Combined kernels

C(Ω, h, s, d, q) for which

Ur(x,ωˆN)≥C(Ω, h, s, d, q)N^s/d, N ∈N. (2.79)

Proof of Theorem 2.10. Follows from Lemma 2.28 and Lemma 2.29.

q is continuous onΩ, andg_s is the hypersingular Riesz kernel, then 1

T(N)E_N(·;g_s, κ, q)−→^Γ S(µ;g_s, κ, q), N → ∞, (2.82) on P(Ω)equipped with the weak^∗ topology.

Proof. Note that for any fixedδ >0, the contribution of pairs (x_i,x_j) for which kx_i,x_jk ≥δ to the LHS of (2.82) is zero, since there are at mostN² T(N) such pairs; this implies only the diagonal valuesκ(x,x) have impact on the Γ-limit. For additional details, see Lemma 2.26.

Since the statement of the proposition is obvious whenκ≡C for some constant C, it suffices to construct a partition of Ω into a metrically separated collection of sets{Ω_m}^M₁ , with a remainder Ω0 of smallH_d measure, and to approximateκwith a constant on every element of the partition.

The metric separation property will then allow to discard interactions of pairs (x_i,x_j) between different Ω_m,1≤m≤M. Lastly, it will be convenient to assume that q(x)>0,x∈Ω; it is not a restrictive assumption since adding a constant toq translates into the same constant being added to both sides of (2.82).

To verify the property 1^Γ of Γ-convergence, first fix a sequence {µ_N} ⊂ P(Ω) converging to µ∈ P(Ω); just as in the proof of Proposition 1.6, it suffices to assume that µN ∈ P_N(Ω). By the same argument as in Proposition 1.6, µmust be absolutely continuous w.r.t. H_d. Fix anε >0;

we shall need a partition of Ω of the form Ω =

M

[

m=0

Ωm

with dist (Ωl,Ωm) ≥σ > 0,1 ≤ l 6= m ≤M; in addition,H_d(∂Ωm) = 0,1 ≤m ≤ M and all Ω_m are closed and such that κ_m−κ_m < ε for 1 ≤ m ≤ M. The latter is achieved by taking diam(Ωm) small enough, due to continuity of κ(x,x). By takingH_d(Ω0) small enough, we shall further guarantee that

Z

Ω0

h

C_s,dκ(x,x)ϕ(x)^1+s/d+q(x)ϕ(x)i

dH_d(x)< ε,

where ϕ:=dµ/dH_d. Such a partition is constructed in the proof of Theorem 2.1 using the Vitali covering lemma. As outlined above, we shall approximateκ(x,x) on Ω_mwithκ_m:= min_Ωmκ(x,x) and κm := max_Ωmκ(x,x); both are well-defined due to the continuity of κon the diagonal.

As discussed above, the case of a constant κ(x,x) can be handled by the argument of Proposition 1.6, thereby giving

1

T(N)E_N(·;g_s, C, q)−→^Γ S(·;g_s, C, q), N → ∞, 1≤m≤M

for any constant C; observe that the continuity of q means that S(·;g_s, C, q) is a continuous perturbation of S(·;gs, C,0), [21, Proposition 2.3]. Denote the N-point supports of µN by ω_N, N ≥2. We shall write

µ^(m) := 1 µ(Ωm)µ

Ωm

, µ^(m)_N := 1

µN(Ωm)µN Ωm

,

the for the renormalized restrictions of µ, µN to Ωm,1 ≤ m ≤ M. In the case of µ(Ωm) = 0 for some m, we shall join Ω_m with an Ω_l of positive measure and adjust the value ofM below accordingly. By the property 1^Γ of Γ-convergence, there holds

Nlim→∞

E_N(µ^(m)_N ;g_s, κ_m, q)

T(N µ_N(Ω_N)) ≥S(µ^(m);gs, κ_m, q), 1≤m≤M,

where we used the definition of the counting measures µN and, by the agreement made in Section 1.1, identified a counting measure with its support as an argument of an energy functional.

The second sum of S(·;g_s, κ, q), containing the external field q(x_i) terms, simply adds the R

Ωqϕ dH_dterm to the Γ-limit due to the stability under continuous perturbations [21, Proposition 2.3], so we shall focus on the weighted energy sum instead. Since H_d(∂Ω_m) = 0, alsoµ(∂Ω_m) = 0, and thus all the limits

N→∞lim

#{ω_N ∩Ωm}

N = lim

N→∞µN(Ωm) =µ(Ωm), 1≤m≤M,

exist. This means, we can apply Lemma 3.7 (which is formulated for the unweighted Riesz energy, but applies here as well) to obtain

Nlim→∞

1

T(N)E_N(µ_N;g_s, κ,0)≥

M

X

m=1

µ(Ω_m)^1+s/dlim inf

N→∞

EN(µ^(m)_N ;gs, κ,0) T(N µN(ΩN))

≥

M

X

m=1

µ(Ωm)^1+s/dlim inf

N→∞

E_N(µ^(m)_N ;g_s, κ_m,0) T(N µN(ΩN))

≥

M

X

m=1

µ(Ωm)^1+s/dS(µ^(m);gs, κ_m,0).

To finish the proof of the property 1^Γ, we need to show that the sum in the RHS of the last inequality approximates the value ofS(µ;g_s, κ, q). Indeed, first observe that

S(µ;g_s, κ, q) =

M

X

m=0

Z

Ωm

h

C_s,dκ(x,x)ϕ(x)^1+s/d+q(x)ϕ(x)i

dH_d(x)

≤

M

X

m=1

µ(Ω_m)^1+s/dS(µ^(m);g_s, κ, q) +ε,

since the functional S(·;g_s, κ, q) is defined on P(Ω), and thus the respective densities ϕ_m(x) have to be rescaled. Furthermore, for everyx∈Ωm, 0≤κ(x,x)−κ_m ≤κm−κ_m< ε; by taking εsmall enough and applying the monotone convergence theorem, we complete the proof of the property 1^Γ of Γ-convergence.

Fix aµ∈ P(Ω). Constructing a recovery sequence forµto verify the property2^Γ will involve a partition of Ω constructed to satisfy the same requirements as in the first part of the proof, but instead of approximatingκ from below, we shall useκ_m to approximate it from above. First, for the remaining part of this proof let µ^(m)_N be the recovery sequences ofS(·;gs, κm, q) for the restrictionsµ^(m),1≤m≤M, with the cardinalities #ω_N^(m) adding up toN (thus assuming a notation not used elsewhere in this text; it will be rather helpful here); let also

µ_N := 1 N

X

m

µ^(m)_N #ω_N^(m).

The existence of such recovery sequences µ^(m)_N follows from Γ-convergence on individual Ωm,1≤ m≤M. We thus have

Nlim→∞

E_N(µ^(m)_N ;g_s, κ_m, q)

T(N µ_N(Ω_N)) =S(µ^(m);g_s, κ_m, q), 1≤m≤M, from where

N→∞lim 1

T(N)EN(µN;gs, κ,0) =

M

X

m=1

µ(Ωm)^1+s/d lim

N→∞

E_N(µ^(m)_N ;g_s, κ,0) T(N µ_N(Ω_N))

≤

M

X

m=1

µ(Ω_m)^1+s/d lim

N→∞

EN(µ^(m)_N ;gs, κm,0) T(N µ_N(Ω_N))

=

M

X

m=1

µ(Ω_m)^1+s/dS(µ^(m);g_s, κ_m,0).

Since the positivity ofκ and q implies S(µ;g_s, κ, q)≥

M

X

m=1

µ(Ω_m)^1+s/dS(µ^(m);g_s, κ, q),

taking ε small enough and using the monotone convergence theorem completes the proof of (2.82).

Corollary 2.31. Any sequence {ω_N} that attains the lowest value of

N→∞lim 1

T(N)E_N(ω_N;gs, κ, q)

converges weak^∗ to the probability measure µ^κ,q with density

ϕ^κ,q(x) = dµ^κ,q dH_d(x) :=

L1−q(x) C_s,d(1 +s/d)κ(x,x)

d/s +

,

where L1 is a normalizing constant.

Proof. The proof proceeds by a variational argument identical to the one given in Section 1.5.2.

Uniqueness of the weak^∗ limit follows from convexity of the functionalS(·;g_s, κ, q) on probability measuresP(Ω) and convergence of minimizers, see Theorem 1.4.

Chapter 3

Riesz energy on fractal sets

In the previous chapter we made a smoothness assumption on Ω: namely, that it bed-rectifiable.

Observe that given an infinite set Ω, the Riesz s-energy E_s(ω_N) :=E(x₁, . . . ,x_N;g_s,0) =X

i6=j

kx_i−x_jk^−s, N = 2,3,4, . . .

on discrete subsets Ω⊃ω_N ={x_i: 1≤i≤N} can always be defined, provided that Ω⊂R^p is compact. Indeed, in this case it is a lower semicontinuous functional in the product topology on (R^p)^N, which allows us to consider configurations minimizing it for a fixed N, as well as the asymptotics ofE_s(Ω, N) := min{E_s(ω_N) : ω_N ⊂Ω}when N → ∞.

As has been pointed out in Chapter 1, in the hypersingular case of s > d := dimHΩ for integer d, there exists extensive treatment of both the asymptotics of E_s(Ω.N), and of the minimizing configurations in the weak^∗ topology. Recall that we identify discrete sets ωN ={x_i: 1≤i≤N} ⊂Ω with the corresponding (empirical) probability measures

ν_N := 1 N

N

X

i=1

δ_xi.

Then, as summarized in Theorem 1.3 [61, 17], any sequence of minimizers ofE_s on Ω^N weak^∗ converges to the normalized d-dimensional Hausdorff measureH_d(· ∩Ω)/H_d(Ω) on Ω, provided that

Ω = Ω⁽⁰⁾∪

∞

[

k=0

Ω^(k), (3.1)

where each Ω^(k)isd-rectifiable andH_d(Ω⁽⁰⁾) = 0, and that thed-dimensional Minkowski content of Ω coincides withH_d(Ω). Moreover, under such assumptions the limit ofE_s(Ω, N)/N^1+s/d, N → ∞, exists; i.e., every sequence{ω_N : N ≥2}of discrete subsets of Ω achieving the limit will converge weak^∗ toH_d/H_d(Ω).

The requirement of Ω being representable as (3.1) is the weakest that is known to guarantee the Poppy-seed bagel theorem; on the other hand, it has been established [14, Proposition 2.6]

that for a class of self-similar fractalsF with dim_HF =d, the limit ofE_s(F, N)/N^1+s/ddoes not exist. Using this observation, [32] gives an example of a sequence of minimizers without a weak^∗ limit.

In view of the above developments, it is natural to ask what can be said about weak^∗ cluster points of {ν_N : N ≥2} in the case when the underlying set Ω is not d-rectifiable; similarly, what is there to be said about cluster points of {E_s(Ω, N)/N^1+s/d : N ≥ 2}. We will show

that when Ω belongs to the class of self-similar fractals satisfying the open set condition, any subsequence {ω_N : N ∈ N} achieving the limit inferior must converge weak^∗ to H_d/H_d(Ω).

Furthermore, in the case when contraction ratios of the fractal Ω are all equal, the energy limit E_s(Ω, N)/N^1+s/d, N ∈ N, along a sequence N ⊂ N can be characterized by the behavior of sequence itself.

The following section contains formal definitions and the prerequisites necessary to state our result; Section 3.2 formulates the main results, and Section 3.3 is dedicated to the proof of weak^∗ convergence of sequences corresponding to the lim inf, as well as the results regarding a fractal with equal contraction ratios. This chapter is based on a joint work in progress by Alexander Reznikov and the author.