Global Minimizers of a Large Class of Anisotropic Attractive-Repulsive Interaction Energies in 2D

We study a large family of Riesz-type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse-supported configurations are the global minimizers based on linear convexity arguments. Moreover, for certain anisotropic parts, we prove that for large values of the parameter the global minimizer is only given by vertically concentrated measures corresponding to one dimensional minimizers. We also show that these ellipse-supported configurations generically do not collapse to a vertically concentrated measure at the critical value for convexity, leading to an interesting gap of the parameters in between. In this intermediate range, we conclude by infinitesimal concavity that any superlevel set of any local minimizer in a suitable sense does not have interior points. Furthermore, for certain anisotropic parts, their support cannot contain any vertical segment for a restricted range of parameters, and moreover the global minimizers are expected to exhibit a zigzag behavior. All these results hold for the limiting case of the logarithmic repulsive potential, extending and generalizing previous results in the literature. Various examples of anisotropic parts leading to even more complex behavior are numerically explored.


Introduction
In this paper we study the interaction energy functional W px ´yqρpyq dyρpxq dx (1.1) where ρ is a probability measure on R 2 .For notational convenience, we use ρpxq dx to denote the integral with respect to a general Borel measure ρ.Here, W is a two-dimensional anisotropic attractive-repulsive interaction potential, given by W pxq " |x| ´sΩpθq `|x| 2 , 0 ă s ă 2, ( and W log pxq " ´ln |x| `Ωpθq `|x| 2 , ( that can be seen as a limit s Ñ 0 `of potentials similar to (1).Here θ P S 1 " r´π, πq denotes the angle of x: x |x| " pcos θ, sin θq, and Ω is a function defined on S 1 , modeling the anisotropic effect of the interaction.Throughout this paper, we will assume the angle function Ω satisfies (H): Ω is smooth (i.e., Ω P C 8 pS 1 q), strictly positive, and Ωpθ `πq " Ωpθq.
Anisotropic interactions of the form (1) appear in the modelling of edge dislocations of the same sign [24,16] in material science.Their associated energy minimizers were conjectured to be vertical walls of dislocations, conjecture settled in [25], where the authors proved that the global minimizer, modulo translations, is given by the semi-circle law on the vertical axis.
Finding the unique global minimizers of the interaction energy for particular potentials is a classical problem in potential theory [15,26].More precisely, for the repulsive logarithmic potential with quadratic confinement W pxq " ´ln |x| `|x| 2 in 2D, that is (1) with Ω " 1, it is known [15] that the unique global minimizer is the characteristic function of a suitable Euclidean ball, while for W pxq " |x| ´s `|x| 2 , that is (1) with Ω " 1 and 0 ă s ă 2, it is known [6,5] that the unique global minimizer of the associated interaction energy E is given by ρ 2 pxq " C 2 pR 2 2 ´|x| 2 q s{2 `(1.4) for some positive constants R 2 , C 2 depending on s (see (A) for the explicit formula for R 2 , C 2 ).Here, the uniqueness of minimizer can be easily obtained from the linear interpolation convexity (LIC) property of W , as explained below.For an interaction potential W , we say it has the LIC property, as defined in [12], if 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.This is equivalent to say that d 2 dt 2 Erp1 ´tqρ 0 `tρ 1 s " 2Erρ 0 ´ρ1 s ą 0. It is straightforward to see that LIC implies the uniqueness of energy minimizer.
For anisotropic attractive-repulsive interaction potentials, the behavior of energy minimizers becomes much more interesting because radial symmetry of minimizers is no longer expected.Intuitively, the shape of the minimizer should elongate along the direction where Ω has smaller values.To observe the phase transition phenomena of anisotropic energy minimizers, we consider a family of potentials W α pxq " |x| ´sp1 `αωpθqq `|x| 2 , 0 ă s ă 2, α ě 0 (1.5)where ω is an angle function and α is a scaling parameter.We assume that ω satisfies (h): ω is smooth, nonnegative, not identically zero, ωpθ `πq " ωpθq, and ωp π 2 q " 0, where in the last condition we fix the angle π 2 as the direction with smallest ω value, without loss of generality.In this case the associated energy will be denoted as E α (when ω is clear from context).Using this family of potentials, one can study how the energy minimizers behave as α changes.
A series of recent works [25,10,11,20,22] study a particular family of anisotropic potentials, corresponding to a limiting case of the potential W α in (1) as s Ñ 0 `, see Section 7, given by 1 W log,α pxq " ´ln |x| `αωpθq `|x| 2 , ωpθq " cos 2 θ, α ě 0. ( As already mentioned above, for the isotropic case α " 0, the unique energy minimizer is a constant multiple of the characteristic function of a ball.For α ą 0, [25,10] show the following behavior of energy minimizers: ‚ If 0 ă α ď α L :" 1, then W log,α has the LIC property, and the energy minimizer is unique, being a constant multiple of the characteristic function of an ellipse.This ellipse is elongated along the x 2 -axis.When α " α L , the ellipse degenerates into a vertical segment, meaning that the minimizer becomes ρ 1D pxq " ρ 1 px 2 qδpx 1 q with ρ 1 pxq " CpR 2 ´|x| 2 q 1{2 `for some positive constants C, R.Here ρ 1 pxq is the unique energy minimizer for the one-dimensional interaction potential ´ln |x| `|x| 2 .‚ If α ą α L , then the energy minimizer is unique and given by ρ 1D .In higher dimensions (d ě 3), a similar potential W α pxq " ´1 |x| d´2 `α x 2 1 |x| d `|x| 2 was studied in [11].Similar conclusion is obtained when W α has the LIC property (i.e., ´1 ă α ď d ´2), but in the case with larger α, the behavior of the energy minimizers remains unknown.Finally, the most recent results in the literature [21] show that the ellipses are still the global minimizers for a class of interaction potentials of the form (1) via a perturbative argument.Notice that in [21] they do not assume that the perturbation ω is positive.However, since the perturbation is bounded below, one can add a constant to (1) making ω positive without changing the minimization problem.
In this work we aim to understand the energy minimizers for W α for the full range of singular repulsive potentials 0 ă s ă 2 and its limiting case s " 0. Our main results find large families of anisotropic parts ω for which we can show the typical behavior illustrated in Figure 1, without any smallness assumption as in [21].Moreover, we give sharp conditions for some of the critical values in this illustration.
Let us elaborate more our previous statement.In fact, one can see that the typical behavior of the energy minimizers can be much more complicated than the previously considered particular case (1).The existence of compactly supported minimizers for very general anisotropic parts is shown for completeness in Appendix B, and thus we may focus on the study of uniqueness and other properties of the minimizers.The main content of the present work will be devoted to the study of the case 0 ă s ă 1.In fact, the limiting case s " 0 (i.e., logarithmic potentials) exhibits similar phenomena as the 0 ă s ă 1 case, while the case 1 ď s ă 2 has relatively simple phenomena.They will be studied in Sections 7 and 8 respectively. 1In the references, the quadratic part of the potential was taken as |x| 2 {2, but the results on the energy minimizers are the same via a rescaling of the spatial variable.An illustrative example using a particle gradient flow simulation with 1600 particles.The interaction potential is W α as in (1) with s " 0.4 and ωpθq " cos 4 θ 0.1 cos 2 θ with the α values 0, 0.2, 0.5, 0.7, 0.8, 0.85, 0.9, 0.95, 1, 1.2, 1.5, 2, 4. The blue ellipses are the predicted shape of the unique minimizer in the LIC cases.The blue dashed lines are the height of ρ 1D .The pink line is the predicted slope of the fragmented segments.See Section 9.3 for more details.
For the 0 ă s ă 1 case, we start by analyzing the LIC property of a general potential W given by (1), in Section 2. By calculating its Fourier transform explicitly, we show that the LIC property for W is equivalent to the nonnegativity of Ŵ away from the origin (Theorem 2.6).For W α in (1), this gives a critical value2 α L ą 0 (see (2.2)), around which the behavior of energy minimizers has a drastic change: ‚ For 0 ď α ď α L , W α has the LIC property, and thus there exists a unique energy minimizer.‚ For α ą α L , W α is not LIC, and we show that W α is infinitesimal concave (Theorem 2.5), a concept introduced in [12, Section 7].One expects the energy minimizers to have a lower dimensional support and/or exhibit complicated behavior.
We first study the case when W has the LIC property (i.e., W α with 0 ď α ď α L ), in Section 3. To our surprise, although the unique energy minimizer is no longer a characteristic function as in [10], the shape of its support is still an ellipse.In fact, the minimizer is given by the push-forward of ρ 2 by a linear transformation (Theorem 3.1).The proof is based on a decomposition of the LIC potential |x| ´sΩpθq into a positive linear combination of one-dimensional potentials (Corollary 2.3), which is a consequence of the explicit formula for its Fourier transform.We also notice in Section 4 that the set of such 'elliptic' distributions is closed under the flow map of the associated Wasserstein-2 gradient flow (Theorem 4.1).For this class of special solutions of the gradient flow, we show its long time convergence to the minimizer (Theorem 4.2).These solutions of the gradient flow are reminiscent of rotating vortex patch solutions in fluid mechanic equations [23,17].
Then we turn to the study of non-LIC potentials (i.e., W α with α ą α L ) in Section 5.For large α, the behavior of the energy minimizers depends on the degeneracy of ωpθq near its minimum point θ " π 2 : ‚ If ω is non-degenerate at π 2 (having a positive second derivative), then is the unique energy minimizer for sufficiently large α (Theorem 5.1).Here ρ 1 pxq is the unique energy minimizer for the one-dimensional interaction potential |x| ´s `|x| 2 , and the positive constants C 1 , R 1 are given in (A).This is similar to the large-α case in [10].Our result is proved by a similar comparison argument as in [10], but requires the design of a special potential (Lemma 5.3) to compare with.Such design is enabled by the tools developed in Section 3. ‚ If ω is degenerate at π 2 , then ρ 1D is never a global energy minimizer for any α (Theorem 5.5), not even a Wasserstein-infinity (d 8 )-local minimizer.However, the width of the support of any global energy minimizer shrinks to zero as α Ñ 8 (Theorem 5.7).Numerical simulation shows that a typical minimizer in this case is a vertical segment with slight zigzags (see Section 9.2).
Finally we study the case when α is slightly greater than α L in Section 6.In fact, if the ellipse-shaped minimizer at α " α L does not collapse to ρ 1D , then neither can happen if α is slightly greater than α L .Due to infinitesimal concavity, it is reasonable to expect the formation of interesting lower-dimensional patterns for the energy minimizers.To understand these structures, we conduct an asymptotic expansion of the generated potential W ˚ρ around a segment-like piece within ρ (Proposition 6.1), showing that its local stability is determined by the sign of the Fourier transform of W in its perpendicular direction.For some particular examples of ω, this gives a critical value α L,0 ą α L , such that for any α P pα L , α L,0 q, the support of any d 8 -local minimizer cannot contain any 'vertical segments'.This guarantees that ρ 1D is not a d 8 -local minimizer, and minimizers are expected to exhibit a zigzag behavior with a collection of tilted slopes.
Finally, Section 7 is devoted to the s " 0 case, in which we derive the logarithmic potential (1) as a limit of potentials similar to (1).This allows us to extend most results for the case 0 ă s ă 1 to the case s " 0. In Section 8 we study the case 1 ď s ă 2, in which we show that LIC always holds, and we conclude that the unique energy minimizer is always supported on some ellipse.In Section 9 we illustrate the main results of the paper by numerical simulations of several examples, and explore cases not covered by our theory.Let us finally remark that some preliminary ideas seem to indicate that our techniques are applicable to higher dimensions.

LIC and Fourier transform
If W is given by (1) with Ω satisfying (H), then it is clear that (W): W is even, locally integrable, lower-semicontinuous, and bounded above and below by positive multiples of |x| ´s near 0.
Then, using the approximation argument in [12,Lemma 2.5], one can justify that for any compactly supported signed measure µ with ş R 2 µpxq dx " ş R 2 xµpxq dx " 0 and Er|µ|s ă 8, where W rep pxq :" |x| ´sΩpθq is the repulsive part of W .Here the quadratic term |x| 2 makes no contribution to Erµs due to the mean-zero properties of µ, as observed in [19].This motivates our detailed study of the Fourier transform of functions of the form |x| ´sΩpθq.
2.1.Fourier transform of the potential.We first compute the Fourier transform of the potential W , ignoring the quadratic part.We will denote the Fourier variable ξ " |ξ| e ϕ (see (A) for the definition of e ϕ ) throughout the paper.
Lemma 2.1.For any smooth Ω with Ωpθ `πq " Ωpθq, we have Here, τ 2´s and c s are explicitly computable, see Appendix A. Ω is also smooth and even.If 1 ă s ă 2, then the formula can be simplified as (2.4) Remark 2.2.Since |x| ´sΩpθq is even, similar formulas work for the inverse Fourier transform, i.e., (2.5) Proof.By scaling arguments, it is easy to show that Fr|x| ´sΩpθqs has to take the form (2.1) as long as it is a locally integrable function.
We first assume 1 ă s ă 2. Then Fr|x| ´sΩpθqs P L 2 `L8 is a locally integrable function since |x| ´sΩpθq P L 2 `L1 .Also, its Fourier transform is given by where the integral is interpreted as an improper integral in the radial direction and Bp0; Rq denotes the ball of radius R centered at 0. Without loss of generality, we consider ξ " e 0 to obtain The integral ş 8 0 r 1´s cos r dr (interpreted as an improper integral) is equal to Γp2 ´sq sin πps´1q 2 by (A), which is positive for 1 ă s ă 2. Therefore (2.1) follows in view of (A).It is clear that (2.1) is equivalent to (2.1) for 1 ă s ă 2 due to the definitions of τ 2´s and c s in (A) and (A).
For the case 0 ă s ď 1, we first show that Fr|x| ´sΩpθqs is a locally integrable function.In fact, we fix a smooth radial cutoff function ψ supported on t1 ď |x| ď 4u such that ψpxq `ψpx{2q " 1 on t2 ď |x| ď 4u analogous to Littlewood-Paley decomposition.Then we decompose |x| ´sΩpθq as where the summation converges in the sense of tempered distributions.We first notice Fr|x| ´sΩpθqp1 ´ř8 k"0 ψp2 ´kxqqs P L 8 since its argument is compactly supported on Bp0; 2q and thus in L 1 .The summation can be written as 2 ´ks gp2 ´kxq, gpxq :" |x| ´sΩpθqψpxq Therefore, we have in the sense of tempered distributions.Notice that ĝ P L 1 since g is smooth and compactly supported, and we have }ĝp2 k ¨q} L 1 " 2 ´2k }ĝ} L 1 .Therefore the above summation also converges in L 1 .This shows k"0 |x| ´sΩpθqψp2 ´kxq ı P L 1 , and thus Fr|x| ´sΩpθqs P L 1 `L8 .In the case 0 ă s ă 1, to prove the formula (2.1) at ξ " e 0 , we first assume Ωp0q " Ωpπ{2q " 0.Then, interpreting as an improper integral in the radial direction, Ωpθq dθ where the last integral in θ is well defined because Ωpθq " Op|θ ´π 2 |q near π{2, and the last equality can be justified by cutting off a small interval rπ{2 ´ , π{2 ` s and integrating by parts.Noticing that ş 8 0 r ´s sin r dr " Γp1 ´sq cos πs 2 by (A), we get the (2.1) in the case Ωp0q " Ωpπ{2q " 0 using the definition (A) for τ s .
The general case of (2.1) follows by noticing that it is true for Ω " 1 and Ω " cos 2 θ with ξ " e 0 .Here, for the case Ω " cos 2 θ, one can first show that Ωpϕq " c s p1 ´p2 ´sq cos 2 ϕq{s by using (2.1) reversely, and then verify that the resulting Ωp0q coincides with (2.1) using (A) and (A).
In the case s " 1, the same calculation up to (2.1) works for any Ω with Ωp0q " 0. Then we get which is (2.1) (noticing that τ 1 " 0, c 1 " 1.The general case of (2.1) follows by noticing that it is true for Ω " 1.
In most parts of this paper, we focus on the range 0 ă s ă 1.We notice that (2.2) gives a decomposition of Ωpθq into a linear combination of functions of the form | cospθ ´ϕq| ´s for various values of ϕ.Notice that |x| ´s| cospθ ´ϕq| ´s " |x ¨ e ϕ | ´s.This gives a decomposition of |x| ´sΩpθq into a linear combination of 1D Riesz potentials along the directions e ϕ .
Corollary 2.3.Let 0 ă s ă 1 and Ω satisfy (H).Let Ω be defined by (2.1).Then (2.7) As a consequence, interaction potentials W of the form (1) with LIC property are exactly given by positive linear combinations of such rotated 1D potentials.

2.2.
Behavior of LIC/non-LIC potentials.It was proved in [12,Theorem 2.4] that the Euler-Lagrange condition for energy minimizers is sufficient for LIC potentials.
Lemma 2.4.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 it is the only probability measure satisfying pW ˚ρqpxq ď essinf pW ˚ρq, ρ a.e.
Using the conclusion of this theorem, the argument of [12,Theorem 7.1] applies, showing that any superlevet set of any d 8 -local minimizer does not have interior points.
Finally, we establish the following equivalence between the LIC property and the nonnegativity of Ŵrep .
Theorem 2.6.Let W is given by (1) with Ω satisfying (H).Then W has the LIC property if and only if Ω given as in (2.1) is nonnegative.
Notice that the LIC property is a notion of strict convexity, while the nonnegativity of Ω is a non-strict inequality.The equivalence of them comes from an argument involving analytic functions, as in the proof below.
We claim that for any compactly supported signed measure µ ‰ 0, μ cannot be identically zero in Bpξ 0 ; q.In fact, μpξq " ş R 2 e ´2πix¨ξ µpxq dx is an analytic function in ξ P C 2 , and having μ identically zero in Bpξ 0 ; q would imply that μpξq " 0 for any ξ P C 2 , contradicting µ ‰ 0.
Conversely, if Ω is negative somewhere, then Proposition 2.5 shows that W is infinitesimal concave, and in particular, W does not have LIC.
For Ωpθq " 1 `αωpθq as given in (1) with ω satisfying (h), we have Ωpϕq " c s p1 `αωpϕqq for some smooth function ω, given by (2.1) applied to ω up to a constant multiple by c s .If 0 ă s ă 1, then ω is necessarily sign-changing because otherwise (2.2) would imply that ω is strictly positive, contradicting the assumption ωp π 2 q " 0. Therefore, there exists a critical value depending on s and ω, around which W α changes from LIC to non-LIC.This naturally leads to a drastic change in the behavior of energy minimizers as stated in the introduction.

Minimizers of LIC potentials
In this section we study the unique global energy minimizer for potentials W given in (1) with the LIC property.Denote R θ " ˆcos θ ´sin θ sin θ cos θ ˙as the rotation matrix, and for a, b ą 0, η P R, where ρ 2 is defined in (1).We also denote ρ 0,b as the weak limit of ρ a,b as a Ñ 0 (similar for ρ 0,b,η and ρ a,0,η ).Notice that ρ a,b is the push-forward of ρ 2 by the linear transformation ˆa 0 0 b ˙, and ρ a,b,η is the push-forward of ρ a,b by R η .supp ρ a,b is an ellipse (possibly degenerate) with axes parallel to the coordinate axes and axis lengths aR 2 and bR 2 , where R 2 is as in (1).supp ρ a,b,η is the previously described ellipse rotated by the angle η counterclockwise.
If Ω ě c ą 0, then item 1 must happen.Remark 3.2.From the proof, it is clear that the same conclusion holds if |x| 2 is replaced by any positive definite quadratic potential Remark 3.3.If one further requires the symmetry condition Ωpθq " Ωp´θq (i.e., symmetry of the potential about the x 1 -axis), then either item 1 happens with η " 0, or item 2 happens with η " 0 or η " π{2.This follows from the fact that Dpa, bq in (3.5) below is always zero under this symmetry condition.
We first prove a lemma on the linear projection of ρ a,b,η onto 1D subspaces.
is a rescaling of ρ 1 (with total mass 1).Therefore, in the general case, (3.4) holds for some λ ą 0. The value of λ is determined by matching the support of the two sides.For ρ a,b,η , the conclusion follows from the fact that ρ a,b is the push-forward of ρ 2 by the composition of R η and ˆa 0 0 b ˙.
To prove Theorem 3.1, the key observation is that (2.3) implies that `|x| ´sΩpθq ˘˚ρ a,b is necessarily a quadratic function in supp ρ a,b .Lemma 3.5.Assume 0 ă s ă 1 and Ω satisfies (H).Assume a, b P p0, 8q.Then for some constant C Ω,a,b , with ¨Apa, bq Bpa, bq Dpa, bq If Ω ě 0 and a " 0, b ą 0, then the same is true provided that the integral in the expression of A is finite.If Ω ě 0 and a ą 0, b " 0, then the same is true provided that the integral in the expression of B is finite.
Remark 3.6.Here, in case a, b ą 0, the steady state condition (3.5) works for possibly non-LIC potentials, showing that ρ a,b is always a steady state for the interaction potential |x| ´sΩpθq `Ax 2 1 `Bx 2 2 2Dx 1 x 2 .It is guaranteed to be the unique (global) minimizer for LIC potentials, but it cannot even be a d 8 -local minimizer for non-LIC potentials, because it violates the necessary condition given by Proposition 2.5 and the sentence after it.Remark 3.7.Although R 1 is not well-defined for 1 ď s ă 2, we notice that is well-defined for 1 ď s ă 2. It turns out that (3.5) and (3.5) are also true for 1 ď s ă 2 (see Lemma 8.2), but requires a different proof.Proof.We first assume a, b P p0, 8q.Notice that for a fixed ϕ, the rotated 1D interaction potential |x ¨ e ϕ | ´s generates a 1D potential in the sense that the last quantity is independent of y 2 .The inner integral ş R ρ a,b pz 1 e ϕ `z2 e K ϕ q dz 2 is exactly the push-forward of ρ a,b by the projection onto the e ϕ direction (as a function of z 1 ).This implies it is a rescaling of ρ 1 , by Lemma 3.4.Since the maximum of See Figure 2 as an illustration.
The fact that ρ 1 minimizes the energy associated to the potential |x| ´s `|x| 2 implies that ż R `|y ´z| ´s `|y ´z| 2 ˘ρ1 pzq dz Rescaling by some λ ą 0, we get ż R `|y ´z| ´s `λ2`s |y ´z| 2 ˘λρ 1 pλzq dz Applying to (3), we see that In particular, `|x ¨ e ϕ | ´s `p R1 rϕ q 2`s |x ¨ e ϕ | 2 ˘˚ρ a,b achieves minimum on supp ρ a,b .
Combining with (2.3) and integrating in ϕ, we see that is constant on supp ρ a,b , and achieves minimum on supp ρ a,b if Ω ě 0. The last integral is which is a quadratic function in x.Therefore we get the conclusion (3.5) with the coefficients A, B, D given by (3.5).
Finally we treat the case Ω ě 0, a " 0, b ą 0 (the case a ą 0, b " 0 is similar).The previous argument clearly works if 0 R supp Ω since the integrals in (3.5) are locally integrable.To treat the general case, we take a sequence of nonnegative functions t Ωn u satisfying (H) such that 0 R supp Ωn , t Ωn pϕqu is increasing in n and converges to Ωpϕq for every ϕ (noticing that the assumption A ă 8 implies Ωp0q " 0, such a sequence can be obtained by multiplying Ω with some mollifiers).Then for any x P supp ρ 0,b , the monotone convergence theorem shows that `|x| ´sΩ n pθq ˚ρ0,b ˘pxq converges to `|x| ´sΩpθq ˚ρ0,b ˘pxq.Also, A n , B n converge to A, B since B ă 8 is clear and A ă 8 is assumed, and then D n converges to D since the integrand of D n is dominated by that of A n `Bn .Therefore `p|x| ´sΩ n pθq `An ρ0,b ˘pxq.This implies C Ωn,0,b converges, say, to C Ω,0,b .This justifies (3.5).To see that `|x| ´sΩpθq `Ax 2 1 `Bx 2 2 `2Dx 1 x 2 ˘˚ρ 0,b achieves its minimal value on supp ρ a,b , we notice that the previous arguments applied to Ω n shows that `p|x| ´sΩ n pθq`A n x 2 1 `Bn x 2 2 `2D n x 1 x 2 q˚ρ 0,b ˘pxq ě C Ωn,0,b for any x.Sending n Ñ 8, we get the conclusion by the monotone convergence theorem.
To see the uniqueness of such pa, bq, we notice that for a, b ą 0, the Jacobian BpA, Bq Bpa, bq " ´ˆa ẇhere the integrals are in ϕ P r´π, πs with respect to the weight ps `2qτ s pR 1 {R 2 q 2`s pa 2 cos 2 ϕ b2 sin 2 ϕq ´ps`4q{2 Ωpϕq dϕ.Since Ω ě c ą 0, the matrices ˆa cos 4 ϕ a cos 2 ϕ sin 2 ϕ b cos 2 ϕ sin 2 ϕ b sin 4 ϕ ȧre semi-positive-definite rank-1 matrices for every ϕ, having different eigenvectors for different ϕ P r0, π{2s, we see that BpA,Bq  Bpa,bq is negative-definite for any a, b ą 0. This implies the injective property of the map pa, bq Þ Ñ pA, Bq, which finishes the proof of the claim.
Notice that Ωp¨`ηq corresponds to Ωp¨`ηq on the Fourier side, and therefore the previous claim applies to Ωp¨`ηq.Then we see that for every η P r0, π{2s, we have a unique pair pa η , b η q such that A η pa η , b η q " B η pa η , b η q " 1.Since the map pa, b, ηq Þ Ñ pA η pa, bq, B η pa, bqq is clearly smooth on p0, 8q 2 ˆR, we see that a η , b η depend continuously on η, and so does gpηq :" D η pa η , b η q.
It is clear that a ă 8 because otherwise one would have as n Ñ 8 contradicting to A ηn pa n , b n q " 1. Similarly b ă 8.It is also clear that pa, bq ‰ p0, 0q because otherwise one would have A ηn pa n , b n q `Bηn pa n , b n q Ñ 8.
Then we separate into cases: ‚ If a, b ą 0, then it is clear that A ηn pa n , b n q Ñ A η pa, bq and similarly for B and D, and we get the conclusion with item i) in the statement.‚ If a " 0, b ą 0, then B ηn pa n , b n q Ñ B η p0, bq by the dominated convergence theorem, and we get B η p0, bq " 1.We will then prove item ii) in the statement of the theorem by proving for any px 1 , x 2 q.In fact, the lower bound follows from ´π pa 2 cos 2 ϕ `b2 sin 2 ϕq ´p2`sq{2 px 1 cos ϕ `x2 sin ϕq 2 Ωpϕ `ηq dϕ ě 0 for any px 1 , x 2 q.To see the upper bound, we use Fatou's Lemma to get ‚ If a ą 0, b " 0, then we get item ii) in the statement of the theorem with η P rπ{2, πq, similar to the previous case.
If we only assume Ω ě 0, then either we have the same situation as before, or we have item ii) as in Lemma 3.8.For the case of item ii), we have some pb, ηq P p0, 8q ˆr0, πq such that B " 1, D " 0 and 0 ď A ď 1 (where B refers to B η p0, bq, and similar for A, D below).Then Lemma 3.5 shows that p|x| ´sΩpθ `ηq`Ax `x2 2 q˚ρ 0,b also achieves its minimum on supp ρ 0,b , i.e., satisfies the condition (2.4).Therefore ρ 0,b is the unique global energy minimizer for Ωp¨`ηq, i.e., ρ 0,b,η is the unique global energy minimizer for Ω.

A special solution to the gradient flow
The Wasserstein-2 gradient flow associated to the energy (1) formally reads where ρpt, xq is a time-dependent particle density function defined on R `ˆR 2 .Lemma 3.5 shows that u is linear if ρ has the form ρ a,b,η at some t.As a consequence, the class of functions ρ a,b,η is invariant under the gradient flow (4).For simplicity, we will further assume Ωpθq " Ωp´θq, so that D " 0 always holds.In this case, we will focus on the invariance of the class of functions ρ a,b .The general case will be briefly discussed in Remark 4.3.
The global wellposedness of (4.1) for initial condition pa 0 , b 0 q P p0, 8q 2 follows from the uniform upper and lower bounds on paptq, bptqq, as will be shown later in the proof of Theorem 4.2.
Proof.Lemma 3.5 shows that If ρpt, ¨q " ρ aptq,bptq with aptq and bptq satisfying (4.1), then we compute each term in the continuity equation to obtain Summing these quantities and using (4.1), we see that (4) holds in supp ρpt, ¨q.
Epa, bq is a constant multiple of the original total energy (1), see Remark 4.4 for details.
Proof.From the integral expression, it is clear that Therefore, we deduce " ´s`a a 1 pA ´1q `bb 1 pB ´1q ˘" ´2s `a2 pA ´1q 2 `b2 pB ´1q 2 ˘, ( i.e., E is decreasing in time. Then, from the expression of E, it is clear that aptq, bptq ă C unif for some constant C unif uniformly in time.Also, notice that Apa, bq ě ca ´s´1 for a close to 0 if b is bounded from above.In fact, this can be seen from the expression of Apa, bq in (3.5), in which the integrand is well-approximated by Ωp0qpa 2 `b2 ϕ 2 q ´p2`sq{2 near ϕ " 0 for small a. Therefore a 1 ą 0 for sufficiently small a, i.e., aptq ą c unif uniformly in time.Similarly bptq ą c unif uniformly in time.
In the proof of Lemma 3.8, we have shown that the Jacobian BpA,Bq Bpa,bq is negative-definite.Since the eigenvalues of BpA,Bq  Bpa,bq are continuous functions of a, b, there exists λ ą 0 such that BpA, Bq Bpa, bq ď ´λI 2 , @pa, bq P rc unif , C unif s 2 .
Therefore, we conclude that which gives the exponential convergence in energy as t Ñ 8, as well as the convergence of paptq, bptqq to pa 8 , b 8 q.
Remark 4.  where A η , B η , D η are as in Lemma 3.8.We leave to the interested reader to check that ρ aptq,bptq,ηptq solves (4) by a direct computation as in Theorem 4.1.From Lemma 3.5, it is clear that ρ a,b,η , a, b ą 0 cannot be a steady state for the gradient flow (4) unless A η pa, bq " B η pa, bq " 1, D η pa, bq " 0. With the further assumption Ω ě c ą 0, this would imply that a steady state of the form ρ a,b,η has to be the unique energy minimizer.As a consequence, the unique steady state of (4.3) is given by the parameters pa, b, ηq for the energy minimizer.The analysis of wellposedness and long time behavior of this ODE system is left as future work.
Remark 4.4.We show that E is a constant multiple of the original total energy (1).By Lemma 3.5, we have p|x| ´sΩpθq `Ax 2 1 `Bx 2 2 q ˚ρa,b " V 1 pR 1 {R 2 q ´2Mpa, bq, x P supp ρ a,b .Therefore, we deduce that Using Lemma 3.4 and the explicit formula of ρ 1 in (1), we see that and similarly ż Therefore, combined with (4), we get s .Therefore we see that Erρ a,b s " pR 1 {R 2 q ´2V 1 1 2`s E. This is another way to check that E is decreasing in time along the flow of (4).

Behavior of minimizers for large α
In this section we discuss the behavior of minimizers for the potential W α given by (1), for α large.We will focus on the case when ωpθq achieves its minimal value at the only point ωp π 2 q " 0, and thus intuitively minimizers tend to concentrate along the x 2 -axis.5.1.ρ 1D is the minimizer of strongly coercive potentials.Theorem 5.1.For 0 ă s ă 1, there exists a constant C ˚" C ˚psq such that the following holds.Let W be given by (1) with Ω satisfying (H), Ωp π 2 q " 1 and Ωpθq ě 1 `C˚ˇˇθ ´π 2 ˇˇ2 , @θ P r0, πs.
Remark 5.2.The non-degeneracy condition (5.1) is satisfied if ωpθq ą 0 for any θ P r0, πszt π 2 u and ω 2 p π 2 q ą 0. We point out that finding an exact formula for α ˚or approximating it numerically seems to be very hard.
Proof.Consider Ω ˚satisfying (H) with 0 R supp Ω˚, Ω ˚p π 2 q " 1 and Ω ˚pθq " Ω ˚p´θq.We apply Lemma 3.5 with a " 0, b " R 1 {R 2 , which is allowed since A and B take finite values.Since ρ 1 is the minimizer of the energy with the 1D potential |x| ´s `|x| 2 , we see that W ˚˚ρ 1D is constant on supp ρ 1D .This implies B " 1 in Lemma 3.5.The constraint Ω ˚p π 2 q " 1 is equivalent to

`x2
2 q ˚ρ1D achieves minimum on supp ρ 1D , and the same is clearly also true if Ax 2  1 is replaced by x 2 1 .Finally, since Ω˚ě 0, we apply Lemma 2.4 and Theorem 2.6 to see that ρ 1D is the unique minimizer of the associated interaction energy.This implies Ω ˚ě Ω ˚p π 2 q " 1 because otherwise a rotated version of ρ 1D would have smaller energy.
Proof of Theorem 5.1.Let Ω ˚given by Lemma 5.3, and the associated potential and energy W ˚and E ˚.The assumptions on Ω ˚implies the existence of C ˚ą 0 such that 1 `C˚| θ ´π 2 | 2 ě Ω ˚pθq for any θ P r0, πs, and equality only holds for θ " π 2 .Also, as seen in the previous proof, ρ 1D is the unique minimizer of E ˚.
Since Ωp π 2 q " 1 and Ωpθq ě 1 `C˚| θ ´π 2 | 2 for θ P r0, πs, for any compactly supported probability measure ρ we have Erρs ě E ˚rρs ě E ˚rρ 1D s " Erρ 1D s (5.2) by Ω ě Ω ˚, the minimizing property of ρ 1D for E ˚, and the fact that Erρ 1D s only involves the values of W with θ " π 2 , the latter being the same as those in W ˚. This shows that ρ 1D is a minimizer of E. Furthermore, if ρ is not supported on a vertical line, then the first inequality in (5.1) is strict since Ωpθq ą Ω ˚pθq whenever θ ‰ π{2.If ρ is supported on a vertical line, then the uniqueness of energy minimizer for the 1D potential |x| ´s `|x| 2 shows that the second inequality in (5.1) is strict unless ρ " ρ 1D .Therefore we conclude that ρ 1D is the unique minimizer of E (up to translation).
For the statement on W α , we notice that the assumptions for ω implies that 1 `αωpθq satisfies the assumptions on Ω for the previous part for α " C ˚{c ω , and it follows that ρ 1D is the unique minimizer of E α .The same comparison argument also shows that if α 1 ă α 2 and ρ 1D is a minimizer of E α1 , then it is the unique minimizer of E α2 .
By Lemma 2.1, ω is smooth, and thus Ωα " c s `αω ě c ą 0 if α is sufficiently small.In this case Theorem 3.1 shows that ρ 1D is not a minimizer of E α .
Therefore, we define that is a positive number, with the property that ρ 1D is the unique minimizer of E α for any α ą α ˚.For any α ă α ˚, ρ 1D cannot be a minimizer of E α , because otherwise we would get that ρ 1D is the unique minimizer of E pα`α˚q{2 , contradicting the definition of α ˚.Therefore the desired properties of α ˚are proved.
Remark 5.4.It is clear that ρ 1D is a minimizer of E α˚.In fact, for any α ą α ˚, we have E α rρs ě E α rρ 1D s for any probability measure ρ.Sending α Ñ α ˚gives the conclusion.However, it is not clear whether ρ 1D is the unique minimizer of E α˚.
Proof.Let M be a positive integer to be chosen, and consider ą 0 small.Define ρ pxq " ρ 1 px 2 qδpx 1 ´ sinpM x 2 qq as a perturbation of ρ 1D .See Figure 3 as an illustration.We claim that there holds the asymptotic expansion E iso rρ s " E iso rρ 1D s `cM 2 `OM p 4 q (5.3) where E iso denotes the energy with Ω " 1 (the isotropic part), and (5.4) The perturbed distribution ρ .
To prove (5.2), we compute the difference of the isotropic energies where the constant in the last Op 4 q is independent of x, y.Therefore we obtain (5.2) with the stated coefficient c M .It is clear that the second integral in (5.2) is no more than 4, and the first integral goes to infinity as M Ñ 8 (by observing that | sinpM xq´sinpM yq x´y | ě cM whenever |x ´y| ă c{M and M x ´π{2 is not close to an integer multiple of π).Therefore, for sufficiently large M (depending only on s), we have c M ă 0.
Next we analyze the energy from the anisotropic part.Denote Ω 1 " Ω ´1, and E ani as the energy with Ω replaced by Ω ´1.Then E ani rρ 1D s " 0, and ¨Ω1 ´tan

Therefore, we conclude
Erρ s ´Erρ 1D s ď c M 2 `C κ `Op 4 q with c M ă 0, which implies that Erρ s ă Erρ 1D s for all sufficiently small because κ ą 2. Since d 8 pρ , ρ 1D q ď , we see that ρ 1D is not a d 8 -local minimizer of E.

5.3.
Estimate of the width of support.As a complementary result to Theorem 5.5, we will show that any global minimizer has to have narrow support in x 1 for large α, even if ω behaves like |θ ´π 2 | κ with a large κ.
Theorem 5.7.Assume 0 ă s ă 1, W α given by (1) with ω satisfying (h) and for some κ ě 2.Then, for any minimizer ρ of E α with zero center of mass, with C depending on s and ω.
Then ρ p2q is a probability measure.We aim to show that E α rρ p2q s ă E α rρs for any with the above smallness conditions, which would lead to a contradiction.First notice that the mean-zero condition for ρ and Lemma B.1 imply that where R ą 0 is such that supp ρ Ă Bp0; Rq given by Lemma B.1 and depending only on s.In fact, if this was not true, then combining with α ą 1, one would have where we used that x 1 ě ´R on the first integral, contradicting the mean-zero condition for ρ.
Then, since W α ą 0, W α px ´yqρpyq dyρpxq dx since in the integrand we always have x 1 ´y1 ą C 1 α ´1{pκ`1q {2 and |x ´y| ď C (by Lemma B.1, C independent of α) and thus, the angles of x ´y corresponding to the integration set are bounded away from π{2.As a consequence, W α px ´yq ě cαpC 1 α ´1{pκ`1q q κ by the lower bound assumption (5.7) on ω.

Intermediate α: possible complex behavior
In this section we analyze the 2D potential generated by 1D vertical distributions.This enables us to determine whether the vertical one dimensional distribution ρ 1D satisfies the first Euler-Lagrange condition.Proposition 6.1.Let 0 ă s ă 1 and W be given by (1) with Ω satisfying (H).Let ρpxq " ψpx 2 qδpx 1 q (6.1) where ψ is a nonnegative continuous function on R with compact support and ş R ψpxq dx " 1. Assume ψ is C 1 near a neighborhood of 0. Then pW ˚ρqp , 0q ´pW ˚ρqp0, 0q " 1 2τ 2´s Ωp0qψp0q 1´s `Op q (6.2) where τ 2´s is a negative number, given by (A).When Ω achieves a local minimum at π{2, then the error term can be improved to Op 2´s q.
In particular, if Ωp0q ą 0 and ψp0q ą 0, then ρ is not a d 8 -local minimizer of E.
Proof.We first express the difference of the potentials as pW ˚ρqp , 0q ´pW ˚ρqp0, 0q where the remainder term R is given by R " ż R ´p1 `y2 q ´s{2 Ωptan ´1 yq ´|y| ´sΩ ´π 2 ¯¯pψp yq ´ψp0qq dy .
To calculate the integral in the main term, we use a change of variable y " tan θ to get ż R ´p1 `y2 q ´s{2 Ωptan ´1 yq ´|y| ´sΩ ´π 2 ¯¯dy Let us interpret the result of the Proposition 6.1.It is clear that this result can be used with translation or rotation.Also, if ρ is the sum of (6.1) and a measure whose support does not contain 0, then (6.1) is still true for small enough since the measure away from 0 can only contribute Op 2 q to pW ˚ρqp , 0q ´pW ˚ρqp0, 0q.In particular, if Ωpϕq ą 0, then the support of any d 8 -local minimizer cannot contain any isolate segment along the e K ϕ direction with a C 1 profile on it.This is particularly interesting if Ωp0q ą 0 but Ωpϕq " Ωp´ϕq ă 0 for some other angle ϕ.In this case, Proposition 6.1 implies that ρ 1D cannot be a d 8 -local minimizer.One expects to see the support of an energy minimizer to present a zigzag behavior, consisting of segments with angles ϕ `π 2 and ´ϕ `π 2 with Ωpϕq " Ωp´ϕq ă 0. For the parametrized potential W α in (1), if c s ωpϕq, the angle function for the Fourier transform of |x| ´sωpϕq, achieves minimum at some ϕ ‰ 0, then such zigzag happens for α L ă α ă α L,0 , where α L is defined in (2.2), and α L,0 (depending on s and ω) is defined as In fact, if ω satisfies the assumption of Theorem 5.5, it may happen that the minimum of ω is only achieved at ωp π 2 q " 0. In this case, the formula (2.1) guarantees ωp0q ă 0, which implies Ωα p0q ă 0 for sufficiently large α.In this case, Proposition 6.1 applies to ρ 1D for Ω α (at least at the points p0, x 2 q, ´R1 ă x 2 ă R 1 ), but ρ 1D is actually not a d 8 -local minimizer by Theorem 5.5.

The s " 0 (logarithmic potential) case
In this section we consider the logarithmic anisotropic potential W log pxq " ´ln |x| `Ωpθq `|x| 2 (7.1) with Ω satisfying (H).The following lemma gives a viewpoint of W log as a limit of potentials like (1) as s Ñ 0 `.
Lemma 7.1.Let Ω satisfy (H) and W log be given by (7) .Define Then W psq pxq Ñ W log pxq uniformly on any compact subset of R 2 not containing 0. Furthermore, where Ωlog is given by a uniform-in-ϕ limit Ωlog pϕq :" lim sÑ0 As s Ñ 0 `, we have τ 2´s Ñ τ 2 " ´p2πq ´2, c s " p2πq ´1s from the explicit formulas (A) and (A) as Γpsq " 1 s as s Ñ 0 `.To take the limit of the integral in (7), one can cutoff the domain at rϕ `π 2 ´ , ϕ `π 2 ` s and use a Taylor expansion of Ω inside this interval.The linear term from the Taylor expansion makes no contribution because | cospϕ ´θq| ´2`s is symmetric around ϕ `π 2 .In this way, we get the convergence lim sÑ0 `Ω psq pϕq " Ωlog pϕq given as in (7.1) as a principal value integral, uniform in ϕ.It follows that lim sÑ0 `Ŵ psq pξq " |ξ| ´2 Ωlog pϕq, uniformly on any compact subset of R 2 not containing 0. Combined with the distributional convergence Ŵpsq Ñ Ŵlog , we obtain ( for some constant Cpsq depending on s.As s Ñ 0 `, we have the limit sτ s Ñ 1 and the uniform-in-ϕ limit Ωpsq Ñ Ωlog .We also have the limit , by an estimate similar to (7).Then it is straightforward to pass to the limit in the above integral and obtain (7.1).
Remark 7.3.Notice that the previous result only gives us the pointwise values of the Fourier transform of the potential Ωpθq away from the origin.This information is enough for our purposes.
For the interaction energy E log associated to W log , one can also use the approximation argument in [12,Lemma 2.5] to justify the analogue of (2): for any compactly supported signed measure µ with ş R 2 µpxq dx " ş R 2 xµpxq dx " 0 and E log r|µ|s ă 8. Then one can show that Theorem 2.6 also holds for W log .
Corollary 7.4.Let W log is given by (7) with Ω satisfying (H).Then W log has the LIC property if and only if Ωlog given as in (7.1) is nonnegative.
We now first extend Lemma 3.5 to the logarithmic case by taking the limit s Ñ 0 `.For fixed a, b P p0, 8q, we notice that supp ρ a{R2,b{R2 is an ellipse with axes a and b: Bp0; a, bq :" independent of s.We will pass to the limit as s Ñ 0 `for the potential generated by ρ a{R2,b{R2 , which converges to a constant multiple of χ Bp0;a,bq .
If Ωlog ě 0 and a " 0, b ą 0, then the same is true provided that the integral in the expression of A is finite.If Ωlog ě 0 and a ą 0, b " 0, then the same is true provided that the integral in the expression of B is finite.
If we further assume Ωlog ě c ą 0, then the uniform-in-ϕ limit in (7.1) shows that Ωpsq ě 0 for sufficiently small s ą 0.Then, the same application of Lemma 3.5 gives that the LHS of ( 7) achieves minimum on Bp0; a, bq.Then taking the limit s Ñ 0 `gives the conclusion that p´ln |x| `Ωpθq `Ax 2 1 Bx 2 2 `2Dx 1 x 2 q ˚1 |Bp0;a,bq| χ Bp0;a,bq achieves its minimal value on Bp0; a, bq.If we only assume Ωlog ě 0, then the same is true because one can apply the previous result to tΩ, 0 ă t ă 1 (whose strict positivity of Fourier transform is guaranteed by (7.1)) and pass to the limit t Ñ 1 ´.
For the case Ωlog ě 0 and a " 0, b ą 0, the conclusion can be obtained in the same way as before if 0 R supp Ωlog .In fact, in this case A psq , B psq , D psq are finite, and thus Lemma 3.5 still gives (7).For every fixed x 1 , the potential ´|x| ´s´1 s `Ωpθq ¯´1 s `Apsq x 2 1 `Bpsq x 2 2 `2D psq x 1 x 2 ¯, as a function of x 2 , also admit the uniform-on-compact-set convergence away from the origin and a uniform-in-s L 2 estimate near the origin.Therefore, by viewing ρ 0,b{R2 pxq " Cδpx 1 qpb 2 ´x2 2 q p1`sq{2 as a function of x 2 , one can pass to the limit s Ñ 0 `and obtain the same conclusion.
For general Ωlog ě 0 and a " 0, b ą 0, one can proceed similar to the proof of Lemma 3.5.In fact, we approximate Ωlog by an increasing sequence of nonnegative smooth functions Ωlog,n with 0 R supp Ωlog,n .If we denote m n :" ş Ωlog,n dϕ, then tm n u is an increasing sequence inside p0, 1q with lim nÑ8 m n " ş Ωlog dϕ " 1. Due to Remark 7.2, we cannot apply (7.1) to Ωlog,n , but we can instead apply it to 1 mn Ωlog,n and construct the corresponding In other words, the potential ´mn ln |x| `Ωn pθq has Fourier transform Ωlog,n pξq away from ξ " 0. Since each potential ´mn ln |x| `Ωn pθq verifies 0 R supp Ωlog,n , then it satisfies the desired conclusion that p´m n ln |x| `Ωn pθq `An x 2 1 `Bn x 2 2 `2D n x 1 x 2 q ˚1 |Bp0; 0, bq| χ Bp0;0,bq achieves its minimal value on Bp0; 0, bq, where A n , B n , D n are obtained from (7.5) using Ωlog,n .The same holds if one adds a constant to the potential.Since (7.1) has a nonnegative convolution kernel ´ln | cospϕ ´θq|, we see that Ω n pθq is increasing for each fixed θ and converges to Ωpθq.Also, tm n u is increasing and converges to 1. Therefore, on any ball Bp0; Rq, the potential ´mn ln |x| `Ωn pθq `mn ln R is pointwise increasing in n.This allows us to pass to the n Ñ 8 limit by the monotone convergence theorem in the relation (7.5) for ´mn ln |x|`Ω n pθq`m n ln R and obtain the conclusion for ´ln |x|`Ωpθq, combining with pA n , B n , D n q Ñ pA, B, Dq by the dominated convergence theorem since A, B ă 8 by assumption.
The case Ωlog ě 0 and a ą 0, b " 0 can be treated similarly.
Denote Bp0; a, b, ηq as the counterclockwise rotation of Bp0; a, bq around the origin by the angle η.In other words, χ Bp0;a,b,ηq pxq " χ Bp0;a,bq pR ´η xq.We notice that an analogue of Lemma 3.8 for the formula (7.5) can be proved in a similar way.Therefore, using a similar proof as Theorem 3.1, we obtain the following result.
As mentioned in the introduction, the previous result generalizes [21].Notice that no smallness assumption is needed on Ω and the condition Ωlog is sharp due to Theorem 2.6.Finally we give an analogue of Theorem 5.1 for logarithmic potentials.Theorem 7.8.There exists a constant C ˚such that the following holds.Let W log be given by (7) with Ω satisfying (H) and Ωpθq ě Ωp π 2 q `C˚ˇˇθ ´π 2 ˇˇ2 , @θ P r0, πs.
Then there exists a unique 0 ă α ˚ď C ˚{c ω (depending on ω), such that for any α ą α ˚, ρ 1D is the unique minimizer of E log,α (up to translation), and for any α ă α ˚, ρ 1D is not a minimizer of E log,α .
Since the proof is similar to Theorem 5.1, we only give a sketch of the proof.Using (7.1), one can first construct Ω ˚, such that W log,˚: " ´ln |x| `Ω˚p θq `|x| 2 satisfies Ŵ˚p ξq ě 0 for any ξ ‰ 0 and W ˚˚ρ 1D achieves minimum on supp ρ 1D , similar to Lemma 5.3.This can be done by taking a smooth Ωlog,˚, sufficiently concentrated near ϕ " π{2, with the properties 0 R supp Ωlog,˚, ş π ´π Ωlog,˚d ϕ " 1.Then the same comparison argument as in the proof of Theorem 5.1 gives the proof of Theorem 7.8.
Then we proceed to study the unique energy minimizer.For simplicity, we will add the extra symmetry condition Ωpθq " Ωp´θq.We will show that the minimizer is given by some ρ a,b defined in (3) where C1 denotes the constant C 1 defined in (A) with s replaced by s ´1 P p0, 1q in the case 1 ă s ă 2.
It can be shown that (8.2) is equivalent to (3.5) (with D " 0 due to the symmetry Ωpθq " Ωp´θq) for any 1 ď s ă 2, with the understanding in Remark 3.7.The details of this equivalence lead to a cumbersome exercise with change of variables and special functions left to the interested reader.In fact, one can start from a change of variable tan θ 1 " b a tan θ in (8.2) to convert it into an integral against Ωpθ 1 q.Then use Ω " τ 2´s | cosp¨q| ´2`s ˚Ω to write (3.5) as a double integral and change the order of integrals.Then we have an outer integral with weight Ωpθq, and the result can be obtained by calculating the inner integral in ϕ explicitly.
Since Ω ě c ą 0 holds by (2.1) (for 1 ă s ă 2) and (2.1) (for s " 1), STEP 1 of the proof of Lemma 3.8 works for 1 ď s ă 2. In fact, most of the proof works in the same way, except for the justification of the limit of f pbq as b Ñ 0 `.In the case 1 ď s ă 2, both numerator and denominator of (3) diverge to infinity, however the numerator is much smaller than the denominator, leading to the same result.We obtain the following consequence.Theorem 8.3.Let W be given by (1) with 1 ď s ă 2 and Ω satisfying (H) and Ωpθq " Ωp´θq.Then the unique energy minimizer is given by ρ a,b for some a, b ą 0.
Proof of Lemma 8.2.We first treat the case a " b " 1 and 1 ă s ă 2. We write the repulsive part of the potential as W rep pxq " |x| ´sΩpθq.We compute its contribution for the potential generated by the density ρ Notice also that 1 and a calculation using special functions shows that V1´C1,R " 1 s´1 .Therefore, we get (for any where the u 2 1 and u 2 2 terms have positive coefficients.Since the LHS is continuous in pu 1 , u 2 q and increasing in |u 2 | for |u 2 | ě R 2 , we see that (for any x) Then integrating in Ωpθq dθ we get the conclusion, since u 2 1 " px 1 cos θ `x2 sin θq 2 , u 2 2 " p´x 1 sin θ x2 cos θq 2 are quadratic functions in x 1 , x 2 , and the x 1 x 2 terms are cancelled during integration due to the symmetry property Ωpθq " Ωp´θq.
For the case a " b " 1 and s "

Numerical examples
In this section we give some numerical examples for the energy minimizers of E α , the interaction energy associated to the potential W α given by (1).For this purpose, we consider the associated particle gradient flow whose formal mean-field limit is the Wasserstein-2 gradient flow (4).The particle-level total energy Epx 1 , . . ., x N q " 1 2N 2 ÿ j‰k W px j ´xk q corresponding to the energy functional (1), is decreasing along the solution of (9), and it is expected that a long time simulation of (9) will minimize the energy, at least locally.
We take a few examples of W α as the interaction potential and solve (9) numerically.We take the number of particles N " 1600.We start from a random initial data (i.i.d.uniform distribution on r´1{2, 1{2s 2 ) and solve (9) by the forward Euler method 3 .The time steps are chosen adaptively, which guarantees the stability of the time integrator.The numerical simulation is terminated when the total energy stabilizes (up to a tolerance level 10 ´5 between adjacent time steps) or the total number of time steps is greater than n max " 20000.It is expected that the final state of the particles is approximately an energy minimizer, at least locally.
In the rest of this section, we present the examples and the numerical results, as well as the explanation for them based on our theory from previous sections.We always take s " 0.4 in the simulations unless mentioned otherwise.In the figures, the blue ellipses are the predicted shape of the unique minimizer in the LIC cases, based on Theorem 3.1 and Lemma 3.5.The blue dashed lines indicate the height of ρ 1D as in (1).Different regimes for the parameter α are marked by different colors on the α-axis.9.1.A classical example ωpθq " cos 2 θ.This example is the direct generalization of the anisotropic logarithmic potential in [10] to 0 ă s ă 1.The Fourier transform of W α , in its angle variable, is given by Ωα pϕq " c s ´1 `α 1 ´p2 ´sq cos 2 ϕ s ¯.
Therefore, W α is LIC if and only if α ď α L " s 1 ´s where α L is as in (2.2).At α " α L , we have Ωα L pϕq " C sin 2 ϕ.In this case, the function f pbq defined in (3), taking b Ñ 8, can be computed by f p8q " 3 For a more accurate simulation of (9), higher order time integrators are preferred.However, this is not crucial in the current paper because our main interest is the energy minimizers instead of the gradient flow dynamics.Therefore, item 1 of Theorem 3.1 holds, i.e., the support of the unique minimizer is a proper ellipse.This implies that ρ 1D is not a minimizer for W α L , and thus not a minimizer for W α if α is slightly larger than α L .Therefore the minimizers behave as the following (see Figure 4): ‚ For 0 ď α ď α L (marked blue on the α-axis), the minimizer is some ρ a,b supported on a proper ellipse.‚ For α L ă α ă α ˚(with α L ă α ˚, marked green on the α-axis), any minimizer is not an ellipse or ρ 1D , and any of its superlevel set has no interior point due to Proposition 2.5.‚ For α ě α ˚(marked red on the α-axis), ρ 1D is the unique minimizer due to Theorem 5.1.Remark 9.1.It is worth noticing that the anisotropic logarithmic potential considered in [10] (i.e., the s " 0 limiting case of the current example) does not admit the 'green' phase as in Figure 4.In other words, as α increases, the minimizer starts as ellipses and transits into ρ 1D at α L directly.It satisfies the conditions of Lemma 5.3 for Ω ˚(under a suitable generalization to logarithmic potentials).
In the above example, we see that the natural generalization of the anisotropic logarithmic potential in [10] to 0 ă s ă 1 admits the 'green' phase, and thus its corresponding 1 `αL ω does not satisfy the requirements for Ω ˚in Lemma 5.3.Therefore, even if one is only interested in the case ωpθq " cos 2 θ with 0 ă s ă 1, it does not seem possible to prove Theorem 5.1 for it without considering a more general class of potentials.A bigger pool of potentials enables one to find the correct Ω ˚to compare with, as in the proof of Theorem 5.1.9.2.An example with degeneracy near π{2: ωpθq " cos 4 θ.This example is designed to have a high order degeneracy near θ " π{2.In fact, the condition (5.1) is not satisfied by this ω, and Theorems 5.5 and 5.7 apply with κ " 4.
The Fourier transform of W α , in its angle variable, is given by Ωα pϕq " c s ´1 `α 3 ´6p2 ´sq cos 2 ϕ `p4 ´sqp2 ´sq cos 4 ϕ sps `2q The last fraction ωpϕq has minimum ´6p1´sq sps`2qp4´sq , achieved at cos 2 ϕ " 3 4´s P p0, 1q.Therefore W α is LIC if and only if α ď α L " sps `2qp4 ´sq 6p1 ´sq . .Therefore the minimizers behave as the following (see Figure 5): ‚ For 0 ď α ď α L (marked blue on the α-axis), the minimizer is some ρ a,b supported on a proper ellipse.‚ For α L ă α ă α L,0 (marked green dashed line on the α-axis), any minimizer is not an ellipse or ρ 1D , and any of its superlevel set has no interior point.No 'vertical segments' are allowed.Also, for α close to α L , the slopes of the tilted segments agree well with the predicted angle cos ´1 b 3 4´s .‚ For α ě α L,0 (marked green on the α-axis), similar as the previous case, but we cannot exclude the possibility of 'vertical segments'.Minimizers get closer to being vertical as α increases, but never becomes ρ 1D due to Theorem 5.5.
Remark 9.2.Although our theory guarantees that the zigzag behavior cannot appear for α ď α L in the continuum model, one can indeed observe such phenomenon in the particle simulation when α is slightly smaller than α L (for example, the α " 0.95 case in Figure 5).From this example, one can see that the particle model may behave differently from the continuum model when the LIC condition barely fails.It would be interesting to apply numerical methods for the continuum model and compare with the numerical results for the particle model.
In this case, we have α L ă α L,0 , and there exists α ˚in the sense of Theorem 5.1.The behavior of minimizers can be analyzed similar to the previous two examples using Theorems 3.1 and 5.1, and Propositions 2.5 and 6.1.Therefore the minimizers behave as the following (see Figure 1): ‚ For 0 ď α ď α L (marked blue on the α-axis), the minimizer is some ρ a,b supported on a proper ellipse.‚ For α L ă α ă α L,0 (marked green dashed line on the α-axis), any minimizer is not an ellipse or ρ 1D , and any of its superlevel set has no interior point.No 'vertical segments' are allowed.Also, for α close to α L , the slopes of the tilted segments agree well with the predicted angle cos ´1 b 9.984 2¨5.76 .‚ For α L,0 ď α ă α ˚(marked green on the α-axis), similar as the previous case, but we cannot exclude the possibility of 'vertical segments'.Minimizers is not ρ 1D and exhibit zigzag behavior.‚ For α ě α ˚(marked red on the α-axis), ρ 1D is the unique minimizer.9.4.An example with two preferred directions: ωpθq " cos 2 θ sin 2 θ.In this example, ωpθq achieves minimal value at two different angles θ " 0 and θ " π{2, i.e., both vertical and horizontal directions are preferred when α is large.
The last fraction ωpϕq has minimum ´1´s sps`2q , achieved at cos 2 ϕ " 0, 1. Therefore W α is LIC if and only if α ď α L " sps `2q 1 ´s .For α ď α L , Theorem 3.1 shows that the unique energy minimizer is some ρ a,b .Since ω is symmetric with respect to exchanging x 1 and x 2 , the same is true for the unique energy minimizer.This implies a " b (i.e., the support of the minimizer is a ball), and their value can be determined by setting A " B " 1 in (3.5).
Notice that Theorem 5.1 does not apply because π{2 is not the unique minimum point of ω.In fact, since ω is symmetric with respect to exchanging x 1 and x 2 , suppose ρ 1D " ρ 1 px 2 qδpx 1 q is a minimizer of E α , then ρ 1 px 1 qδpx 2 q is also a minimizer.For large α, we do not know whether ρ 1D is a minimizer.Theorem 5.7 does not apply either because ωp0q " 0 makes (5.7) false.Therefore the minimizers behave as the following (see Figure 6): ‚ For 0 ď α ď α L (marked blue on the α-axis), the minimizer is some ρ a,a supported in a ball.‚ For α ą α L (marked red on the α-axis), any superlevel set of any minimizer has no interior point, but the precise shapes of minimizers are unknown.Numerical results show the formation of some fractal structure, in which particles tend to align either vertically or horizontally at different spatial scales.However, it is likely that such configuration is merely a local minimizer because ρ 1D clearly has smaller energy than it, at least for large α.
The behavior of the minimizers for large α remains open in this case.Also, similar comments regarding the difference between particle and continuum models as in Remark 9.2 apply to this example.9.5.An example with 1 ă s ă 2. In this example we take s " 1.4 and ωpθq " cos 2 θ.The results in Section 8 show that W α is LIC for any α ě 0, and the unique energy minimizer is always some ρ a,b .The numerical results shown in Figure 7 verifies this phenomenon.
It is worth noticing that the gap between the cluster of particles and the predicted ellipse shape is larger than the previous examples with s " 0.4.This is a consequence of the fact that ρ 2 pxq " C 2 pR 2 2 ´|x| 2 q s{2 has smaller values near the boundary of the support if s is larger.   .Numerical results for the energy minimizers, for W α with s " 1.4 and ωpθq " cos 2 θ.The α values are 0, 0.6, 1.5, 3. 9.6.Illustration of the asymptotic limit s Ñ 0 `.Finally we give an example illustrating the asymptotic limit s Ñ 0 `towards the logarithmic potential in Lemma 7.1.We fix the choice Ωpθq " 1`0.5 cos 2 θ and take W psq and W log as in (7.1) and ( 7) respectively.It is easily verified that W psq , 0 ă s ă 1 and its asymptotic limit W log all have the LIC property, and the unique energy minimizer is some ρ a,b (for W psq ) or 1 |Bp0;a,bq| χ Bp0;a,b,ηq (for W log ).See Figure 8 for the numerical results.Also, as s Ñ 0 `, we observe that the minimizer for W psq converges to that for W log , which can be viewed as a consequence of the convergence of ( 7) to (7.5).

Figure 1 .
Figure1.An illustrative example using a particle gradient flow simulation with 1600 particles.The interaction potential is W α as in (1) with s " 0.4 and ωpθq " cos 4 θ 0.1 cos 2 θ with the α values 0, 0.2, 0.5, 0.7, 0.8, 0.85, 0.9, 0.95, 1, 1.2, 1.5, 2, 4. The blue ellipses are the predicted shape of the unique minimizer in the LIC cases.The blue dashed lines are the height of ρ 1D .The pink line is the predicted slope of the fragmented segments.See Section 9.3 for more details.

Figure 2 .
Figure 2. Calculation of the potential generated by ρ a,b via 1D projections.

Figure 8 .
Figure 8. Numerical results for the energy minimizers, for W psq with Ωpθq " 1 0.5 cos 2 θ and various s.The s values are 0 (logarithmic), 0.05, 0.2, 0.6.On the right, the supports of the minimizers for different s are compared.
1 s and Erµs, we start by taking m ą 0 to be chosen, and noticing that | φpxq| ď Cp1 `|x|q ´m,where C may depend on m, since φ is smooth and compactly supported.Therefore, we deduce|µ 1 pxq| ď CR 2 p1 `R|x|q ´m.Therefore, using the estimate |W pxq| ď Cp|x| ´s `|x| 2 q, we obtain Next notice that |ψp yq ´ψp0q| ď C mint |y|, 1u since ψ is C 1 at 0 and bounded on R. Therefore Op s q and the conclusion in the general case follows.When π{2 is a local minimum point of Ω, we have the improved estimate |Ωptan ´1 yq ´Ωp π 2 q| ď C mint|y| ´2, 1u since Ω 1 pπ{2q " 0. Therefore we get |R| ď C and the conclusion follows.
´s¯ |y| dy `C ż 1ď|y|ď1{ ´p1 `y2 q ´s{2 |y| ´1 `|y| ´s|y| ´2u ¯ |y| dy `C ż |y|ą1{ ´p1 `y2 q ´s{2 |y| ´1 `|y| ´s|y| ´2¯d y ďC `C s `C s " Here Ωpsq pϕq denotes the angle function of the Fourier transform of W psq given by (2.1).Notice that the operator Ω Þ Ñ Ωlog as in (7.1) commutes with translation, and maps constant functions to p2πq ´1.Therefore Ωlog always satisfies ş π ´π Ωlog pϕq dϕ " 1, the last quantity representing the amount of logarithmic potential ´ln |x| contained in W log .The formula (7.1) allows one to construct Ω from a given smooth Ωlog with ş π ´π Ωlog pϕq dϕ " 1 modulo a constant.Notice that the constants is irrelevant for minimizing the interaction energy.Proof.To see the convergence W psq pxq Ñ W log pxq for x ‰ 0, first notice that lim sÑ0 `|x| ´sΩpθq " Ωpθq.Both convergences are uniform on any compact subset of R 2 not containing 0.For any 0 ă s ă 1 and 0 ă ă 1, we first estimate Combined with the previously shown convergence, we see that W psq Ñ W log in the space of tempered distributions, which implies Ŵpsq Ñ Ŵlog in the same sense.By (2.1) and (2.1), we have `Ω psq pϕq " ´p2πq ´2p.v.