, N)/T_s,d(N) in terms of

N→∞lim E_s,d^qε(Ω, N)/T_s,d(N) and

Nlim→∞E_s,d^qε(Ω, N)/T_s,d(N).

Firstly, by construction q(x)≥q_ε(x), x∈Ω, which gives

E_s,d^q (Ω, N)≥ E_s,d^qε(Ω, N). (2.50) On the other hand,

E_s,d^q (Ω, N)≤ E_s,d^q (D∪[

j

B_j⁰, N)≤ E_s,d^q (D∪[

j

B_j⁰, N)≤ T(N)

(1−ε)^s/dS(q_ε,Ω), (2.51) where the last inequality follows from (2.46) and (2.38). This proves (2.6).

(ii).It remains to prove equation (2.8) for a sequence {ω_N}_N≥2 satisfying (2.7). Since the probability measures on Ω are weak^∗ compact, one can pick a subsequence{ˆωN}_N∈N⊂ {ω_N}_N≥2 for which the corresponding normalized counting measures have a weak^∗ limit:

1 N

X

x∈ˆωN

δx

* µ,∗ N3N → ∞.

Then µis H_d-absolutely continuous by the Lemma 2.25. Set ρ(x) := _dH^dµ

d(x).

Since the integral R

Ωρ dH_d= 1 is finite, at H_d-a.e. pointx of Ω there holds

r→0lim

1 H_d(B(x, r))

Z

B(x,r)

ρ dH_d=ρ(x). (2.52)

Fix two distinct pointsx₁, x₂ for which both (2.23) for measure H_dand (2.52) hold. Then for an arbitrary fixed 0< ε <min(1/2, ρ(x1), ρ(x2), q(x1), q(x2)) there exist closed disjoint balls B₁ :=B(x₁, r₁), B₂ :=B(x₂, r₂) centered aroundx₁, x₂ such that equations

H_d[{z∈Ω∩B_m :|q(z)−q(x_m)|< ε}]

H_d[Bm] >1−ε, m= 1,2, (2.53)

Z

Bm

|ρ(z)−ρ(xm)|dH_d(z)< εH_d(Bm), m= 1,2, (2.54)

hold for all closed balls concentric with B_m of radius at most r_m. Without loss of generality we will also require H_d(∂B1) =H_d(∂B2) = 0 and that q(x)≥q(xm)−ε for allx∈Bm, m= 1,2 (which can be assumed by lower semicontinuity). Letq_m :=q(x_m), m= 1,2; let also q₁ ≤q₂.

Due toµbeing absolutely continuous with respect toH_d, the assumptionH_d(∂B_m) = 0, m= 1,2, and the limit (2.20), Lemma 2.22 implies

N→∞lim

N∈N

E_s,d^q (ˆωN∩(B1∪B2), N)

T(N) ≥ X

m=1,2

C_s,dµ(B_m)^1+s/d

H_d(Bm)^s/d +µ(Bm)(qm−ε)

!

. (2.55) On the other hand, from Lemma 2.26:

N→∞lim

N∈N

E_s,d^q (ˆω_N ∩(B₁∪B₂), N)

T(N) ≤min





 X

m=1,2

C_s,dαm^1+s/d

((1−ε)H_d(B_m))^s/d + (q_m+ε)α_m







(2.56)

with minimum taken over positive α₁, α₂ satisfying α₁+α₂ =µ(B₁) +µ(B₂). If we denote

( ˆα₁,αˆ₂) := arg min (

X

m=1,2

C_s,dα^1+s/dm

((1−ε)H_d(B_m))^s/d + (q_m+ε)α_m

!

:α₁+α₂=µ(B₁) +µ(B₂) )

,

(2.57)

and argue as in the proof of Lemma 2.27, we obtain, similarly to (2.44), that ( ˆα1,αˆ2) satisfy q2−q1

C_s,d(1 +s/d) =

αˆ1

(1−ε)H_d(B₁) s/d

−

αˆ2

(1−ε)H_d(B₂) s/d

. (2.58)

Inequalities (2.55)–(2.56) and the definition of ( ˆα1,αˆ2) give:

X

m=1,2

Cs,dµ(Bm)^1+s/d

H_d(B_m)^s/d +µ(B_m)(q_m−ε)

!

≤ X

m=1,2

C_s,dαˆ^1+s/dm

((1−ε)H_d(B_m))^s/d + ˆα_m(q_m+ε)

!

≤ X

m=1,2

C_s,dµ(B_m)^1+s/d

((1−ε)H_d(B_m))^s/d +µ(B_m)(q_m+ε)

! .

(2.59)

Observe that if in the above construction we fix the ballB1 and allowr2→0, the first term on the RHS of (2.58) is bounded, so the ratio ˆα₂/H_d(B₂) is bounded as well, say, ˆα₂/H_d(B₂)≤R₂¹.

1due to the assumptionsq1 ≤q2 andHd(B2)/Hd(B1)< ε, the equality ˆα1+ ˆα2=µ(B1) +µ(B2), and equations

Let alsor₂ be such thatH_d(B₂)/H_d(B₁)< ε and ˆα₂/H_d(B₁)< ε. Due to equation (2.54), there holds |µ(B_m)/H_d(Bm)−ρ(xm)|< ε, m= 1,2. Dividing (2.59) through byH_d(B1) for such a choice ofr₂ gives:

C_s,d(ρ(x₁)−ε)^1+s/d+ (ρ(x₁)−ε)(q₁−ε)

≤ C_s,d (1−ε)^s/d

αˆ1

H_d(B1) 1+s/d

+

αˆ1

H_d(B1)

(q1+ε) +

αˆ₂ H_d(B1)

C_s,d (1−ε)^s/d

αˆ₂ H_d(B2)

s/d

+

αˆ₂ H_d(B1)

(q2+ε)

≤ Cs,d

(1−ε)^s/d(ρ(x₁) +ε)^1+s/d+ (ρ(x₁) +ε)(q₁+ε) +ε

C_s,d

(1−ε)^s/d(ρ(x2) +ε)^1+s/d+ (ρ(x2) +ε)(q2+ε)

,

(2.60)

Finally, because ε >0 was arbitrary and the function C_s,dt^1+s/d+q(x₁)t, t≥0 is monotone, inequalities (2.60) yield by the above discussion

ρ(x1) = lim

r1→0

ˆ α1

H_d(B1). (2.61)

We could similarly fix the ballB2 first and ensureH_d(B1)/H_d(B2)< ε, takingr2 →0 afterwards, thus also

ρ(x₂) = lim

r2→0

ˆ α₂

H_d(B₂). (2.62)

In conjunction with (2.58) the last two equations give q₂−q₁

C_s,d(1 +s/d) =ρ(x₁)^s/d−ρ(x₂)^s/d (2.63) for H_d× H_d-a.e. pair (x1,x2)∈Ω×Ω. Due to the normalization propertyR

Ωρ(x)dH_d= 1 and the definition ofL1 in (2.5),

ρ(x) =

L1−q(x) C_s,d(1 +s/d)

d/s +

H_d-a.e. (2.64)

which coincides with the density in formula (2.8).

(iii).Finally, we turn to the case when the function q need not be bounded above. Consider

qC(x) :=







q(x), q(x)≤C, C, otherwise.

(2.54) and (2.58), one can takeR2=ρ(x1) +ρ(x2) + 2 as a rough estimate.

Recall that Ω(C) = {x ∈ Ω : q(x) ≤ C} is a d-rectifiable set as a closed subset of Ω. The Theorem 2.1 is therefore applicable to each functionqC if seen as defined on Ω(C). By Remark 2.18, the value of L₁ is finite. For all C ≥L₁,

supp (µ^qC)∩ {x:q(x)> C}=∅. (2.65) InequalityqC(x)≤q(x) for all x∈Ω implies

E_s,d^qC(Ω, N)≤ E_s,d^q (Ω, N), N ≥2, soS(qC,Ω)≤g^q_s,d(Ω). On the other hand, due to set inclusion,

g^q_s,d(Ω)≤g^q_s,d(Ω)≤g^q_s,d(Ω(C)) =S(qC,Ω(C)) =S(qC,Ω),

where the last two equalities follow from Theorem 2.1 applied to the function qC and sets Ω(C) and Ω respectively, and equation (2.65). To summarize, g^q_s,d(Ω) =g^q_s,d(Ω) =S(q,Ω).

Let now {ω_N}_N≥2 be a sequence satisfying (2.7). Fix a C > L1. BecauseqC(x)≤q(x) for allx∈Ω and S(q,Ω) =S(qC,Ω), the sequence{ω_N}N≥2 is also asymptotically (s, d, qC)-energy minimizing. Then by Theorem 2.1 this sequence converges weak^∗ to dµ^qC, and it remains to observe that forC > L1 it holds dµ^qC = dµ^q, where the two measures are defined in equation (2.5).

Proof of Theorem 2.3. The desired result is an immediate application of Theorem 2.1 since using equation (2.5) for the external field from (2.9) givesL₁ = 0, so the asymptotic distribution is indeed (2.10).

Proof of Proposition 2.5. We have

q⁰(x) = (1 + ∆)q(x).

According to (2.5), the equation Z

Ω

l−q⁰(x) M_s,d

d/s +

dH_d(x) = 1 (2.66)

for variablel has the unique solutionl=L⁰₁. Using (2.9), it can be rewritten as Z

Ω

ρ^s/d+∆ρ^s/d+l M_s,d

!d/s

+

dH_d(x) = 1, (2.67)

which, in view ofR

Ωρ dH_d= 1 and monotonicity of the function (·)₊, shows that the solution of

(2.66) satisfies|l| ≤ |∆| kρk^s/d_∞ , that is,

|L⁰₁| ≤ |∆| kρk^s/d_∞ . We will therefore write L⁰₁=K∆ with|K| ≤ kρk^s/d∞ .

Let us now estimate the difference between densities ρ⁰ = dµ^q0/ dH_d and ρ = dµ^q/ dH_d in terms of ∆. Factor out ρ^s/d from the parentheses in (2.67) and observe that for ∆ <

M_s,d/ 1 + (kρk_∞δ⁻¹)^s/d

the expression inside is nonnegative, which allows to expand it up to o(∆):

ρ⁰ =ρ 1 +∆(1 +K/ρ^s/d) Ms,d

!d/s

=ρ+ ∆d(1 +K/ρ^s/d) sMs,d

+o(∆), ∆→0.