Minimizers of 3D anisotropic interaction energies

We study a large family of axisymmetric Riesz-type singular interaction potentials with anisotropy in three dimensions. We generalize some of the results of our recent work in two dimensions to the present setting. For potentials with linear interpolation convexity, their associated global energy minimizers are given by explicit formulas whose supports are ellipsoids. We show that for less singular anisotropic Riesz potentials, the global minimizer may collapse into one or two dimensional concentrated measures which minimize restricted isotropic Riesz interaction energies. Some partial aspects of these questions are also tackled in the intermediate range of singularities in which one dimensional vertical collapse is not allowed. Collapse to lower dimensional structures is proved at the critical value of the convexity but not necessarily to vertically or horizontally concentrated measures, leading to interesting open problems.


Introduction
In this work, we focus on the analysis of the 3D anisotropic interaction energy W px ´yqρpyq dyρpxq dx where ρ is a probability measure on R 3 and W is the anisotropic interaction potential W pxq " |x| ´sΩpxq `|x| 2 , 0 ă s ă 3 . (1.1) Here we denote x " x |x| P S 2 as the angle variable 1 , and the anisotropic part Ω is defined on S 2 .We will always assume Ω satisfies (H): Ω is smooth, strictly positive, and Ωpxq " Ωp´xq.
For such Ω, W satisfies the condition (W): W is even, locally integrable, lower-semicontinuous, bounded from below, and bounded above and below by positive multiples of |x| ´s near 0.
As a result, Erρs is well-defined with values in R Y t`8u for any probability measure ρ.We will also use the following less strict assumptions (H0): Ω is smooth, and Ωpxq " Ωp´xq.
in some situations, and in particular, we will study the parametrized potential with ω satisfying (h): ω is smooth, min ω " 0 and ωpxq " ωp´xq.In a recent result [8], the authors have analysed the 2D anisotropic interaction energies in detail.The main objective of the present work is to generalize the strategy and techniques developed in [8] to the three dimensional case.Our approaches work for general dimensions d ě 4 with d ´4 ă s ă d, which we will not treat to avoid cumbersome technicalities.
One of the basic tools used in [8] is the Linear Interpolation Convexity (LIC) of the interaction energy functional Erρs defined as: for any two compactly supported probability measures ρ 0 ‰ ρ 1 with the same center of mass, the energy along their linear interpolation curve Erp1 ´tqρ 0 `tρ 1 s, t P r0, 1s is always strictly convex.We proved in [8] that strict LIC is essentially equivalent to nonnegativity of the Fourier transform of W for potentials of the form (1.1).LIC was utilised before for uniqueness of global miminizers and [12,13,17,9,10,11] and uniqueness of Wasserstein-8 local minimizers in [7].
To see whether LIC holds for an anisotropic interaction potential, we follow a similar strategy in 3D as in the 2D case by studying the Fourier transform of the potential W in details.Our main strategy for identifying global minimizers for anisotropic potentials in the LIC case is then quite direct: it suffices to check that the candidate satisfies the Euler-Lagrange conditions obtained in [1] whenever the potential energy is LIC.
Let us recall some of the previous results for anisotropic potentials.The collapse to one dimensional minimizers was shown to happen in 2D for the particular case W log,α pxq " ´ln |x| `α x 2 1 |x| 2 `|x| 2 , α ě 0, in the seminal paper [17] at the value of α for which the potential ceases to be LIC.A series of recent works [5,6,14,16,15] study this particular family of anisotropic potentials and small perturbations, showing that the the collapse to one dimensional vertical minimizers happens for α large enough and showing that the minimizers for smaller values than the critical LIC convexity value are given by the characteristic function of suitable ellipses.
We recently showed in [8] that a similar behavior in 2D is shared by a large family of potentials of the form (1.2) with 0 ă s ă 2 as well as its logarithmic counterpart, generalizing the results in the previous papers.More precisely, we proved that the minimizers in the LIC range are given by push-forward of the global minimizer for isotropic Riesz potentials to suitable ellipses, showing that the ellipse-shaped minimizers are indeed generic in the LIC range.The collapse to the one dimensional vertical minimizer was also obtained for this family, showing that this again happens for α large enough.Moreover, we showed that generically there is a gap in between these two behaviors in which one dimensional structures appear but not necessarily are supported on vertical lines.This is based on the concept of infinitesimal concavity introduced in [7]: finding a signed measure µ with zero mass and center of mass and arbitrarily small support such that Erµs ă 0. Infinitesimal concavity of an interaction energy Erρs is a signature of collapsing to lower dimensions since we proved in [7] that their global minimizers contain no interior points for any superlevel set of their absolute continuous parts.
Concerning the 3D case, the only available result [6] dealt with the one-parameter family of potentials |x| 3 `|x| 2 .It showed that the global minimizer in the LIC range ´1 ă α ď 1 is the characteristic function of an ellipsoid, but the behavior beyond this range was not discussed.The main objective of the present work is to generalize the results in [8] from 2D to 3D, thus treating 3D axisymmetric potentials of the form (1.1) and (1.2) with general singularities s and angular profiles Ω.We are able to generalize two types of results.On one hand, the LIC property of the potentials leads to ellipsoid-shaped global minimizers characterized by the push-forward of the minimizers for 3D isotropic Riesz potentials.On the other hand, infinitesimal concavity results in the collapse to lower dimensional structures.For suitable singularity of the potential and the angular profile, such structures for large values of α are known to be the 1D/2D minimizers for restricted isotropic Riesz potentials.More complicated behavior is also possible, including expansion of the support as α Ñ 8. We now describe the main results of this paper in details.
Most of our results will be obtained using spherical coordinates that we denote as x " ¨sin θ cos µ sin θ sin µ cos θ ‚, 0 ď θ ď π, 0 ď µ ă 2π in the physical space, and ξ " |ξ| ξ, ξ " ¨sin ϕ cos ν sin ϕ sin ν cos ϕ ‚, 0 ď ϕ ď π, 0 ď ν ă 2π (1.3) in the Fourier space.In Section 2 we first derive the formula for the Fourier transform for functions of the form |x| ´sΩpxq.We will show in Lemma 2.1 that for Ω satisfying (H0) and 0 ă s ă 3, Fr|x| ´sΩpxqs " |ξ| ´3`s Ωp ξ; sq for some function Ωp ξ; sq smooth in the variable ξ P S 2 .For 2 ă s ă 3, its explicit formula is given by (2.3) as a convolution-type operator, and we also derive the explicit formulas for other values of s in (2.4) and (B.1).Here, one important observation is the holomorphic property of Ωp ξ; sq with respect to the variable s (when extending to all complex numbers s with 0 ă psq ă 3).It allows us to apply analytic continuation arguments to treat the values of s for which we lack explicit formulas.We also review the results we obtained in [8] related to the LIC property and state their generalizations to 3D.
Then, in Section 3 we study the energy minimizers for the interaction potential W in (1.1), in the case of LIC.For simplicity, as mentioned earlier, we focus on the case when W is axisymmetric.For 0 ă s ă 3, we prove that the unique energy minimizer is necessarily some ρ a,b as in (3.1), an axisymmetric rescaling of the minimizer of the corresponding 3D isotropic interaction energy (Theorem 3.1).In particular, the shape of its support is an ellipsoid.This result is analogous to its counterpart in [8].To prove this result in the case of 0 ă s ă 1, we take a similar approach as in [8].The key step is to show that the potential generated by any ρ a,b is quadratic (Lemma 3.2), which is proved by the decomposition of the potential |x| ´sΩpxq into a convex combination of 1D potentials in different directions as in (2.6).To extend it to a wider range of s, we use an analytic continuation argument based on the holomorphic properties established in the previous section.
The previously studied LIC cases include any potential of the form W α in (1.2) if ‚ Either ω (the angle function of the Fourier transform of |x| ´sωpxq as in (2.2)) is nonnegative.This include all the cases of 2 ď s ă 3, by (2.3).‚ Or ω is not nonnegative but α is small.To be precise, W α is LIC if and only if 0 ď α ď α L , where α L " c s ´min ω P p0, 8q . (1.4) In contrast to the LIC cases, we saw in [8] that for 2D anisotropic potentials with 0 ă s ă 1 (for which ω is always sign-changing) the minimizers tend to collapse on 1D distributions for large α.As an analogue, in Section 4 we exploit the cases of W α for which the minimizers collapse on lower dimensional distributions for large α.Here in the 3D case, the collapse phenomenon is richer than that of the 2D case because minimizers may collapse to 1D or 2D distributions, depending on how singular the potential is and how the function ω achieves its minimum on S 2 .For such collapse to happen, one necessary condition is that s is not too large, so that the energy is finite for such concentrated measures.Therefore, it is not surprising for us to obtain the following results, in the case of axisymmetric potentials with certain nondegeneracy conditions: ‚ (Theorem 4.1) If 0 ă s ă 1 and ω is minimized at θ " 0, then for sufficiently large α, the energy minimizer for W α is unique, given by ρ 1D (c.f.Appendix A). ‚ (Theorem 4.2) If 0 ă s ă 2 and ω is minimized at θ " π{2, then for sufficiently large α, the energy minimizer for W α is unique, given by ρ 2D .
The proof is based on comparison arguments with specially designed potentials, similar to [8].We remark that the case when ω is minimized at some θ " θ 0 P p0, π{2q is still open.This part of our result answers an open problem proposed in [6], which is concerned with the energy minimizer for W α with s " 1, ωpxq " cos 2 θ and large α.In fact, this ω clearly satisfies the assumptions of Theorem 4.2, and we may conclude that the unique energy minimizer for W α is ρ 2D for sufficiently large α.
Finally, in Section 5 we study the case 1 ď s ă 2 and ω minimized at θ " 0, which is in general not covered by the previously stated results.We first notice that the sign of ω is not completely determined (Theorem 5.1): for any fixed 1 ď s ă 2, ‚ There exists axisymmetric ω, minimized at θ " 0, such that ω is nonnegative (and thus W α is always LIC and its minimizer is always ellipsoid-shaped).‚ There also exists axisymmetric ω, minimized at θ " 0, such that ω is sign-changing.In either case, since ω is minimized at θ " 0, one might expect that the minimizers would elongate along the x 3 -direction; also, since the power index 1 ď s ă 2 does not allow the concentration to 1D measures, it would elongate to infinity.However, it turns out that this intuition is far from the truth.We will study the expansion of the minimizers from the following three aspects (for 1 ď s ă 2): ‚ (Theorem 5.5) If ω is strictly positive, then the ellipsoid-shaped minimizer for W α necessarily expands to infinity in all dimensions as α Ñ 8, with the ratio between its axes converging to a positive constant.This result also works in the case 2 ď s ă 3. The proof is based on detailed analysis of the formula (3.3) which determines the axis lengths for the ellipsoids via A " B " 1. ‚ (Theorem 5.9) There exists ω, minimized at θ " 0 with ω ě 0, such that the ellipsoid-shaped minimizer for W α expands in x 1 and x 2 -directions but shrinks in x 3 -direction as α Ñ 8.It is proved by an explicit construction with an argument similar to the previous result.‚ (Theorem 5.10) For general axisymmetric ω, as long as it is positive at θ " π{2, the minimizers for W α have to expand in at least two dimensions as α Ñ 8, i.e., these minimizers cannot be supported inside a fixed infinite cylinder for all α.The proof is based on a comparison argument with isotropic energies, together with a scaling analysis for α.
These results show that the condition that ω minimized at θ " 0 does not imply that x 3 -direction is preferred by the minimizers for W α .As α gets large, the minimizers may or may not expand in x 3 -direction, and they have to expand in at least two dimensions.We remark that the behavior of the energy minimizers for W α with intermediate α remains largely open.This is also the case for large α for 1 ď s ă 2 with ω minimized at θ " π{2 and ω sign-changing, for which the only thing we know is its expansion phenomenon in Theorem 5.10.In these cases the energy is not LIC, and the collapse to lower dimensions happens.In fact, we know that the potential is infinitesimal concave (Proposition 2.7) and thus one expects the minimizers to be supported in lower dimensional sets and/or fractals.Also, one can conduct local analysis for the generated potential, and show that 1D/2D fragments along certain directions are prohibited in any Wasserstein-8 local minimizers, as was done in Section 6 of [8].We expect this behavior to be related to sign information of Ω.
Finally, we point out that the logarithmic case corresponding formally to s " 0 can also be included in 3D following a similar limiting procedure as in [8, where the last summation converges in the sense of distribution, and also pointwisely for every ξ ‰ 0 since ĝ is a Schwartz function.This summation also converges in It is also clear that FrI 2 spξq is smooth in ξ for ξ ‰ 0. Furthermore, for every ξ ‰ 0, one can differentiate the above summation with respect to s and obtain Therefore, FrI 2 spξq is holomorphic in s for any ξ ‰ 0. Furthermore, this expression shows that B s FrI 2 spξq is also smooth in ξ for ξ ‰ 0.
Combining the above results, we see that for s P S, Fr|x| ´sΩpxqs is in L 1 `L8 , smooth in ξ for any ξ ‰ 0, and holomorphic in s for any fixed ξ ‰ 0. Scaling argument (by replacing x with λx, λ ą 0) shows that Fr|x| ´sΩpxqs has to take the form (2.2) for some function Ω, and then we see that Ωp ξ; sq is smooth in ξ and holomorphic in s, and B s Ωp ξ; sq is also smooth in ξ.
To prove (2.3), we fix 2 ă s ă 3, take any ξ ‰ 0, and calculate Fr|x| ´sΩpxqspξq as an improper integral where we use the fact that Fr|x| ´sΩpxqs is real (since |x| ´sΩpxq is even) in the second equality, and dominated convergence theorem in the last equality.Formula (A.4) gives the value of the improper integral ş 8 0 r ´s`2 cos r dr " ´Γp3 ´sq sin p´s`2qπ
It is clear that if ψ is axisymmetric, so are Ω and Ω.
For the purpose of later applications, we take ψ as the rescalings of a fixed mollifier in the previous lemma and analyze the behavior of Ω and Ω.
Here " means bounded above and below by positive constants.Furthermore, for properly chosen ψ 1 (to be specified in the proof ) and sufficiently small , we have ‚ Ω , as a function of ϕ, is decreasing in ϕ P r0, π{2s.‚ There exist positive constants c ψ,1 , c ψ,2 , independent of , such that See Figure 1 for illustration.
Proof.Item 1 is clear.To see item 2, it suffices to notice that Ω pxq is an average of the spherical function δpp¨q ¨p0, 0, 1qq on a ball of radius centered at x, up to a negligible curvature effect from the sphere.To see item 3, we notice that Ω p ξq is an average of the spherical function cp1 ´|p¨q ¨p0, 0, 1q| 2 q p´3`sq{2 on a ball of radius centered at ξ.The last function has a singularity cp1 ´ξ2 3 q p´3`sq{2 " cpsin ϕq ´3`s near ϕ " 0. Therefore item 3 follows.
Then item 4 follows from (2.10) and the fact that the input ψ is decreasing in θ by construction, due to the application of the previous fact, with g 1 ptq " p1 ´t2 q p´3`sq{2 and g 2 pcos θq " ψ pθq.
since the integral operator on S 2 given by the integral kernel p1´|x¨ξ| 2 q p´3`sq{2 commutes with ∆ S 2 .By our choice of ψ , the function ∆ S 2 ψ pxq is axisymmetric, strictly negative on θ P r0, τ q, strictly positive on θ P pτ , q.It is also mean-zero on tx P S 2 : θ ď u because ψ is compactly-supported on this set.The function p1´|x¨p0, 0, 1q| 2 q p´3`sq{2 is axisymmetric, positive and decreasing in θ for θ P p0, π{2s.Therefore we see that ∆ S 2 Ω p0q ă 0. In fact, by analyzing the scaling that p1 ´|x ¨p0, 0, 1q| 2 q p´3`sq{2 " θ ´3`s and |∆ S 2 ψ pxq| " ´4 for θ ď c , one can quantify it as for sufficiently small .Combined with a similar scaling argument for the spherical gradient of ∆ S 2 Ω , one can show that the above inequality is also true for ∆ S 2 Ω pϕq with ϕ ď c .This gives Ω pϕq ď Ω p0q ´c ´5`s ϕ 2 for ϕ ď c .Combined with item 3, we get item 5.

Results on the LIC property.
In this subsection we state some results on the existence of energy minimizers and the LIC property of the potential W given by (1.1), as generalization of those in [8] to 3D.The proofs are similar to those in [8] and thus omitted.
Lemma 2.5.Assume 0 ă s ă 3. Then for any W in (1.1) with Ω satisfying (H), there exists a compactly supported energy minimizer in the class of probability measures on R 3 .The same is true for If we further assume 0 ă s ă 1, then any minimizer for W α with zero center of mass is supported in Bp0; Rq for some R ą 0 independent of α.The same is true if we instead assume 1 ď s ă 2 and ωpxq " 0 for any x with θ " π{2.
In the second part of the above lemma, the extra conditions guarantee that min ρ E α rρs is uniformly bounded in α (for 0 ă s ă 1, a possibly rotated ρ 1D has energy independent of α; for 1 ď s ă 2 and ω| θ"π{2 " 0, ρ 2D has energy independent of α).This is crucial in the proof of the uniform-in-α bound for the support of minimizers, as done in [8,Lemma B.1].
The following lemma shows that for LIC potentials, an Euler-Lagrange condition for the energy minimizer is also sufficient.Lemma 2.6.Assume W satisfies (W), and W has the LIC property.Assume there exists a compactly supported global energy minimizer (which has to be unique up to translation).Then for any probability measure ρ with Erρs ă 8, the following are equivalent: (i) ρ is the unique energy minimizer for W up to translation.
(ii) ρ satisfies the condition A direct application of the results in [7] as fully detailed in [8, Section 2] leads to the following consequences.
Theorem 2.8.Let W be given by (1.1) with 0 ă s ă 3 and Ω satisfying (H).Then W has the LIC property if and only if Ω given as in (2.2) is nonnegative.
For the class of potentials W α in (1.2) with ω satisfying (h), the corresponding angle function for the Fourier transform of |x| ´sp1 `αωpxqq is Ωα " c s `αω, where c s is given in (A.3).According to the sign of ω, the behavior of its energy minimizers can be categorized as: ‚ If ω is nonnegative, then W α has the LIC property for any α ě 0. ω is necessarily nonnegative if 2 ď s ă 3. ‚ If ω is sign-changing, then W α has the LIC property if 0 ď α ď α L where α L is defined in (1.4).If α ą α L , then W α does not have the LIC property, and thus infinitesimal concave by Proposition 2.7.ω is necessarily sign-changing if 0 ă s ă 1.We will see in Theorem 5.1 that both cases can happen if 1 ď s ă 2. Notice that this is the most novel case compared to the two dimensional results in [8].

Ellipsoid-shaped minimizers for LIC potentials
For simplicity, we will focus on the potentials W in (1.1) with the extra assumption (Hx): Ω satisfies (H) and axisymmetric with respect to the x 3 -axis.
In other words, Ω is a function of θ P r0, πs with Ωpθq " Ωpπ ´θq (abusing notation, denoting Ωpxq " Ωpθq).Its Fourier transform is also axisymmetric with respect to the ξ 3 -axis, and we may write Ωp ξq " Ωpϕq.For ω, we also introduce a similar assumption (hx): ω satisfies (h) and axisymmetric with respect to the x 3 -axis.
For a, b ą 0, denote as an axisymmetric rescaling of ρ 3 (defined in (A.1)).ρ 0,b is understood as the weak limit of ρ a,b as a Ñ 0 `, which is supported on the x 3 -axis.Similarly ρ a,0 is supported on the x 1 x 2 -plane.The support of ρ a,b with pa, bq P r0, 8q 2 ztp0, 0qu is a possibly degenerate ellipsoid with axis lengths aR 3 , aR 3 , bR 3 in the x 1 , x 2 , x 3 directions respectively.
Then there exists a unique pair pa, bq P p0, 8q 2 such that a nondegenerate ellipsoid ρ a,b is the unique energy minimizer for W (up to translation).Here the assumption Ω ě c ą 0 is automatically satisfied if 2 ď s ă 3.
If Ω ě 0, then there exists a unique pair pa, bq P r0, 8q 2 ztp0, 0qu such that a possibly degenerate ellipsoid ρ a,b is the unique energy minimizer for W (up to translation).
To prove the theorem, the key is to show that the potential generated by any ρ a,b is quadratic in its support.
Lemma 3.2.Assume 0 ă s ă 3 and Ω satisfies (Hx) and a, b ą 0. Then where then the same is true provided that B is finite.Remark 3.4.As observed in [8], although R 1 is only defined for 0 ă s ă 1, is defined for any complex number s with psq ą 0, holomorphic in s, and positive for real number s ą 0. This quantity appeared in (3.3) is understood in this way.
To prove this lemma, we first treat the case 0 ă s ă 1, in which we have the decomposition (2.6) and we may apply the strategy of 1D projections similar to [8].Then we extend it to the full range of s by analytic continuation.
3.1.The case 0 ă s ă 1.We first give a lemma on the projection of ρ a,b onto one-dimensional subspaces.Lemma 3.5.Assume 0 ă s ă 1.Let T be a linear transformation on R 3 with 1D image, spanned by a unit vector x1 .Let x1 , x2 , x3 be an orthonormal basis of R 3 .Then The proof is similar to [8, Section 3] and thus omitted.
Proof of Lemma 3.2, in the case 0 ă s ă 1.We first treat the case a, b ą 0. We aim to apply (2.6) to compute the generated potential.For fixed ξ1 P S 2 , let ξ1 , ξ2 , ξ3 be an orthonormal basis.Then The inner double integral ť R 2 ρ a,b pz 1 ξ1 `z2 ξ2 `z3 ξ3 q dz 2 dz 3 is the push-forward of a linear transformation onto the 1D subspace spanned by ξ1 .By Lemma 3.5, we get ĳ where (similar to the quantity r ϕ in [8, Section 3]) with ϕ 1 being the angle for ξ1 as in (1.3).The fact that ρ 1 minimizes the 1D interaction energy with potential |x| ´s `|x| , we obtain and it holds in supp ρ a,b in particular.Integrating in ξ1 as in (2.6), we obtain (3.2) with the quadratic parts in the potential being |x ¨ξ| 2 pa 2 sin 2 ϕ `b2 cos 2 ϕq ´p2`sq{2 Ωp ξq d ξ ¨pa 2 sin 2 ϕ `b2 cos 2 ϕq ´p2`sq{2 Ωpϕq sin ϕ dϕ using the axisymmetry Ωp ξq " Ωpϕq.This gives the expressions of A and B as in (3.3).We also observe that the LHS of (3.5) is greater than the RHS constant for y 1 R r´rξ 1 , rξ 1 s.Therefore, if Ω ě 0, the LHS of (3.2) achieves minimum on supp ρ a,b .
If a " 0, b ą 0, the the previous argument applies if π  2 R supp Ω where the last Ω is interpreted as a function of ϕ.For general Ω ě 0 with A ă 8, we necessarily have Ωp π 2 q " 0. Then one can approximate Ω by an increasing sequence of smooth nonnegative axisymmetric functions Ωn with π 2 R supp Ωn .The corresponding Ω n satisfies (Hx) since the positivity of Ω n is given by (2.5).The conclusion holds for each Ωn , and we obtain conclusion for Ω by the monotone convergence theorem.
If a ą 0, b " 0, one can use a similar approximation with 0 R supp Ωn .

3.2.
Extending the range of s.Next we aim to prove Lemma 3.2 for 1 ď s ă 3. Our argument is based on analytic continuation for the variable s.
We will first assume a, b ą 0, and explain why the proof also works for the exceptional case b " 0 at the end.
For this purpose, we first apply Lemma 3.2 with 0 ă s ă 1 (which is already proved in the previous subsection) and pa, bq replaced by pa{R 3 , b{R 3 q.Notice that D :" supp ρ a{R3,b{R3 " is independent of s.Thus we obtain for 0 ă s ă 1, where Here we emphasized the dependence of Ω on s with Ω being a fixed function on S 2 satisfying (H0).In other words, Ωpx; sq is determined by Fr|x| ´sΩpx; sqs " |ξ| ´3`s Ωp ξq, which is well-defined by applying Lemma 2.1 reversely.We will fix a choice of p Ω, a, bq and view every quantity above as a function in s (this includes Ωp¨; sq, A, B, τ s R 2`s 1 , ρ a{R3,b{R3 and the constant C in (3.7)).We recall from Remark 3.4 that τ s R 2`s 1 is an holomorphic function in s P S (recalling the definition of S in (2.1)), and thus A, B are well-defined and holomorphic for such s P S. Lemma 2.1 shows that Ωpx; sq is well-defined for s P S.
We claim that for any x P R 3 , the value of the generated potential at x f px; sq :" ´p|x| ´sΩpx; sq `Ax 2 1 `Ax 2 2 `Bx 2 3 q ˚ρa{R3,b{R3 ¯pxq, 0 ă s ă 3 can be extended to an holomorphic function in s P S. To see this, we will treat the part with the repulsive potential |x| ´sΩpx; sq (which we denote as f rep psq, suppressing the x dependence for a moment), and the quadratic part is straightforward since A, B are holomorphic.In fact, for any s P p0, 3q, ,p1`sq{2q is a normalization factor, being holomorphic in s P S. Therefore, this formula for ρ a{R3,b{R3 pxq can be extended holomorphically to any complex number s P S, for any fixed x P D. We write f rep psq as We already know from (3.7) that f px 1 ; sq " f px 2 ; sq for any x 1 , x 2 P D for 0 ă s ă 1.By the holomorphic property of f in s, the same is true for any s P S. In particular, it is true for s P r1, 3q, which gives (3.2) with (3.3) for s P r1, 3q.
Finally we treat the case Ω ě 0, a ą 0, b " 0, B ă 8 for some s " s 0 P r1, 2q.In this case the finiteness of B for s 0 implies the same property of B for any complex s with psq P p0, s 0 q.As a result, the formula (3.7) still holds for 0 ă s ă 1 by the previous subsection.Then we apply the same procedure to extend f px; sq.First, A, B are holomorphic for psq P p0, s 0 q by the dominated convergence theorem, using the assumption that B ă 8 for s " s 0 .It is easy to verify that for b " 0, ρ a{R3,b{R3 is supported on the x 1 x 2 -plane, given by ¯s{2 às a rescaling of ρ 2 .Then the generated potential from the repulsive part is Then we see that f px; sq can be extended holomorphically to psq P p0, s 0 q by the same argument as before.This holomorphic property implies f px 1 ; sq " f px 2 ; sq for any x 1 , x 2 P D and s P p0, s 0 q as before.Then the same is true for s 0 by taking the limit s Ñ s 0 using the dominated convergence theorem for the formulas for A, B and f rep .
To prove the 'inequality part' of Lemma 3.2 for general s, we first need a lemma on the analytic continuation of 1D generated potentials.Lemma 3.6.For 0 ă s ă 1, define gpx; sq " rpR ´s´2 1 |x| ´s `|x| 2 q ˚ρ 1 spxq ´rpR ´s´2 1 |x| ´s `|x| 2 q ˚ρ 1 sp1q for x P R, where ρ1 " R 1 ρ 1 pR 1 xq is a rescaling of ρ 1 , being the unique energy minimizer for the 1D potential R ´s´2 1 |x| ´s `|x| 2 .Then for every fixed x, gpx; sq can be extended holomorphically to s P S. gp¨; sq and B s gp¨; sq are bounded on p1, T s for any fixed s P S and fixed T ą 1. g also satisfies gpx; sq ą 0, @x ą 1, 1 ď s ă 3 Proof.Notice that supp ρ1 " r´1, 1s.Clearly gp¨; sq is an even function in x and vanishes on r´1, 1s for any s P p0, 1q by the minimizing property of ρ1 .Therefore it suffices to treat the case x ą 1.
Then we estimate Gpt; sq for t close to 1 and s P S. It is clear that Gpt; sq is bounded near t " 1 if psq P p0, 1q.We claim that |Gpt; sq| `|B s Gpt; sq| ď Cpt ´1q p´ psq`1q{2 p1 `| lnpt ´1q|q 2 for any T ą 1, 1 ă t ă T and psq P r1, 3q, where C depends on T and uniform on compact sets for s.
To see this, we first notice that the prefactors C 1 R 2`s 1 and sR ´s´2 1 in (3.10) have no singularity in s, and thus can be ignored.Also, it suffices to treat the case of real s.In this case, the integral Gpt; sq has the same type of singularity as ż 1 ´1pt ´yq ´s´1 p1 ´yq p1`sq{2 dy " ż 2 0 p `yq ´s´1 y p1`sq{2 dy, where we denote " t ´1 ą 0. In the nontrivial case of sufficiently small , the last integral can be estimated by cutting at y " : the part 0 ă y ă can be bounded by C ş 0 ´s´1 p1`sq{2 dy " C p´s`1q{2 ; the part ă y ă 2 can be bounded by C ş 2 y ´s´1 y p1`sq{2 dy ď C p´s`1q{2 up to an extra factor | ln | at s " 1.The estimate of B s G can be done similarly because the integrand only has extra logarithmic singularities in t ´y and 1 ´y (with the notations in (3.10)).
Therefore Gp¨; sq and B s Gp¨; sq are absolutely integrable in t P p1, T s for any s P S. This allows us to define the analytic continuation of gpx; sq to s P S by the original formula (3.9), and also shows the claimed boundedness of g and B s g.
The conclusion gpx; sq ą 0 for x ą 1 and s P r1, 3q follows from the fact that Gpt; sq ą 0 for t ą 1 and s P r1, 3q, which can be seen from (3.10) since the integrand and the prefactor are positive.Remark 3.7.In the last paragraph of the above proof, the positivity of G seems to be subtle.In fact, the coefficient ´sR ´s´2 1 " ´2 cos sπ 2 ps`1qπ βp 1 2 , 3`s 2 q is nonnegative for 1 ď s ď 3 but becomes negative for 0 ă s ă 1 or 3 ă s ă 4. On one hand, the fact that ρ1 being an energy minimizer for 0 ă s ă 1 implies gpx; sq ą 0, but this cannot be obtained from the above proof; on the other hand, this proof does not provide a nice way of extending g to s ě 3, or indicate its positivity there in case it is extended.
Proof of Lemma 3.2 for general s, 'inequality part'.We follow the notations in the proof of the 'equality part' of Lemma 3.2.For Ω ě 0, s P r1, 3q and a, b ą 0, we will prove that the generated potential p|x| ´sΩpx; sq `Ax 2 1 `Ax 2 2 `Bx 2 3 q ˚ρa{R3,b{R3 achieves minimum on D (here A, B follow (3.8)).At the end of the proof, we will also show the same conclusion for the case a ą 0, b " 0, s P r1, 2q, B ă 8.
We first assume a, b ą 0. For x P R 3 and s P S, define hpx; sq "rp|x| ´sΩpx; sq `Ax Therefore the integral on the RHS of (3.12) is well-defined for any such s, and one can differentiate with respect to s by differentiating the integrand (noticing that rξ is always bounded from below for a, b ą 0).We recall from Remark 3.4 that τ s R 2`s 1 is an holomorphic function in s P S. Therefore the RHS of (3.12) is holomorphic in s, and must agree with h as in its definition (3.11) for any s P S since the latter is also holomorphic in s.Since the integrand and prefactor in (3.12) are nonnegative for s P r1, 3q by Lemma 3.6 and Remark 3.4 respectively, we see that hpx; sq ě 0 as in (3.11), which finishes the proof in the case a, b ą 0.
In the case a ą 0, b " 0, s P r1, 2q, B ă 8, we will take the limit b Ñ 0 `of the previous case.Fix a and s, and denote D b , A b , B b , h b as the domains/quantities depending on b.
Notice that D 0 Ă D b for any b ą 0 due to its definition (3.6).We showed in the 'equality part' of Lemma 3.2 that p|x| ´sΩpx; sq `Ab x 2 1 `Ab x 2 2 `Bb x 2 3 q ˚ρa{R3,b{R3 is constant in D b for any b ě 0, and thus the same is true in D 0 .We denote this constant as C b , which is indeed twice the total energy of ρ a{R3,b{R3 for the potential |x| ´sΩpx; sq `Ab x 2 1 `Ab x 2 2 `Bb x 2 3 .We also have A b Ñ A 0 and B b Ñ B 0 as b Ñ 0 `by the dominated convergence theorem, since A 0 , B 0 ă 8. Therefore, by the weak lower-semicontinuity of the energy functional, we see that C 0 ď lim inf bÑ0 `Cb since ρ a{R3,b{R3 converges weakly to ρ a{R3,0 .
On the other hand, for x 1 R D 0 , we have x 1 R D b for sufficiently small b ą 0. Therefore rp|x| ´sΩpx; sqÀ b x 2 1 `Ab x 2 2 `Bb x 2 3 q ˚ρa{R3,b{R3 spx 1 q, as a function of b, is continuous at b " 0. The above two limits allow us to pass to the limit on the previously obtained result h b px 1 ; sq ě 0 and obtain the desired result h 0 px 1 ; sq ě 0.
When b Ñ 0 `, the numerator converges to ş π 0 | sin ϕ| 1´s Ωpϕq dϕ and thus remains bounded (for 0 ă s ă 2), while the denominator goes to infinity because it has a singularity like | sin ϕ| ´1´s and Ω ě c ą 0. Therefore lim bÑ0 `f pbq " 0. For the case 2 ď s ă 3, both the numerator and the denominator go to infinity, but it is clear that the denominator is much larger near the singularities ϕ " 0, π, and thus lim bÑ0 `f pbq " 0 is still true.Similarly lim bÑ8 f pbq " 8. Since f is continuous, we see that there exists b ą 0 with f pbq " 1.
If lim bÑ0 `f pbq ě 1, then Ap1, 0q ě Bp1, 0q, and thus one can find a ą 0 such that item 3 in the statement of the lemma holds by using the homogeneity.If lim bÑ8 f pbq ď 1, then one can find b ą 0 such that item 2 holds.If lim bÑ0 `f pbq ă 1 and lim bÑ8 f pbq ą 1, then item 1 holds as in the previously considered case Ω ě c ą 0.
Proof of Theorem 3.1.If Ω ě c ą 0, then Lemma 3.8 gives a pair pa, bq P p0, 8q 2 such that Apa, bq " Bpa, bq " 1.Then Lemma 3.2 and Lemma 2.6 show that ρ a,b is the unique energy minimizer for W in (1.1), since W has the LIC property by Theorem 2.8.
Then we treat the general case of Ω ě 0. We may assume 0 ă s ă 2 because Ω is always strictly positive in the case 2 ď s ă 3 due to (2.3), (2.4) and the assumed strict positivity of Ω.One of the three items of Lemma 3.8 must happen, and the conclusion can be obtained as before in the case of item 1.If item 2 happens, i.e., there exists b P p0, 8q such that Ap0, bq ď 1, Bp0, bq " 1, then Lemma 3.2 shows that p|x| ´sΩpxq `Ap0, bqx 2  1 `Ap0, bqx 2 2 `x2 3 q ˚ρ0,b achieves minimum on supp ρ 0,b .Since Ap0, bq ď 1, the same is true for W ˚ρ0,b " p|x| ´sΩpxq `x2 3 q ˚ρ0,b .Therefore Lemma 2.6 show that ρ a,b is the unique energy minimizer for W .The case of item 3 of Lemma 3.8 is similar.Remark 3.9.As a byproduct of the uniqueness part of Theorem 3.1, the pair pa, bq as in Lemma 3.8 is unique.

Collapse to lower dimensions for large α
In this section we prove that the energy minimizers for the potential W α (defined in (1.2)) will collapse into 1D/2D for sufficiently large α, for certain ranges of s and under certain non-degeneracy conditions.The candidates of the lower-dimensional minimizers ρ 1D and ρ 2D , as given by (A.2), are the minimizers of restricted isotropic Riesz interaction energies.Theorem 4.1.For fixed 0 ă s ă 1, there exists C ˚such that the following holds: For Ω satisfying (Hx), if the minimum of Ω is achieved at θ " 0 with Ωp0q " 1 and the non-degeneracy condition then ρ 1D is the unique energy minimizer.As a result, if ω satisfies (hx), achieves minimum at ωp0q " 0, and satisfies the non-degeneracy condition ωpθq ě c ω |θ| 2 , @θ P " 0, π 2 ı for some c ω ą 0. Then there exists a unique 0 ă α ˚ď C ˚{c ω , such that for any α ą α ˚, ρ 1D is the unique energy minimizer for W α , and for any 0 ď α ă α ˚, ρ 1D is not an energy minimizer for W α .
Theorem 4.2.For fixed 0 ă s ă 2, there exists C ˚such that the following holds: For Ω satisfying (Hx), if the minimum of Ω is achieved at θ " π 2 with Ωp π 2 q " 1 and the non-degeneracy condition ı then ρ 2D is the unique energy minimizer.As a result, if ω satisfies (hx), achieves minimum at ωp π 2 q " 0, and satisfies the non-degeneracy condition for some c ω ą 0. Then there exists a unique 0 ă α ˚ď C ˚{c ω , such that for any α ą α ˚, ρ 2D is the unique energy minimizer for W α , and for any 0 ď α ă α ˚, ρ 2D is not an energy minimizer for W α .
Remark 4.3.In the above two theorems, the requirements on the range of s are necessary, because a larger s would make ρ 1D (respectively, ρ 2D ) having infinite energy, and cannot be an energy minimizer for any α.In particular, they are not applicable to the case 1 ď s ă 2 and ω achieves minimum at ωp0q " 0, which will be studied in detail in the next section.
Remark 4.4.The 'non-degeneracy condition' in Theorem 4.1 is necessary.In fact, if instead one has Ωpθq ď 1 `C|θ| κ for some κ ą 2, then one can apply [8, Theorem 5.5] in the x 1 x 3 -plane to show that ρ 1D is not a Wasserstein-8 local minimizer.We expect similar result also holds for Theorem 4.2.
In this case, one is allowed to apply Lemma 3.2 with a " 0 and b " R 1 {R 3 (for which ρ a,b " ρ 1D ) since Ω˚,1 vanishes near π{2 and A is clearly finite.The condition Ω ˚,1 p0q " 1 is equivalent to ρ 1D being a steady state for |x| ´sΩ ˚,1 pxq `|x| 2 , i.e., B " 1.This gives the requirement Then p|x| ´sΩ ˚,1 pxq `|x| 2 q ˚ρ1D achieves minimum on supp ρ 1D as long as Let us now show that indeed we can find a function Ω˚,1 such that A ă 1.Compared with the last integral in (4.2), there is an extra factor tan 2 ϕ in the definition of A, which is close to zero near ϕ " 0, π.Therefore, by taking | cos ϕ| ´s sin ϕ Ω˚,1 pϕq as a mollifier supported in ϕ P r {2, s Y rπ ´ , π ´ {2s with ą 0 small and satisfying (4.2), we can guarantee that A ă 1 is also satisfied.Then we may divide by | cos ϕ| ´s sin ϕ and obtain the desired Ω˚,1 .

Expansion of energy minimizers for large α
In this section we consider the cases where the potential is too singular to allow the energy minimizer to collapse to lower dimension distributions.In particular, we are interested in the energy minimizers for W α in (1.2) in the case 1 ď s ă 2, ω satisfying (hx) and achieving minimum at θ " 0. In this case, the minimizers are not allowed to concentrate on the preferred x 3 -direction.As the parameter α in (1.2) gets large, we will analyze the expansion of the energy minimizer as α increases.
If s " 1, then the same are true with 'strictly positive' in item 1 replaced by 'nonnegative' (item 1 would be false without this replacement).
Remark 5.2.As a complement of this theorem, we notice that it is not possible to have ω satisfying (hx), not identically zero, with ω ď 0. In fact, for such ω, we have the Fourier formula for the interaction energy ż at least for compactly supported smooth ρ.If ω ě 0 and not identically zero, then the LHS is positive provided that ρ is nonnegative and compactly supported.This shows ω ď 0 cannot hold because otherwise the RHS would be nonpositive.
In the rest of the proof, we assume 1 ă s ă 2.
One can take ω 1 as the Ω in Lemma 2.4 with ď θ 0 , and all the desired properties are consequences of Lemma 2.4 and the explicit expression (2.10).
For the purpose of a later application, we also give an existence result which is a variant of ω 1 in Theorem 5.1.
In the rest of this proof, we assume 1 ă s ă 2. We define δp ξ ¨ȳqpψ 1 pȳq ´κψ 2 pȳqq dȳ where ψ is as defined in Lemma 2.4 so that items 4 and 5 of the lemma are satisfied, and the parameters 0 ă 2 2 ď 1 ď ϕ 0 and κ ą 0 to be chosen.Notice that this strategy is similar to the one in Theorem 5.1 but in Fourier space.Then we apply Lemma 2.3 reversely and obtain p1 ´|x ¨ȳ| 2 q ´s{2 pψ 1 pȳq ´κψ 2 pȳqq dȳ Item 1 of Lemma 2.4 shows that ω3 pϕq " 0 on ϕ P r0, π{2 ´ϕ0 s.Item 2 of Lemma 2.4, together with the condition 2 2 ď 1 , shows that ω3 ě 0 with ω3 pπ{2q ą 0 as long as (5.3) Item 3 of Lemma 2.4 shows that ω 3 p0q " 0 as long as ´s 1 " cp 1 , 2 qκ ´s 2 (5.4) where cp 1 , 2 q is a positive constant depending on 1 and 2 , being uniformly bounded from above and below.Choosing κ according to (5.4), we see that (5.3) reduces to We fix the choice 1 " ϕ 0 , and then this condition, together with 2 2 ď 1 , are automatically satisfied as long as 2 is sufficiently small.
Remark 5.4.As a byproduct, this lemma shows that item 1 in Lemma 4.6 is not a consequence of items 2 and 3 therein.In fact, the function ω 3 in Lemma 5.3 achieves its minimum at ω 3 p0q " 0 and satisfies ω 3 pπ{2q ą 0 and ω3 ě 0. By taking a constant multiple, we may assume ω 3 pπ{2q " 1, and thus we have A " 1 as in the proof of Lemma 4.6.Also we have B ă 1 because Lemma 5.3 guarantees that ω3 is concentrated near π{2.Therefore ω 3 satisfies items 2 and 3 of Lemma 4.6 but not item 1.This is in contrast with Lemma 4.5, in which item 1 is a consequence of items 2 and 3.

The LIC case.
Then we analyze the expansion phenomenon in case Ω α is always LIC for any α ě 0, i.e., the case ω ě 0. In item 1 of Theorem 5.1, we have seen that for 1 ď s ă 2 there exists such ω satisfying (hx) with its minimum achieved at θ " 0, and strict positivity of ω is possible if 1 ă s ă 2.
We will first treat the case when ω is strictly positive.The following theorem describes how the ellipsoid-shaped energy minimizer expands as α gets large for a wider range of s.
The uniqueness can also be obtained by analyzing the derivative of f pt, αq, and this approach works for α P p0, 8s.In fact, after tedious but simple computations one gets where the integrals are with respect to the positive weight psin 2 ϕ `t2 cos 2 ϕq ´p2`sq{2´1 pc s α ´1 `ωpϕqq sin ϕ dϕ.
Notice that the numerator is negative by the Cauchy-Schwarz inequality.Therefore, we see B t f pt, αq ą 0 for any t P p0, 8q and α P p0, 8s.
Since the denominator in (5.6) is positive and away from zero, it is clear that lim αÑ8 f pt, αq " f pt, 8q and the limit is uniform on compact sets for t P p0, 8q.Therefore, using B t f pt, 8q ą 0, we may apply the implicit function theorem to conclude that lim αÑ8 t α " t 8 i.e., b α {a α Ñ t 8 as stated in the theorem.
To see the limiting behavior of a α as α Ñ 8 (which also implies the same for b α ), we recall the A " 1 equation in (5.5) (writing b α " a α t α ) Sending α Ñ 8, the LHS converges to a positive constant.Therefore so does the RHS.
Remark 5.7.This result shows that a α , b α scale (as α Ñ 8) in the same way as if ω " 1, i.e., an isotropic potential whose repulsive part has strength α.One can easily show that this agreement is also true for the minimal total energy, i.e., scales like α 2{p2`sq both the stated ω and ω " 1.In particular, if one considers 1 ă s ă 2 and ω as in item 1 of Theorem 5.1, the smallness of ω near θ " 0 does not affect the minimal energy in terms of its α-scaling.
Remark 5.8.A similar result can also be proved for 2D anisotropic potentials studied in [8].
In the case 1 ď s ă 2 with ω nonnegative but not having a positive lower bound, it is possible that b α Ñ 0 as α Ñ 8, as shown in the following theorem.Theorem 5.9.Assume 1 ď s ă 2. Then there exists ω satisfying (hx), achieving minimum only at ωp0q " 0, with ω ě 0, such that the parameters a α , b α as given in Theorem 3.1 for Ω α , satisfying b α Ñ 0 as α Ñ 8.
This asymptotic behavior of minimizers is counter-intuitive, because ω achieves minimum only at ωp0q " 0 (i.e., the x 3 -direction), but the minimizers expand in its perpendicular directions, the x 1 x 2plane.
Proof.We follow the notations in the proof of Theorem 5.5.We take ω as in Lemma 5.3 with ϕ 0 small.Then ω satisfies (hx), achieving minimum only at ωp0q " 0, and ω is nonnegative and concentrated near ϕ " π{2.For such ω, we have f pt, 8q, as in (5.7), well-defined and continuous for t P r0, 8q as discussed in the previous proof.Moreover, one can ensure that f p0, 8q ě 2 if ϕ 0 is sufficiently small since the support of ω lies inside ϕ P rπ{2 ´ϕ0 , π{2 `ϕ0 s.As a result, f pt, 8q ě 2 for any t P r0, 8q since we showed that f pt, 8q is increasing in t.

5.3.
The general case: expansion of minimizers for large α.In the general case when 1 ď s ă 2, ω achieving its minimum at θ " 0 and ω is possibly sign-changing, we will apply delicate comparison arguments to show that any energy minimizer cannot be constrained in a fixed infinite cylinder as α increases.In other words, the minimizer has to expand in at least two dimensions.
We also notice that the expansion in all three dimensions is not true in general, due to the example in Theorem 5.9.Here a |x| 2 ´|x ¨ξ| 2 is the distance between x and the line containing the vector ξ.Therefore the result shows that supp ρ pαq cannot be contained in a fixed infinite cylinder for all α.Let us define C R,ξ to be the cylinder of radius R ą 0 and direction ξ P S 2 defined by the inequality a |x| 2 ´|x ¨ξ| 2 ď R. The assumption ωpπ{2q ą 0 is sharp, because Lemma 2.5 shows that if ωpπ{2q " 0 then any energy minimizer for W α with zero center of mass has uniformly bounded support.
To prove this theorem, we will argue by contradiction and assume that every ρ pαq is supported in a fixed cylinder of radius R. Then we will use a comparison argument with isotropic energies, based on the following lemma.
Lemma 5.11.Assume 1 ď s ă 2. For any α ą 0, let E α denote the total energy for W α pxq " α|x| ´s `|x| 2 .Fix R ą 0 and ξ P S 2 .Denote P as the set of probability measures on R Notice that the minimal value of E α rρs for ρ P P scales like α 2{p2`sq where the exponent 2{p2`sq ă 2{3 for 1 ă s ă 2. Therefore the above result shows that restricting to an infinite cylinder of radius R makes the minimal energy larger in terms of its α-scaling (which degenerates to a logarithmic factor for s " 1).
Proof.We first treat the case 1 ă s ă 2. Let ρ P P R,ξ .Define where the extra '´1' takes account of the n " 0 term in ř n n 2 a n .It is straightforward to show that inf u F 1 rus ą 0, and inf u F α rus " α 2{3 inf u F 1 rus by rescaling.Therefore we conclude that E α rρs ě cpα 2{3 ´1q, which implies the desired result for any sufficiently large α.The case of smaller α ě 1 is clearly true up to switching to a smaller constant c.
Finally we treat the case s " 1 which is more delicate.We will imitate the previous proof but consider cylinder pieces of various heights.Define where the extra '´2 2k ' takes account of the n " 0 term in ř n 2 2k n 2 a k,n .Then, using inf u F α rus " cα 2{3 as before, we obtain αK2 ´k ÿ n a 2 k,n `ÿ n 2 2k n 2 a k,n ě cpcpαKq 2{3 ´22k q .
Therefore, we conclude By taking K " ln α with properly chosen constant multiple, one can absorb the last negative term and obtain the conclusion.

Remark 3 . 3 .
If we do not assume the axisymmetry, then we expect a result similar to Lemma 3.2 with the quadratic form Ax 2 1 `Ax 2 2 `Bx 2 3 replaced by a general quadratic form in x, and a result similar to Theorem 3.1 with ρ a,b replaced by the push-forward of ρ 3 by a general linear transformation.In the 2D case, such results were obtained in [8,Section 3].

2 ¯ps´1q{2 3 R 2 `s 3 a 2 b
allowed to take B s f rep psq by differentiating the integrand.In fact, differentiating |x ´y| produces an extra logarithmic singularity at x or the boundary, which does not affect the integrability for s P S; differentiating C is clearly allowed; differentiating Ωpx ´y; sq does not affect the integrability due to the smoothness of B s Ωpx; sq in x as shown in Lemma 2.1.This proves the holomorphic property of f rep , and thus that of f .

ρR 3 p| ¨|´s ˚ρqρ dx ě ÿ n ĳ |x¨ξ|,|y¨ξ|Prn´1{2,n`1{2q |x ´y| ´sρpyq dyρpxq dx ě c ÿ n a 2 n 3 p| ¨|2 ˚ρqρ dx " 2 ż R 3 |x| 2 ρpxq dx ě c ÿ n n 2 E α rρs ě c ˜α ÿ n a 2 nF 2 n
dx, n P Z as the mass of ρ inside each piece of the cylinder of height 1.Then the repulsive part of the energy satisfies ż since one always has |x ´y| ´s ě c in the last integrand, c depending on R. The quadratic part of the energy is estimated by ż R a n using the mean-zero assumption.Therefore we see that α rus " α ż R u 2 dx `żR x 2 upxq dx (5.11) for nonnegative u P L 2 pRq with compact support and ş R u dx " 1, then it is clear that α ÿ n a `ÿ n n 2 a n ě cF α " ÿ n a n χ rn´1{2,n`1{2q ı ´1 ě c inf u F α rus ´1,

8 ÿ k" 0 2 3 p| ¨|´1 ˚ρqρ dx ě c 8 ÿ 3 p| ¨|2 ˚ρqρ dx " 2 ż R 3 |x| 2 ρpxq dx ě c ÿ n 2 n 2 ´k ÿ n a 2 k,n `ÿ n 2
a k,n " ż |x¨ξ|Pr2 k pn´1{2q,2 k pn`1{2qq ρ dx, k P Z ě0 , n P Z as the mass of ρ inside each piece of the cylinder of height 2 k .Notice that |x| ´1 ě c ´kχ |x|ďp1`2Rq2 k with c depending on R. It is clear that |x 1 ´x2 | ď p1 `2Rq2 k whenever x 1 , x 2 are in the same piece of the cylinder of height 2 k .Therefore ż R For any k P Z ě0 , the quadratic part of the energy is estimated by ż R 2k n 2 a k,n using the mean-zero assumption (with c independent of k).Therefore we see that E 2k n 2 a k,n for any K P Z ě1 .Later we will specify the choice of K. Using the notation F α rus in (5.11), for every k P Z ě0 , we have αK2 2k n 2 a k,n ě c ´FαK " ÿ n a k,n 2 ´kχ r2 k pn´1{2q,2 k pn`1{2qq ı ´22k

E 3 W 2 ż R 3 ż R 3 |x ´y| ´sω 4
α rρ pαq s " min ρ E α rρs ď C min ρ E α rρs " Cα 2{p2`sq (5.12) using the isotropic energy functional E α defined in Lemma 5.11.On the other hand, we define the energy for W α,˚p xq " |x| ´sp1 `αω ˚pxqq `|x| 2 as E α,˚r ρs " α,˚p x ´yqρpyq dyρpxq dx "E α rρs ´ α px ´yqρpyq dyρpxq dx where the last integral is well-defined as long as E α rρs ă 8, due to dominated convergence.Then E α,˚r ρs ď E α rρs (5.13) for any compactly supported ρ P P with E α rρs ă 8, since Fr|x| ´sω 4 pxqs " |ξ| ´3`s ω4 p ξq ą 0 by the construction of ω 4 .Since ω ˚ě c ą 0, we have E α,˚r ρ pαq s ě cE α rρ pαq s (5.14) Fourier transform.We first give the formula for the Fourier transform for functions of the form |x| ´sΩpxq.We will state it for complex numbers s in the region Proof.We first show that Fr|x| ´sΩpxqs is a locally integrable function for any complex number s P S. We fix the Littlewood-Paley cutoff function ψpxq, which is radial, smooth, nonnegative, supported on t1 ď |x| ď 4u and ψpxq `ψpx{2q " 1 on t2 ď |x| ď 4u.Then we may decompose |x| ´sΩpxq as converges in the sense of distributions.Since psq P p0, 3q, I 1 is locally integrable.I 1 is supported on t|x| ď 2u, and thus FrI 1 spξq is in L 8 and smooth in ξ (for every fixed s P S).Also, one can differentiate FrI 1 spξq with respect to s by B s FrI 1 spξq " ψp2 ´kxq ¯|x| ´sp´ln |x|qΩpxqe ´2πix¨ξ dx since the last integral converges absolutely.This shows FrI 1 spξq is holomorphic in s and B s FrI 1 spξq is smooth in ξ.Each term ψp2 ´kxq|x| ´sΩpxq in the summation I 2 in (2.8) can be written as 2 ´ks gp2 ´kxq, where gpxq " ψpxq|x| ´sΩpxq is a compactly supported smooth function.Therefore its Fourier transform can be computed as | cos θ| ´3`s sin θ dθ " 2 s´2 .Since | cos θ| ´3`s sin θ concentrates near θ " π{2 as s Ñ 2 `, we see that 2τ 3´s | cos θ| ´3`s sin θ forms an approximation of identity in θ P r0, πs.Therefore we obtain replaced by a{R 3 , b{R 3 ) and tracing the constants carefully, we get (the last quantity denoting the constant value of the generated potential on D).For every fixed x P R 3 , hpx; sq is holomorphic in s P S by the proof of the 'equality' part of Lemma 3.2.If 0 ă s ă 1, we already proved the positivity of hpx; sq in subsection 3.1.Moreover, following (3.5)(with a, b 1 ; s where rξ 1 :" pa 2 sin 2 ϕ 1 `b2 cos 2 ϕ 1 q 1{2 .Here gpx; sq is defined by the generated potential in 1D Lemma 3.6 gives the boundedness of gpx; sq and B s gpx; sq for fixed s P S and x in any fixed compact set.