Ground States in the Diffusion-Dominated Regime
José A. Carrillo, Franca Hoffmann, Edoardo Mainini, Bruno Volzone
GGROUND STATES IN THE DIFFUSION-DOMINATED REGIME
JOSÉ A. CARRILLO , FRANCA HOFFMANN , , EDOARDO MAININI , AND BRUNO VOLZONE Abstract.
We consider macroscopic descriptions of particles where repulsion is modelledby non-linear power-law diffusion and attraction by a homogeneous singular kernel leadingto variants of the Keller-Segel model of chemotaxis. We analyse the regime in which diffu-sive forces are stronger than attraction between particles, known as the diffusion-dominatedregime, and show that all stationary states of the system are radially symmetric decreasingand compactly supported. The model can be formulated as a gradient flow of a free energyfunctional for which the overall convexity properties are not known. We show that globalminimisers of the free energy always exist. Further, they are radially symmetric, compactlysupported, uniformly bounded and C ∞ inside their support. Global minimisers enjoy cer-tain regularity properties if the diffusion is not too slow, and in this case, provide stationarystates of the system. In one dimension, stationary states are characterised as optimisers of afunctional inequality which establishes equivalence between global minimisers and stationarystates, and allows to deduce uniqueness. Introduction
We are interested in the diffusion-aggregation equation(1.1) ∂ t ρ = ∆ ρ m + χ ∇ · ( ρ ∇ S k ) for a density ρ ( t, x ) of unit mass defined on R + × R N , and where we define the mean-fieldpotential S k ( x ) := W k ( x ) ∗ ρ ( x ) for some interaction kernel W k . The parameter χ > denotesthe interaction strength. Since (1.1) conserves mass, is positivity preserving and invariant bytranslations, we work with solutions ρ in the set Y := (cid:26) ρ ∈ L ( R N ) ∩ L m ( R N ) , || ρ || = 1 , (cid:90) R N xρ ( x ) dx = 0 (cid:27) . The interaction W k is given by the Riesz kernel W k ( x ) = | x | k k , k ∈ ( − N, . Let us write k = 2 s − N with s ∈ (cid:0) , N (cid:1) . Then the convolution term S k is governed by afractional diffusion process, c N,s ( − ∆) s S k = ρ , c N,s = (2 s − N ) Γ (cid:0) N − s (cid:1) π N/ s Γ( s ) = k Γ ( − k/ π N/ k + N Γ (cid:0) k + N (cid:1) . Department of Mathematics, Imperial College London, South Kensington Campus, LondonSW7 2AZ, UK. Email: [email protected] . DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road,Cambridge CB3 0WA, UK. Email: [email protected] . Dipartimento di Ingegneria Meccanica, Università degli Studi di Genova, Piazzale Kennedy,Pad. D, 16129, Genova, Italia. Email: [email protected] . Dipartimento di Ingegneria Università degli Studi di Napoli “Parthenope”, Napoli, 80143,Italia. Email: [email protected] . a r X i v : . [ m a t h . A P ] M a y GROUND STATES IN THE DIFFUSION-DOMINATED REGIME
For k > − N the gradient ∇ S k := ∇ ( W k ∗ ρ ) is well defined locally. For k ∈ ( − N, − N ] however, it becomes a singular integral, and we thus define it via a Cauchy principal value,(1.2) ∇ S k ( x ) := ∇ ( W k ∗ ρ ) ( x ) , if − N < k < , (cid:90) R N ∇ W k ( x − y ) ( ρ ( y ) − ρ ( x )) dy , if − N < k ≤ − N .
Here, we are interested in the porous medium case m > with N ≥ . The correspondingenergy functional writes(1.3) F [ ρ ] = H m [ ρ ] + χ W k [ ρ ] with H m [ ρ ] = 1 m − (cid:90) R N ρ m ( x ) dx , W k [ ρ ] = 12 (cid:90) (cid:90) R N × R N | x − y | k k ρ ( x ) ρ ( y ) dxdy . Given ρ ∈ Y , we see that H m and W k are homogeneous by taking dilations ρ λ ( x ) := λ N ρ ( λx ) .More precisely, we obtain F [ ρ λ ] = λ N ( m − H m [ ρ ] + λ − k χ W k [ ρ ] . In other words, the diffusion and aggregation forces are in balance if N ( m −
1) = − k . This isthe case for choosing the critical diffusion exponent m c := 1 − k/N called the fair-competitionregime . In the diffusion-dominated regime we choose m > m c , which means that the diffusionpart of the functional (1.3) dominates as λ → ∞ . In other words, concentrations are notenergetically favourable for any value of χ > and m > m c . The range < m < m c is referred to as the attraction-dominated regime . In this work, we focus on the diffusion-dominated regime m > m c .Further, we define below the diffusion exponent m ∗ that will play an important role for theregularity properties of global minimisers of F :(1.4) m ∗ := (cid:40) − k − N − k − N , if N ≥ and − N < k < − N , + ∞ if N ≥ and − N ≤ k < . The main results in this work are summarised in the following two theorems:
Theorem 1.
Let N ≥ , χ > and k ∈ ( − N, . All stationary states of equation (1.1) are radially symmetric decreasing. If m > m c , then there exists a global minimiser ρ of F on Y . Further, all global minimisers ρ ∈ Y are radially symmetric non-increasing, compactlysupported, uniformly bounded and C ∞ inside their support. Moreover, all global minimisers of F are stationary states of (1.1) , according to Definition 3, whenever m c < m < m ∗ . Finally,if m c < m ≤ , we have ρ ∈ W , ∞ (cid:0) R N (cid:1) . Theorem 2.
Let N = 1 , χ > , k ∈ ( − , and m > m c . All stationary states of (1.1) areglobal minimisers of the energy functional F on Y . Further, stationary states of (1.1) in Y are unique. ROUND STATES IN THE DIFFUSION-DOMINATED REGIME 3 d i ff u s i o n - d o m i n a t e d r e g i m e porousmediumregimefastdiffusionregime W k singular W k non-singular-N 1-N 2-N 0 N ∇ W k / ∈ L loc ( R N ) kmW − N = Newtonian potentialfair-competition regime m c = 1 − k/N m ∗ = − k − N − k − N a ttr a c t i o n - d o m i n a t e d r e g i m e Figure 1.
Overview of the parameter space ( k, m ) for N ≥ : fair-competitionregime ( m = m c , red line), diffusion-dominated regime ( m > m c , yellow re-gion) and attraction-dominated regime ( m < m c , blue region). For m = m c ,attractive and repulsive forces are in balance (i.e. in fair competition ). For m c < m < m ∗ in the diffusion-dominated regime, global minimisers of F arestationary states of (1.1), see Theorem 1, a result which we are not able toshow for m ≥ m ∗ (striped region).Aggregation-diffusion equations of the form (1.1) are ubiquitous as macroscopic modelsof cell motility due to cell adhesion and/or chemotaxis phenomena while taking into accountvolume filling constraints [29, 44, 10]. The non-linear diffusion models the very strong localisedrepulsion between cells while the attractive non-local term models either cell movement towardchemosubstance sources or attractive interaction between cells due to cell adhension by longfilipodia. They encounter applications in cancer invasion models, organogenesis and patternformation [28, 24, 45, 41, 18].The archetypical example of the Keller-Segel model in two dimensions corresponding to thelogarithmic case ( m = 1 , k = 0) has been deeply studied by many authors [31, 32, 43, 30, 42,23, 6, 46, 5, 2, 3, 15, 19], although there are still plenty of open problems. In this case, thereis an interesting dichotomy based on a critical parameter χ c > : the density exists globallyin time if < χ < χ c (diffusion overcomes self-attraction) and expands self-similarly [14, 27],whereas blow-up occurs in finite time when χ > χ c (self-attraction overwhelms diffusion),while for χ = χ c infinitely many stationary solutions exist with intricated basins of attraction[3]. The three-dimensional configuration with Newtonian interaction ( m = 1 , k = 2 − N ) appears in gravitational physics [20, 21], although it does not have this dichotomy, belongingto the attraction-dominated regime. However, the dichotomy does happen for the particularexponent m = 4 / of the non-linear diffusion for the 3D Newtonian potential as discovered GROUND STATES IN THE DIFFUSION-DOMINATED REGIME in [4]. This was subsequently generalised for the fair-competition regime where m = m c for agiven k ∈ ( − N, in [12, 13].In fact, as mentioned before two other different regimes appear: the diffusion-dominatedcase when m > m c and the attraction-dominated case when m < m c . In Figure 1, we makea sketch of the different regimes including cases related to non-singular kernels for the sake ofcompleteness. Note that non-singular kernels k > allow for values of m < correspondingto fast-diffusion behaviour in the diffusion-dominated regime m > m c . We refer to [12, 13]and the references therein for a full discussion of the state of the art in these regimes.In the diffusion-dominated case, it was already proven in [16] that global minimisers existin the particular case of m > m c for the logarithmic interaction kernel k = 0 . Theiruniqueness up to translation and mass normalisation is a consequence of the important sym-metrisation result in [17] asserting that all stationary states to (1.1) for − N ≤ k < areradially symmetric. We will generalise this result to our present framework for the range − N < k < − N not included in [17] due to the special treatment needed for the arising sin-gular integral terms. This is the main goal of Section 2 where we remind the reader the precisedefinition and basic properties of stationary states for (1.1). In short, we show that stationarysolutions are continuous compactly supported radially non-increasing functions with respectto their centre of mass. Some of these results are in fact generalisations of previous results in[12, 17] and we skip some of the details.Let us finally comment that the symmetrisation result reduces the uniqueness of stationarystates to uniqueness of radial stationary states that eventually leads to a full equivalencebetween stationary states and global minimisers of the free energy (1.3). This was used in [17]to solve completely the 2D case with m > m c for the logarithmic interaction kernel k = 0 ,and it was the new ingredient to fully characterise the long-time asymptotics of (1.1) in thatparticular case.In view of the main results already announced above, we show in Section 3 the existence ofglobal minimisers for the full range m > m c and k ∈ ( − N, which are steady states of theequation (1.1) as soon as m < m ∗ . This additional constraint on the range of non-linearitiesappears only in the most singular range − N < k < − N and allows us to get the right Hölderregularity on the minimisers in order to make sense of the singular integral in the gradient ofthe attractive non-local potential force (1.2).Besides existence of minimisers, Section 3 contains some of the main novelties of this paper.First, in order to prove boundedness of minimisers, we develop a fine estimate on the interactionterm based on the asymptotics of the Riesz potential of radial functions, and show that thisestimate is well suited exactly for the diffusion dominated regime (see Lemma 15 and Theorem16). Moreover, thanks to the Schauder estimates for the fractional Laplacian, we improve theregularity results for minimisers in [12] and show that they are smooth inside their support,see Theorem 21. This result applies both to the diffusion dominated and fair competitionregime.These global minimisers are candidates to play an important role in the long-time asymp-totics of (1.1). We show their uniqueness in one dimension by optimal transportation tech-niques in Section 4. The challenging open problems remaining are uniqueness of radiallynon-increasing stationary solutions to (1.1) in its full generality and the long-time asymptoticsof (1.1) in the whole diffusion-dominated regime, even for non-singular kernels within the fastdiffusion case.Plan of the paper: In Section 2 we define and analyse stationary states, showing that theyare radially symmetric and compactly supported. Section 3 is devoted to global minimisers.We show that global minimisers exist, are bounded and we provide their regularity properties.Eventually, Section 4 proves uniqueness of stationary states in the one-dimensional case. ROUND STATES IN THE DIFFUSION-DOMINATED REGIME 5 Stationary states
Let us define precisely the notion of stationary states to the diffusion-aggregation equation(1.1).
Definition 3.
Given ¯ ρ ∈ L (cid:0) R N (cid:1) ∩ L ∞ (cid:0) R N (cid:1) with || ¯ ρ || = 1 and letting ¯ S k = W k ∗ ¯ ρ ,we say that ¯ ρ is a stationary state for the evolution equation (1.1) if ¯ ρ m ∈ W , loc (cid:0) R N (cid:1) , ∇ ¯ S k ∈ L loc (cid:0) R N (cid:1) , and it satisfies (2.1) ∇ ¯ ρ m = − χ ¯ ρ ∇ ¯ S k in the sense of distributions in R N . If − N < k ≤ − N , we further require ¯ ρ ∈ C ,α (cid:0) R N (cid:1) forsome α ∈ (1 − k − N, . In fact, as shown in [12] via a near-far field decomposition argument of the drift term, thefunction S k and its gradient defined in (1.2) satisfy even more than the regularity ∇ S k ∈ L loc (cid:0) R N (cid:1) required in Definition 3: Lemma 4.
Let ρ ∈ L (cid:0) R N (cid:1) ∩ L ∞ (cid:0) R N (cid:1) with || ρ || = 1 and k ∈ ( − N, . Then the followingregularity properties hold:(i) S k ∈ L ∞ (cid:0) R N (cid:1) .(ii) ∇ S k ∈ L ∞ (cid:0) R N (cid:1) , assuming additionally ρ ∈ C ,α (cid:0) R N (cid:1) with α ∈ (1 − k − N, in therange k ∈ ( − N, − N ] . Lemma 4 implies further regularity properties for stationary states of (1.1). For preciseproofs, see [12].
Proposition 5.
Let k ∈ ( − N, and m > m c . If ¯ ρ is a stationary state of equation (1.1) and ¯ S k = W k ∗ ¯ ρ , then ¯ ρ is continuous on R N , ¯ ρ m − ∈ W , ∞ (cid:0) R N (cid:1) , and (2.2) ¯ ρ ( x ) m − = m − m (cid:0) C [ ¯ ρ ]( x ) − χ ¯ S k ( x ) (cid:1) + , ∀ x ∈ R N , where C [ ¯ ρ ]( x ) is constant on each connected component of supp ( ¯ ρ ) . It follows from Proposition 5 that ¯ ρ ∈ W , ∞ (cid:0) R N (cid:1) in the case m c < m ≤ .2.1. Radial Symmetry of Stationary States.
The aim of this section is to prove thatstationary states of (1.1) are radially symmetric. This is one of the main results of [17], and isachieved there under the assumption that the interaction kernel is not more singular than theNewtonian potential close to the origin. As we will briefly describe in the proof of the nextresult, the main arguments continue to hold even for the more singular Riesz kernels W k . Theorem 6 (Radiality of stationary states) . Let χ > and m > m c . If ¯ ρ ∈ L ( R N ) ∩ L ∞ ( R N ) with (cid:107) ¯ ρ (cid:107) = 1 is a stationary state of (1.1) in the sense of Definition 3, then ¯ ρ is radiallysymmetric non-increasing up to a translation.Proof. The proof is based on a contradiction argument, being an adaptation of that in [17,Theorem 2.2], to which we address the reader the more technical details. Assume that ¯ ρ is not radially decreasing up to any translation. By Proposition 5, we have(2.3) (cid:12)(cid:12) ∇ ¯ ρ m − ( x ) (cid:12)(cid:12) ≤ c for some positive constant c in supp ( ¯ ρ ) . Let us now introduce the continuous Steiner symmetri-sation S τ ¯ ρ in direction e = (1 , , · · · , of ¯ ρ as follows. For any x ∈ R , x (cid:48) ∈ R N − , h > ,let S τ ¯ ρ ( x , x (cid:48) ) := (cid:90) ∞ M τ ( U hx (cid:48) ) ( x ) dh , GROUND STATES IN THE DIFFUSION-DOMINATED REGIME where U hx (cid:48) = { x ∈ R : ¯ ρ ( x , x (cid:48) ) > h } and M τ ( U hx (cid:48) ) is the continuous Steiner symmetrisation of the U hx (cid:48) (see [17] for the precisedefinitions and all the related properties). As in [17], our aim is to show that there existsa continuous family of functions µ ( τ, x ) such that µ (0 , · ) = ¯ ρ and some positive constants C > , c > and a small δ > such that the following estimates hold for all τ ∈ [0 , δ ] :(2.4) F [ µ ( τ, · )] − F [ ¯ ρ ] ≤ − c τ (2.5) | µ ( τ, x ) − ¯ ρ ( x ) | ≤ C ¯ ρ ( x ) τ for all x ∈ R N (2.6) (cid:90) Ω i ( µ ( τ, x ) − ¯ ρ ( x )) dx = 0 for any connected component Ω i of supp ( ¯ ρ ) . Following the arguments of the proof in [17, Proposition 2.7], if we want to construct a contin-uous family µ ( τ, · ) for (2.5) to hold, it is convenient to modify suitably the continuous Steinersymmetrisation S τ ¯ ρ in order to have a better control of the speed in which the level sets U hx (cid:48) are moving. More precisely, we define µ ( τ, · ) = ˜ S τ ¯ ρ as ˜ S τ ¯ ρ ( x , x (cid:48) ) := (cid:90) ∞ M v ( h ) τ ( U hx (cid:48) ) ( x ) dh with v ( h ) defined as v ( h ) := (cid:40) h > h , < h ≤ h , for some sufficiently small constant h > to be determined. Note that this choice of thevelocity is different to the one in [17, Proposition 2.7] since we are actually keeping the levelsets of ˜ S τ ¯ ρ ( · , x (cid:48) ) frozen below the layer at height h . Next, we note that inequality (2.3) andthe Lipschitz regularity of ¯ S k (Lemma 4) are the only basic ingredients used in the proof of[17, Proposition 2.7] to show that the family µ ( τ, · ) satisfies (2.5) and (2.6). Therefore, itremains to prove (2.4). Since different level sets of ˜ S τ ¯ ρ ( · , x (cid:48) ) are moving at different speeds v ( h ) , we do not have M v ( h ) τ ( U h x (cid:48) ) ⊂ M v ( h ) τ ( U h x (cid:48) ) for all h > h , but it is still possible toprove that (see [17, Proposition 2.7]) H m [ ˜ S τ ¯ ρ ] ≤ H m [ ¯ ρ ] for all τ ≥ . Then, in order to establish (2.4), it is enough to show(2.7) W k [ ˜ S τ ¯ ρ ] ≤ W k [ ¯ ρ ] − χc τ for all τ ∈ [0 , δ ] , for some c > and δ > .As in the proof of [17, Proposition 2.7], proving (2.7) reduces to show that for sufficientlysmall h > one has(2.8) (cid:12)(cid:12)(cid:12) W k [ ˜ S τ ¯ ρ ] − W k [ S τ ¯ ρ ] (cid:12)(cid:12)(cid:12) ≤ cχτ for all τ . To this aim, we write S τ ¯ ρ ( x , x (cid:48) ) = (cid:90) ∞ h M τ ( U hx (cid:48) ) ( x ) dh + (cid:90) h M τ ( U hx (cid:48) ) ( x ) dh =: f ( τ, x ) + f ( τ, x ) and we split ˜ S τ ¯ ρ similarly, taking into account that v ( h ) = 1 for all h > h : ˜ S τ ¯ ρ ( x , x (cid:48) ) = f ( τ, x ) + (cid:90) h M v ( h ) τ ( U hx (cid:48) ) ( x ) dh =: f ( τ, x ) + ˜ f ( τ, x ) . Note that f = S τ ( T h ¯ ρ ) , ROUND STATES IN THE DIFFUSION-DOMINATED REGIME 7 where T h ¯ ρ is the truncation at height h of ¯ ρ . Since v ( h ) = 0 for h ≤ h , we have ˜ f = T h ¯ ρ. If we are in the singular range k ∈ ( − N, − N ] , we have ¯ ρ ∈ C ,α (cid:0) R N (cid:1) for some α ∈ (1 − k − N, . Since the continuous Steiner symmetrisation decreases the modulus of continuity(see [8, Theorem 3.3] and [8, Corollary 3.1]), we also have S τ ¯ ρ, f , ˜ f ∈ C ,α (cid:0) R N (cid:1) . Further,Lemma 4 and the arguments of [17, Proposition 2.7] guarantee that the expressions A ( τ ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) f ( W k ∗ f ) − ˜ f ( W k ∗ f ) dx (cid:12)(cid:12)(cid:12)(cid:12) and A ( τ ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) f ( W k ∗ f ) − ˜ f ( W k ∗ ˜ f ) dx (cid:12)(cid:12)(cid:12)(cid:12) can be controlled by || ¯ ρ || ∞ and the α -Hölder seminorm of ¯ ρ . Hence, we can apply the argumentin [17, Proposition 2.7] to conclude for the estimate (2.8). Now it is possible to proceed exactlyas in the proof of [17, Theorem 2.2] to show that for some positive constant C , we have thequadratic estimate |F [ µ ( τ, · )] − F [ ¯ ρ ] | ≤ C τ , which is a contradiction with (2.4) for small τ . (cid:3) Stationary States are Compactly Supported.
In this section, we will prove that allstationary states of equation (1.1) have compact support, which agrees with the propertiesshown in [33, 16, 17]. We begin by stating a useful asymptotic estimate on the Riesz potentialinspired by [49, §4]. For the proof of Proposition 7, see Appendix A.
Proposition 7 (Riesz potential estimates) . Let k ∈ ( − N, and let ρ ∈ Y be radially sym-metric.(i) If − N < k < , then | x | k ∗ ρ ( x ) ≤ C | x | k on R N .(ii) If − N < k ≤ − N and if ρ is supported on a ball B R for some R < ∞ , then | x | k ∗ ρ ( x ) ≤ C T k ( | x | , R ) | x | k , ∀ | x | > R , where (2.9) T k ( | x | , R ) := (cid:16) | x | + R | x |− R (cid:17) − k − N if k ∈ ( − N, − N ) , (cid:16) (cid:16) | x | + R | x |− R (cid:17)(cid:17) if k = 1 − N Here, C > and C > are explicit constants depending only on k and N . From the above estimate, we can derive the expected asymptotic behaviour at infinity.
Corollary 8.
Let ρ ∈ Y be radially non-increasing. Then W k ∗ ρ vanishes at infinity, withdecay not faster than that of | x | k .Proof. Notice that Proposition 7(i) entails the decay of the Riesz potential at infinity for − N < k < . Instead, let − N < k ≤ − N . Let r ∈ (1 − k − N, and notice that | y | k ≤ | y | k + r if | y | ≥ , so that if B is the unit ball centered at the origin we have | x | k ∗ ρ ( x ) ≤ (cid:90) B ρ ( x − y ) | y | k dy + (cid:90) B C ρ ( x − y ) | y | k + r dy ≤ (cid:32) sup y ∈ B ρ ( x − y ) (cid:33) (cid:90) B | y | k dy + ( W k + r ∗ ρ )( x ) . The first term in the right hand side vanishes as | x | → ∞ , since y (cid:55)→ | y | k is integrable at theorigin, and since ρ is radially non-increasing and vanishing at infinity as well. The second termgoes to zero at infinity thanks to Proposition 7(i), since the choice of r yields k + r > − N . GROUND STATES IN THE DIFFUSION-DOMINATED REGIME
On the other hand, the decay at infinity of the Riesz potential can not be faster than thatof | x | k . To see this, notice that there holds | x | k ∗ ρ ( x ) ≥ (cid:90) B ρ ( y ) | x − y | k dy ≥ ( | x | + 1) k (cid:90) B ρ ( y ) dy with (cid:82) B ρ > since ρ ∈ Y is radially non-increasing. (cid:3) As a rather simple consequence of Corollary 8, we obtain:
Corollary 9.
Let ¯ ρ be a stationary state of (1.1) . Then ¯ ρ is compactly supported.Proof. By Theorem 6 we have that ¯ ρ is radially non-increasing up to a translation. Since thetranslation of a stationary state is itself a stationary state, we may assume that ¯ ρ is radiallysymmetric with respect to the origin. Suppose by contradiction that ¯ ρ is supported on thewhole of R N , so that equation (2.2) holds on the whole R N , with C k [ ¯ ρ ]( x ) replaced by a uniqueconstant C . Then we necessarily have C = 0 . Indeed, ¯ ρ m − vanishes at infinity since it isradially decreasing and integrable, and by Corollary 8 we have that ¯ S k = W k ∗ ¯ ρ vanishes atinfinity as well. Therefore ¯ ρ = (cid:18) χ ( m − m ¯ S k (cid:19) / ( m − . But Corollary 8 shows that W k ∗ ρ decays at infinity not faster than | x | k and this wouldentail, since m > m c , a decay at infinity of ρ not faster than that of | x | − N , contradicting theintegrability of ρ . (cid:3) Global Minimisers
We start this section by recalling a key ingredient for the analysis of the regularity of the driftterm in (1.1), i.e. certain functional inequalities which are variants of the Hardy-Littlewood-Sobolev (HLS) inequality, also known as the weak Young’s inequality [36, Theorem 4.3]: forall f ∈ L p ( R N ) , g ∈ L q ( R N ) there exists an optimal constant C HLS = C HLS ( p, q, k ) > suchthat (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:90) R N × R N f ( x ) | x − y | k g ( y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) ≤ C HLS (cid:107) f (cid:107) p (cid:107) g (cid:107) q , (3.1) if p + 1 q = 2 + kN , p, q > , k ∈ ( − N, . The optimal constant C HLS is found in [35]. In the sequel, we will make use of the followingvariations of above HLS inequality:
Theorem 10.
Let k ∈ ( − N, , and m > m c . For f ∈ L ( R N ) ∩ L m ( R N ) , we have (3.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:90) R N × R N | x − y | k f ( x ) f ( y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ∗ || f || ( k + N ) /N || f || m c m c , where C ∗ = C ∗ ( k, m, N ) is the best constant.Proof. The inequality is a direct consequence of the standard sharp HLS inequality and ofHölder’s inequality. It follows that C ∗ is finite and bounded from above by the optimalconstant in the HLS inequality. (cid:3) ROUND STATES IN THE DIFFUSION-DOMINATED REGIME 9
Existence of Global Minimisers.Theorem 11 (Existence of Global Minimisers) . For all χ > and k ∈ ( − N, , there existsa global minimiser ρ of F in Y . Moreover, all global minimisers of F in Y are radially non-increasing. We follow the concentration compactness argument as applied in Appendix A.1 of [33].Our proof is based on [37, Theorem II.1, Corollary II.1]. Let us denote by M p ( R N ) theMarcinkiewicz space or weak L p space. Theorem 12. (see [37, Theorem II.1] ) Suppose W ∈ M p ( R N ) , < p < ∞ , and consider theproblem I M = inf ρ ∈Y q,M (cid:26) m − (cid:90) R N ρ m dx + χ (cid:90) (cid:90) R N × R N W ( x − y ) ρ ( x ) ρ ( y ) dxdy (cid:27) . where Y q,M = (cid:26) ρ ∈ L q ( R N ) ∩ L ( R N ) , ρ ≥ a.e., (cid:90) R N ρ ( x ) dx = M (cid:27) , q = p + 1 p < m . Then there exists a minimiser of problem ( I M ) if the following holds: (3.3) I M < I M + I M − M for all M ∈ (0 , M ) . Proposition 13. (see [37, Corollary II.1] ) Suppose there exists some λ ∈ (0 , N ) such that W ( tx ) ≥ t − λ W ( x ) for all t ≥ . Then (3.3) holds if and only if (3.4) I M < for all M > . Proof of Theorem 11.
First of all, notice that our choice of potential W k ( x ) = | x | k /k is indeedin M p ( R N ) with p = − N/k . Further, it can easily be verified that Proposition 13 applies with λ = − k . Hence we are left to show that there exists a choice of ρ ∈ Y q,M such that F [ ρ ] < .Let us fix R > and define ρ ∗ ( x ) := M Nσ N R N B R ( x ) , where B R denotes the ball centered at zero and of radius R > , and where σ N = 2 π ( N/ / Γ( N/ denotes the surface area of the N -dimensional unit ball. Then H m [ ρ ∗ ] = 1 m − (cid:90) R N ρ m ∗ dx = ( M N ) m σ − mN N ( m − R N (1 − m ) , W k [ ρ ∗ ] = 12 (cid:90) (cid:90) R N × R N W k ( x − y ) ρ ∗ ( x ) ρ ∗ ( y ) dxdy = ( M N ) kσ N R N (cid:90) (cid:90) R N × R N | x − y | k B R ( x ) B R ( y ) dxdy ≤ ( M N ) kσ N R N (2 R ) k σ N N R N = 2 k − M R k k < . We conclude that F [ ρ ∗ ] = H m [ ρ ∗ ] + χ W k [ ρ ∗ ] ≤ M m N m − σ − mN ( m − R N (1 − m ) + 2 k − M χ R k k . Since we are in the diffusion-dominated regime N (1 − m ) < k < , we can choose R > largeenough such that F [ ρ ∗ ] < , and hence condition (3.4) is satisfied. We conclude by Proposition13 and Theorem 12 that there exists a minimiser ¯ ρ of F in Y q,M with q = ( p +1) /p = ( N − k ) /N . It can easily be seen that in fact ¯ ρ ∈ L m ( R N ) using the HLS inequality (3.1): −W k [ ρ ] = 12 (cid:90) (cid:90) R N × R N | x − y | k ( − k ) ρ ( x ) ρ ( y ) dxdy ≤ C HLS ( − k ) || ρ || r , where r = 2 N/ (2 N + k ) = 2 p/ (2 p − . Using Hölder’s inequality, we find −W k [ ρ ] ≤ C HLS ( − k ) || ρ || qq || ρ || − q . Hence, since F [ ¯ ρ ] < , || ¯ ρ || mm ≤ − χ ( m − W k [ ¯ ρ ] ≤ χ ( m − (cid:18) M − q C HLS ( − k ) (cid:19) || ¯ ρ || qq < ∞ . Translating ¯ ρ so that its centre of mass is at zero and choosing M = 1 , we obtain a minimiser ¯ ρ of F in Y . Moreover, by Riesz’s rearrangement inequality [36, Theorem 3.7], we have W k [ ρ ] ≤ W k [ ρ ] , ∀ ρ ∈ Y , where ρ is the Schwarz decreasing rearrangement of ρ . Thus, if ¯ ρ is a global minimiser of F in Y , then so is ¯ ρ , and it follows that W k [ ¯ ρ ] = W k [ ¯ ρ ] . We conclude from [36, Theorem 3.7] that ¯ ρ = ¯ ρ , and so all global minimisers of F in Y areradially symmetric non-increasing. (cid:3) Global minimisers of F satisfy a corresponding Euler–Lagrange condition. The proof canbe directly adapted from [16, Theorem 3.1] or [12, Proposition 3.6], and we omit it here. Proposition 14.
Let k ∈ ( − N, and m > m c . If ρ is a global minimiser of the free energyfunctional F in Y , then ρ is radially symmetric and non-increasing, satisfying (3.5) ρ m − ( x ) = (cid:18) m − m (cid:19) ( D [ ρ ] − χW k ∗ ρ ( x )) + a.e. in R N . Here, we denote D [ ρ ] := 2 F [ ρ ] + (cid:18) m − m − (cid:19) || ρ || mm , ρ ∈ Y . Boundedness of Global Minimisers.
This section is devoted to showing that all globalminimisers of F in Y are uniformly bounded. In the following, for a radial function ρ ∈ L ( R N ) we denote by M ρ ( R ) := (cid:82) B R ρ dx the corresponding mass function, where B R is a ball of radius R , centered at the origin. We start with the following technical lemma: Lemma 15.
Let χ > , − N < k < , m > and ≤ q < m/N . Assume ρ ∈ Y is radiallydecreasing. For a fixed H > , the level set { ρ ≥ H } is a ball centered at the origin whoseradius we denote by A H . Then we have the following cross-range interaction estimate: thereexists H > , depending only on q, N, m, (cid:107) ρ (cid:107) m , such that, for any H > H , (cid:90) B CAH (cid:90) B AH | x − y | k ρ ( x ) ρ ( y ) dx dy ≤ C k,N M ρ ( A H ) K k,q,N ( H ) , where K k,q,N ( H ) := (cid:26) H − q ( k + N ) + H − kq if k ∈ ( − N, , k (cid:54) = 1 − N ,H − q (2 + log(1 + H q )) + H q ( N − if k = 1 − N and C k,N is a constant depending only on k and N . ROUND STATES IN THE DIFFUSION-DOMINATED REGIME 11
Proof.
Notice that the result is trivial if ρ is bounded. The interesting case here is ρ unbounded,implying that A H > for any H > .First of all, since ρ ∈ L m ( R N ) and ρ ≥ H on B A H , the estimate σ N A NH N H m = (cid:90) B AH H m ≤ (cid:90) B AH ρ m ≤ || ρ || mm implies that H q A H is vanishing as H → + ∞ as soon as q < m/N , and in particular that wecan find H > , depending only on q, m, N, || ρ || m , such that H − q ≥ A H for any H > H . We fix q ∈ [0 , m/N ) and H > H as above from here on.Let us make use of Proposition 7, which we apply to the compactly supported function ρ H := ρ { ρ ≥ H } /M ρ ( A H ) .Case − N < k < Proposition 7(i) applied to ρ H gives the estimate (cid:90) B AH | x − y | k ρ ( y ) dy ≤ C M ρ ( A H ) | x | k , ∀ x ∈ R N , and hence, integrating against ρ on B CA H and using ρ ≤ H on B CA H , (cid:90) B CAH (cid:90) B AH | x − y | k ρ ( x ) ρ ( y ) dx dy ≤ C M ρ ( A H ) (cid:90) B CAH | x | k ρ ( x ) dx = C M ρ ( A H ) (cid:32)(cid:90) B CAH ∩ B H − q | x | k ρ ( x ) dx + (cid:90) B CAH \ B H − q | x | k ρ ( x ) dx (cid:33) ≤ C M ρ ( A H ) (cid:32) H (cid:90) B CAH ∩ B H − q | x | k dx + H − kq (cid:90) B CAH \ B H − q ρ ( x ) dx (cid:33) ≤ C M ρ ( A H ) (cid:32) Hσ N (cid:90) H − q A H r k + N − dr + H − kq (cid:33) ≤ C M ρ ( A H ) (cid:18) σ N k + N H − q ( k + N ) + H − kq (cid:19) , which conludes the proof in that case.Case − N < k ≤ − N In this case, we obtain from Proposition 7(ii) applied to ρ H theestimate (cid:90) B AH | x − y | k ρ ( y ) dy ≤ C M ρ ( A H ) T k ( | x | , A H ) | x | k , ∀ x ∈ B CA H , and integrating against ρ ( x ) over B CA H , we have(3.6) (cid:90) B CAH (cid:90) B AH | x − y | k ρ ( x ) ρ ( y ) dx dy ≤ C M ρ ( A H ) (cid:90) B CAH T k ( | x | , A H ) | x | k ρ ( x ) dx . We split the integral in the right hand side as I + I , where I := (cid:90) B CAH ∩ B H − q T k ( | x | , A H ) | x | k ρ ( x ) dx, I := (cid:90) B CAH \ B H − q T k ( | x | , A H ) | x | k ρ ( x ) dx . Let us first consider I , where we have | x | ≥ H − q ≥ A H on the integration domain. Sincethe map | x | (cid:55)→ | x | + A H | x |− A H is monotonically decreasing to in ( A H , + ∞ ) , it is bounded above by on (2 A H , + ∞ ) . We conclude from (2.9) that T k ( | x | , A H ) ≤ for | x | ∈ ( H − q , + ∞ ) . Thisentails(3.7) I ≤ (cid:90) B CAH \ B H − q | x | k ρ ( x ) dx ≤ H − kq , where we used once again | x | ≥ H − q , recalling that k < .Concerning I , we have ρ ≤ H on B CA H which entails(3.8) I ≤ H (cid:90) B CAH ∩ B H − q T k ( | x | , A H ) | x | k dx = σ N H (cid:90) H − q A H T k ( r, A H ) r k + N − dr. If − N < k < − N , we use (2.9) and ( r + 2 A H ) / ( r + A H ) < for r ∈ (0 , + ∞ ) , so that(3.9) (cid:90) H − q A H T k ( r, A H ) r k + N − dr ≤ (cid:90) H − q (cid:18) r + 2 A H r + A H (cid:19) − k − N r k + N − dr ≤ − k − N k + N H − q ( k + N ) . If k = 1 − N we have from (2.9), since A H ≤ H − q < ,(3.10) (cid:90) H − q A H T k ( r, A H ) r k + N − dr = (cid:90) H − q A H (cid:18) (cid:18) r + A H r − A H (cid:19)(cid:19) dr ≤ (cid:90) H − q (cid:18) (cid:18) r + 1 r (cid:19)(cid:19) dr = H − q + H − q log(1 + H q ) + log(1 + H − q ) ≤ H − q (2 + log(1 + H q )) . Combining (3.8), (3.9), (3.10) we conclude I ≤ σ N − k + N k + N H − q ( k + N ) if − N < k < − N , and I ≤ σ N H − q (2 + log(1 + H q )) if k = 1 − N . These information together with the estimate(3.7) can be inserted into (3.6) to conclude. (cid:3) We are now in a position to prove that any minimiser of F is bounded. Theorem 16.
Let χ > , k ∈ ( − N, and m > m c . Then any global minimiser of F over Y is uniformly bounded and compactly supported.Proof. Since ρ is radially symmetric decreasing by Proposition 14, it is enough to show ρ (0) < ∞ . Let us reason by contradiction and assume that ρ is unbounded at the origin. We willshow that F [ ρ ] − F [ ˜ ρ ] > for a suitably chosen competitor ˜ ρ , ˜ ρ ( x ) = ˜ ρ H,r ( x ) := N M ρ ( A H ) σ N r N D r ( x ) + ρ ( x ) B CAH ( x ) , where B A H and q are defined as in Lemma 15, B CA H denotes the complement of B A H and D r is the characteristic function of a ball D r := B r ( x ) of radius r > , centered at some x (cid:54) = 0 and such that D r ∩ B A H = ∅ . Note that A H ≤ H − q / < H − q / < / . Hence, we can take r > and D r centered at the point x = (2 r, , . . . , ∈ R N . Notice in particular that since ρ is unbounded, for any H > we have that B A H has non-empty interior. On the other hand, B A H shrinks to the origin as H → ∞ since ρ is integrable. ROUND STATES IN THE DIFFUSION-DOMINATED REGIME 13 As D r ⊂ B CA H and ρ = ˜ ρ on B CA H \ D r , we obtain ( m −
1) ( H m [ ρ ] − H m [ ˜ ρ ]) = (cid:90) B AH ρ m dx + (cid:90) B CAH ρ m dx − (cid:90) B CAH (cid:18) ρ + N M ρ ( A H ) σ N r N D r (cid:19) m dx = (cid:90) B AH ρ m dx + (cid:90) D r (cid:20) ρ m − (cid:18) ρ + N M ρ ( A H ) σ N r N (cid:19) m (cid:21) dx . We bound ε r : = (cid:90) D r (cid:20) ρ m − (cid:18) ρ + N M ρ ( A H ) σ N r N (cid:19) m (cid:21) dx ≤ M ρ ( A H ) m (cid:16) σ N N (cid:17) − m r N (1 − m ) , where we use the convexity identity ( a + b ) m ≥ | a m − b m | for a, b > . Hence, ε r goes to as r → ∞ . Summarising we have for any r > ,(3.11) ( m −
1) ( H m [ ρ ] − H m [ ˜ ρ ]) = (cid:90) B AH ρ m dx + ε r , with ε r vanishing as r → ∞ .To estimate the interaction term, we split the double integral into three parts:(3.12) k ( W k [ ρ ] − W k [ ˜ ρ ]) = (cid:90) (cid:90) R N × R N | x − y | k ( ρ ( x ) ρ ( y ) − ˜ ρ ( x ) ˜ ρ ( y )) dxdy = (cid:90) (cid:90) B AH × B AH | x − y | k ρ ( x ) ρ ( y ) dxdy + 2 (cid:90) (cid:90) B AH × B CAH | x − y | k ρ ( x ) ρ ( y ) dxdy + (cid:90) (cid:90) B CAH × B CAH | x − y | k ( ρ ( x ) ρ ( y ) − ˜ ρ ( x ) ˜ ρ ( y )) dxdy =: I + I + I ( r ) . Let us start with I . By noticing once again that ρ = ˜ ρ on B CA H \ D r for any r > , we have I ( r ) = (cid:90) (cid:90) D r × D r | x − y | k ( ρ ( x ) ρ ( y ) − ˜ ρ ( x ) ˜ ρ ( y )) dxdy + 2 (cid:90) (cid:90) D r × ( B CAH \ D r ) | x − y | k ( ρ ( x ) ρ ( y ) − ˜ ρ ( x ) ˜ ρ ( y )) dxdy =: I ( r ) + I ( r ) . Since ˜ ρ = ρ + NM ρ ( A H ) σ N r N on D r , we have I ( r ) = − N M ρ ( A H ) σ N r N (cid:90) (cid:90) D r × ( B CAH \ D r ) | x − y | k ρ ( y ) dxdy . By the HLS inequality (3.1), we have | I ( r ) | ≤ N M ρ ( A H ) σ N r N (cid:90) (cid:90) D r × R N | x − y | k ρ ( y ) dxdy ≤ C HLS
N M ρ ( A H ) σ N r N (cid:107) D r (cid:107) a (cid:107) ρ (cid:107) b if a > , b > and /a + 1 /b − k/N = 2 . We can choose b ∈ (1 , min { m , N/ ( k + N ) } ) , whichis possible as − N < k < , m > , and then we get a > , ρ ∈ L b ( R N ) as < b < m , and | I ( r ) | ≤ C HLS || ρ || b M ρ ( A H ) (cid:18) σ N r N N (cid:19) a − . The latter vanishes as r → ∞ . For the term I , we have I ( r ) = − N M ρ ( A H ) σ N r N (cid:90) (cid:90) D r × D r | x − y | k ρ ( y ) dxdy − (cid:18) N M ρ ( A H ) σ N r N (cid:19) (cid:90) (cid:90) D r × D r | x − y | k dxdy . With the same choice of a, b as above, the HLS inequality implies | I ( r ) | ≤ N M ρ ( A H ) σ N r N (cid:90) (cid:90) D r × R N | x − y | k ρ ( y ) dxdy + (cid:18) N M ρ ( A H ) σ N r N (cid:19) (cid:90) (cid:90) D r × D r | x − y | k dxdy ≤ C HLS M ρ ( A H ) (cid:32) || ρ || b (cid:18) σ N r N N (cid:19) a − + M ρ ( A H ) (cid:18) σ N r N N (cid:19) a + b − (cid:33) , which vanishes as r → ∞ since a > and b > . We conclude that I ( r ) → as r → ∞ .The integral I can be estimated using Theorem 10, and the fact that ρ ≥ H > on B A H together with m > m c , I = (cid:90) (cid:90) B AH × B AH | x − y | k ρ ( x ) ρ ( y ) dxdy ≤ C ∗ M ρ ( A H ) k/N (cid:90) B AH ρ m c ( x ) dx ≤ C ∗ M ρ ( A H ) k/N (cid:90) B AH ρ m ( x ) dx . (3.13)On the other hand, the HLS inequalities (3.1) and (3.2) do not seem to give a sharp enoughestimate for the cross-term I , for which we instead invoke Lemma 15, yielding(3.14) I ≤ C k,N M ρ ( A H ) K k,q,N ( H ) , for given q ∈ [0 , m/N ) and large enough H as specified in Lemma 15.In order to conclude, we join together (3.11), (3.12), (3.13) and (3.14) to obtain for any r > and any large enough H , F [ ρ ] − F [ ˜ ρ ] = H m [ ρ ] − H m [ ˜ ρ ] + χ ( W k [ ρ ] − W k [ ˜ ρ ]) ≥ (cid:18) m − χ C ∗ k M ρ ( A H ) k/N (cid:19) (cid:90) B AH ρ m dx + χ C k,N k M ρ ( A H ) K s,q,N ( H )+ ε r m − χ k I ( r ) . (3.15)Now we choose q . On the one hand, notice that for a choice η > small enough such that m > m c + η , we have(3.16) − m + ηk + N < m − − η ( − k ) . On the other hand, − N < k < implies − k/N > N/ (2 N + k ) . Since m > m c , this gives theinequality m > N/ (2 N + k ) . Hence, for small enough η > such that m > N (2+ η ) / (2 N + k ) , ROUND STATES IN THE DIFFUSION-DOMINATED REGIME 15 we have(3.17) − m + ηk + N < mN .
Thanks to (3.16) and (3.17) we see that we can fix a non-negative q such that(3.18) − m + ηk + N < q < min (cid:26) mN , ( m − − η )( − k ) (cid:27) . Since q satisfies (3.18), it follows that − kq < m − − η and at the same time − q ( k + N )
This section is devoted to the regu-larity properties of global minimisers. With enough regularity, global minimisers satisfy theconditions of Definition 3, and are therefore stationary states of equation (1.1). This will allowus to complete the proof of Theorem 1.We begin by introducing some notation and preliminary results. As we will make use of theHölder regularising properties of the fractional Laplacian, see [47, 50], the notation c N,s ( − ∆) s S k = ρ , s ∈ (0 , N/ is better adapted to the arguments that follow, fixing s = ( k + N ) / , and we will thereforestate the results in this section in terms of s .One fractional regularity result that we will use repeatedly in this section follows directly fromthe HLS inequality (3.1) applied with k = 2 s − N : for any s ∈ (0 , N/ , < p < N s , q = N pN − sp , we have(3.19) ( − ∆) s f ∈ L p (cid:0) R N (cid:1) ⇒ f ∈ L q (cid:0) R N (cid:1) . Further, for ≤ p < ∞ and s ≥ , we define the Bessel potential space L s,p ( R N ) as madeby all functions f ∈ L p ( R N ) such that ( I − ∆) s f ∈ L p ( R N ) , meaning that f is the Bessel potential of an L p ( R N ) function (see [51, pag. 135]). Since we are working with the operator ( − ∆) s instead of ( I − ∆) s , we make use of a characterisation of the space L s,p ( R N ) in termsof Riesz potentials. For < p < ∞ and < s < we have(3.20) L s,p ( R N ) = (cid:8) f ∈ L p ( R N ) : f = g ∗ W s − N , g ∈ L p ( R N ) (cid:9) , see [48, Theorem 26.8, Theorem 27.3], see also exercise 6.10 in Stein’s book [51, pag. 161].Moreover, for ≤ p < ∞ and < s < / we define the fractional Sobolev space W s,p ( R N ) by W s,p (cid:0) R N (cid:1) := (cid:26) f ∈ L p ( R N ) : (cid:90) (cid:90) R N × R N | f ( x ) − f ( y ) | p | x − y | N +2 sp dx dy < ∞ (cid:27) . We have the embeddings(3.21) L s,p ( R N ) ⊂ W s,p ( R N ) for p ≥ , s ∈ (0 , / , (3.22) W s,p (cid:0) R N (cid:1) ⊂ C ,β (cid:0) R N (cid:1) for β = 2 s − N/p, p > N/ s, s ∈ (0 , / , see [51, pag. 155] and [22, Theorem 4.4.7] respectively.Let s ∈ (0 , and α > such that α + 2 s is not an integer. Since c N,s ( − ∆) s S k = ρ holdsin R N , then we have from [47, Theorem 1.1, Corollary 3.5] (see also [9, Proposition 5.2]) that(3.23) (cid:107) S k (cid:107) C ,α +2 s ( B / (0)) ≤ c (cid:16) (cid:107) S k (cid:107) L ∞ ( R N ) + (cid:107) ρ (cid:107) C ,α ( B (0)) (cid:17) , with the convention that if α ≥ for any open set U in R N , then C ,α ( U ) := C α (cid:48) ,α (cid:48)(cid:48) ( U ) ,where α (cid:48) + α (cid:48)(cid:48) = α , α (cid:48)(cid:48) ∈ [0 , and α (cid:48) is the greatest integer less than or equal to α . With thisnotation, we have C , ( R N ) = C , ( R N ) = W , ∞ ( R N ) . In particular, using (3.23) it followsthat for α > , s ∈ (0 , and α + 2 s not an integer,(3.24) (cid:107) S k (cid:107) C ,α +2 s ( R N ) ≤ c (cid:16) (cid:107) S k (cid:107) L ∞ ( R N ) + (cid:107) ρ (cid:107) C ,α ( R N ) (cid:17) . Moreover, rescaling inequality (3.23) in any ball B R ( x ) where R (cid:54) = 1 we have the estimate(3.25) α (cid:88) (cid:96) =0 R (cid:96) (cid:107) D (cid:96) S k (cid:107) L ∞ ( B R/ ( x )) + R α +2 s [ D α S k ] C ,α +2 s − α ( B R/ ( x )) ≤ C (cid:34) (cid:107) S k (cid:107) L ∞ ( R N ) + α (cid:88) (cid:96) =0 R s + (cid:96) (cid:107) D (cid:96) ρ (cid:107) L ∞ ( B R ( x )) + R α +2 s [ D α ρ ] C ,α − α ( B R ( x )) (cid:35) where α , α are the greatest integers less than α and α + 2 s respectively. In (3.25) thequantities (cid:107) D (cid:96) S k (cid:107) L ∞ and [ D (cid:96) ρ ] C ,α denote the sum of the L ∞ norms and the C ,α seminormsof the derivatives D ( β ) S k , D ( β ) ρ of order (cid:96) (that is | β | = (cid:96) ).Finally, we recall the definition of m c and m ∗ in (1.4) in terms of s : m c := 2 − sN and m ∗ := − s − s if N ≥ and s ∈ (0 , / , + ∞ if N ≥ and s ∈ [1 / , N/ . Let us begin by showing that global minimisers of F enjoy the good Hölder regularity inthe most singular range, as long as diffusion is not too slow. Theorem 17.
Let χ > and s ∈ (0 , N/ . If m c < m < m ∗ , then any global minimiser ρ ∈ Y of F satisfies S k = W k ∗ ρ ∈ W , ∞ ( R N ) , ρ m − ∈ W , ∞ ( R N ) and ρ ∈ C ,α ( R N ) with α = min { , m − } . ROUND STATES IN THE DIFFUSION-DOMINATED REGIME 17
Proof.
Recall that the global minimiser ρ ∈ Y of F is radially symmetric non-increasing andcompactly supported by Theorem 11 and Theorem 16. Since ρ ∈ L (cid:0) R N (cid:1) ∩ L ∞ (cid:0) R N (cid:1) byTheorem 16, we have ρ ∈ L p (cid:0) R N (cid:1) for any < p < ∞ . Since ρ = c N,s ( − ∆) s S k , it followsfrom (3.19) that S k ∈ L q ( R N ) , q = NpN − sp for all < p < N s , that is S k ∈ L p ( R N ) for all p ∈ ( NN − s , ∞ ) . Then, if s ∈ (0 , , since S k is the Riesz potential of the density ρ in L p , bythe characterisation (3.20) of the Bessel potential space, we conclude that S k ∈ L s,p ( R N ) forall p > NN − s . Let us first consider s < / , as the cases / < s < N/ and s = 1 / follow asa corollary. < s < / In this case, we have the embedding (3.21) and so S k ∈ W s,p ( R N ) for all p ≥ > NN − s if N ≥ and for all p > max { , − s } if N = 1 . Using (3.22), we conclude that S k ∈ C ,β (cid:0) R N (cid:1) with β := 2 s − N/p, for any p > N s > if N ≥ and for any p > max { s , − s } if N = 1 . Hence ρ m − ∈ C ,β (cid:0) R N (cid:1) for the same choice of β using the Euler–Lagrange condition (3.5) since ρ m − is the truncationof a function which is S k up to a constant.Note that m c ∈ (1 , and m ∗ > . In what follows we split our analysis into the cases m c < m ≤ and < m < m ∗ , still assuming s < / . If m ≤ , the argument follows alongthe lines of [12, Corollary 3.12] since ρ m − ∈ C ,α ( R N ) implies that ρ is in the same Hölderspace for any α ∈ (0 , . Indeed, in such case we bootstrap in the following way. Let us fix n ∈ N such that(3.26) n + 1 < s ≤ n and let us define(3.27) β n := β + ( n − s = 2 ns − N/p.
Form (3.26) and (3.27) we see that by choosing large enough p there hold − s < β n < .Note that S k ∈ L ∞ (cid:0) R N (cid:1) by Lemma 4, and if ρ ∈ C ,γ (cid:0) R N (cid:1) for some γ ∈ (0 , such that γ + 2 s < , then S k ∈ C ,γ +2 s (cid:0) R N (cid:1) by (3.24), implying ρ m − ∈ C ,γ +2 s (cid:0) R N (cid:1) using theEuler–Lagrange conditions (3.5). Therefore ρ ∈ C ,γ +2 s (cid:0) R N (cid:1) since m ∈ ( m c , . Iteratingthis argument ( n − times starting with γ = β gives ρ ∈ C ,β n (cid:0) R N (cid:1) . Since β n < and β n + 2 s > , a last application of (3.24) yields S k ∈ W , ∞ ( R N ) , so that ρ m − ∈ W , ∞ ( R N ) ,thus ρ ∈ W , ∞ ( R N ) . This concludes the proof in the case m ≤ .Now, let us assume < m < m ∗ and s < / . Recall that ρ m − ∈ C ,γ (cid:0) R N (cid:1) for any γ < s , and so ρ ∈ C ,γ (cid:0) R N (cid:1) for any γ < sm − . By (3.24) we get S k ∈ C ,γ (cid:0) R N (cid:1) for any γ < sm − + 2 s , and the same for ρ m − by the Euler–Lagrange equation (3.5). Once more witha bootstrap argument, we obtain improved Hölder regularity for ρ m − . Indeed, since(3.28) + ∞ (cid:88) j =0 s ( m − j = 2 s ( m − m − and since m < m ∗ means s ( m − m − > , after taking a suitably large number of iterations weget S k ∈ W , ∞ ( R N ) and ρ m − ∈ W , ∞ ( R N ) . Hence, ρ ∈ C , / ( m − (cid:0) R N (cid:1) . N ≥ , / ≤ s < N/ We start with the case s = 1 / . We have S k ∈ L p ( R N ) for any p > NN − as shown at the beginning of this proof. By (3.20) we get S k ∈ L ,p (cid:0) R N (cid:1) for all p > NN − . Then we also have S k ∈ L r,p ( R N ) for all p > NN − and for all r ∈ (0 , / by the embeddings between Bessel potential spaces, see [51, pag. 135]. Noting that ≥ NN − for N ≥ , by (3.21) and (3.22) we get S k ∈ C , r − N/p ( R N ) for any r ∈ (0 , / and any p > N r . That is, S k ∈ C ,γ ( R N ) for any γ ∈ (0 , . By the Euler–Lagrange equation (3.5), ρ ∈ C ,γα ( R N ) with α = min { , m − } , and so (3.24) for s = 1 / implies S k ∈ W , ∞ ( R N ) .Again by the Euler–Lagrange equation (3.5), we obtain ρ m − ∈ W , ∞ ( R N ) .If / < s < N/ on the other hand, we obtain directly that S k ∈ W , ∞ ( R N ) by Lemma 4,and so ρ m − ∈ W , ∞ ( R N ) .We conclude that ρ ∈ C ,α ( R N ) with α = min { , m − } for any / ≤ s < N/ . (cid:3) Remark 18. If m ≥ m ∗ and s < / , we recover some Hölder regularity, but it is not enoughto show that global minimisers of F are stationary states of (1.1). More precisely, m ≥ m ∗ means s ( m − m − ≤ , and so it follows from (3.28) that ρ ∈ C ,γ (cid:0) R N (cid:1) for any γ < sm − . Notethat m ≥ m ∗ also implies sm − ≤ − s , and we are therefore not able to go above the desiredHölder exponent − s . Remark 19.
In the arguments of Theorem 17 one could choose to directly bootstrap onfractional Sobolev spaces. In fact, for < s < / and m > we have that ρ m − ∈ W s,p ( R N ) implies ρ ∈ W sm − ,p ( m − ( R N ) . Indeed, let α < and u ∈ W α,p ( R N ) , where and p ∈ [1 , ∞ ) .By the algebraic inequality || a | α − | b | α | ≤ C | a − b | α we have (cid:90) (cid:90) R N × R N || u ( x ) | α − | u ( y ) | α | p/α | x − y | N + α s ( p/α ) dxdy ≤ c (cid:90) (cid:90) R N × R N | u ( x ) − u ( y ) | p | x − y | N +2 sp dxdy , thus | u | α ∈ W αs,p/α ( R N ) . This property is also valid for Sobolev spaces with integer order, see[40]. In particular, thanks to this property, in case m ≥ m ∗ we may obtain ρ m − ∈ W α,p ( R N ) for any α < s ( m − m − and any large enough p , hence (3.22) implies that ρ has the Hölderregularity stated in Remark 18.We are now ready to show that global minimisers possess the good regularity properties tobe stationary states of equation (1.1) according to Definition 3. Theorem 20.
Let χ > , s ∈ (0 , N/ and m c < m < m ∗ . Then all global minimisers of F in Y are stationary states of equation (1.1) according to Definition 3.Proof. Note that m < m ∗ means − s < / ( m − , and so thanks to Theorem 17, S k and ρ satisfy the regularity conditions of Definition 3. Further, since ρ m − ∈ W , ∞ (cid:0) R N (cid:1) , we cantake gradients on both sides of the Euler–Lagrange condition (3.5). Multiplying by ρ andwriting ρ ∇ ρ m − = m − m ∇ ρ m , we conclude that global minimisers of F in Y satisfy relation(2.1) for stationary states of equation (1.1). (cid:3) In fact, we can show that global minimisers have even more regularity inside their support.
Theorem 21.
Let χ > , m c < m and s ∈ (0 , N/ . If ρ ∈ Y is a global minimiser of F ,then ρ is C ∞ in the interior of its support.Proof. By Theorem 17 and Remark 18, we have ρ ∈ C ,α ( R N ) for some α ∈ (0 , . Since ρ isradially symmetric non-increasing, the interior of supp ( ρ ) is a ball centered at the origin, whichwe denote by B . Note also that ρ ∈ L ( R N ) ∩ L ∞ ( R N ) by Theorem 16, and so S k ∈ L ∞ ( R N ) by Lemma 4.Assume first that s ∈ (0 , ∩ (0 , N/ . Applying (3.25) with B R centered at a point within B and such that B R ⊂⊂ B , we obtain S k ∈ C ,γ ( B R/ ) for any γ < α + 2 s . It follows fromthe Euler–Lagrange condition (3.5) that ρ m − has the same regularity as S k on B R/ , andsince ρ is bounded away from zero on B R/ , we conclude ρ ∈ C ,γ ( B R/ ) for any γ < α + 2 s .Repeating the previous step now on B R/ , we get the improved regularity S k ∈ C ,γ ( B R/ ) ROUND STATES IN THE DIFFUSION-DOMINATED REGIME 19 for any γ < α + 4 s by (3.25), which we can again transfer onto ρ using (3.5), obtaining ρ ∈ C ,γ ( B R/ ) for any γ < α + 4 s . Iterating, any order (cid:96) of differentiability for S k (and thenfor ρ ) can be reached in a neighborhood of the center of B R . We notice that the argumentcan be applied starting from any point x ∈ B , and hence ρ ∈ C ∞ ( B ) .When N ≥ and s ∈ [1 , N/ , we take numbers s , . . . , s l such that s i ∈ (0 , for any i = 1 , . . . , l and such that (cid:80) li =1 s i = s . We also let S l +1 k := S k , S jk := Π li = j ( − ∆) s j S k , ∀ j ∈ { , . . . , l } . Then S k = ρ . Note that Lemma 4(i) can be restated as saying that ρ ∈ Y ∩ L ∞ ( R N ) implies ( − ∆) − δ ρ ∈ L ∞ ( R N ) for all δ ∈ (0 , N/ . Taking δ = s − r for any r ∈ (0 , s ) , we have ( − ∆) r S k = ( − ∆) r − s ρ ∈ L ∞ . In particular, this means S jk ∈ L ∞ ( R N ) for any j = 1 , . . . , l + 1 .Moreover, there holds ( − ∆) s j S j +1 k = S jk , ∀ j ∈ { , . . . , l } . Therefore we may recursively apply (3.25), starting from S k = ρ ∈ C ,α ( B R ) , where the ball B R is centered at a point within B such that B R ⊂⊂ B , and using the iteration rule S jk ∈ C ,γ ( B σ ) ⇒ S j +1 k ∈ C ,γ +2 s j (cid:0) B σ/ (cid:1) ∀ j ∈ { , . . . , l } , ∀ γ > s.t. γ + 2 s j is not an integer, ∀ B σ ⊂⊂ B. We obtain S l +1 k = S k ∈ C ,γ ( B R/ (2 l ) ) for any γ < α + 2 s , and as before, the Euler–Lagrangeequation (3.5) implies that ρ ∈ C ,γ ( B R/ (2 l ) ) for any γ < α + 2 s . If we repeat the argument,we gain s in Hölder regularity for ρ each time we divide the radius R by l . In this way, wecan reach any differentiability exponent for ρ around any point of B , and thus ρ ∈ C ∞ ( B ) . (cid:3) Remark 22.
We observe that the smoothness of minimisers in the interior of their supportalso holds in the fair competition regime m = m c . In such case global Hölder regularity wasobtained in [12].The main result Theorem 1 follows from Theorem 6, Corollary 9, Theorem 11, Proposition14, Theorem 16, Theorem 20 and Theorem 21.4. Uniqueness
Optimal Transport Tools.
Optimal transport is a powerful tool for reducing functionalinequalities onto pointwise inequalities. In other words, to pass from microscopic inequalitiesbetween particle locations to macroscopic inequalities involving densities. This sub-sectionsummarises the main results of optimal transportation we will need in the one-dimensionalsetting. They were already used in [11, 13], where we refer for detailed proofs.Let ˜ ρ and ρ be two probability densities. According to [7, 38], there exists a convex function ψ whose gradient pushes forward the measure ˜ ρ ( a ) da onto ρ ( x ) dx : ψ (cid:48) ρ ( a ) da ) = ρ ( x ) dx .This convex function satisfies the Monge-Ampère equation in the weak sense: for any testfunction ϕ ∈ C b ( R ) , the following identity holds true (cid:90) R ϕ ( ψ (cid:48) ( a )) ˜ ρ ( a ) da = (cid:90) R ϕ ( x ) ρ ( x ) dx . The convex map is unique a.e. with respect to ρ and it gives a way of interpolating measuresusing displacement convexity [39]. On the other hand, regularity of the transport map is acomplicated matter. Here, as it was already done in [11, 13], we will only use the fact that ψ (cid:48)(cid:48) ( a ) da can be decomposed in an absolute continuous part ψ (cid:48)(cid:48) ac ( a ) da and a positive singularmeasure [52, Chapter 4]. In one dimension, the transport map ψ (cid:48) is a non-decreasing function, therefore it is differentiable a.e. and it has a countable number of jump singularities. For anymeasurable function U , bounded below such that U (0) = 0 we have [39](4.1) (cid:90) R U ( ˜ ρ ( x )) dx = (cid:90) R U (cid:18) ρ ( a ) ψ (cid:48)(cid:48) ac ( a ) (cid:19) ψ (cid:48)(cid:48) ac ( a ) da . The following Lemma proved in [11] will be used to estimate the interaction contribution inthe free energy.
Lemma 23.
Let K : (0 , ∞ ) → R be an increasing and strictly concave function. Then, forany a, b ∈ R (4.2) K (cid:18) ψ (cid:48) ( b ) − ψ (cid:48) ( a ) b − a (cid:19) ≥ (cid:90) K (cid:0) ψ (cid:48)(cid:48) ac ([ a, b ] s ) (cid:1) ds , where the convex combination of a and b is given by [ a, b ] s = (1 − s ) a + sb . Equality is achievedin (4.2) if and only if the distributional derivative of the transport map ψ (cid:48)(cid:48) is a constantfunction. Functional Inequality in One Dimension.
In what follows, we will make use of acharacterisation of stationary states based on some integral reformulation of the necessarycondition stated in Proposition 14. This characterisation was also the key idea in [11, 13] toanalyse the asymptotic stability of steady states and the functional inequalities behind.
Lemma 24 (Characterisation of stationary states) . Let N = 1 , χ > and k ∈ ( − , . If m > m c with m c = 1 − k , then any stationary state ¯ ρ ∈ Y of system (1.1) can be written inthe form (4.3) ¯ ρ ( p ) m = χ (cid:90) R (cid:90) | q | k ¯ ρ ( p − sq ) ¯ ρ ( p − sq + q ) dsdq . The proof follows the same methodology as for the fair-competition regime [13, Lemma 2.8]and we omit it here.
Theorem 25.
Let N = 1 , χ > , k ∈ ( − , and m > m c . If (1.1) admits a stationarydensity ¯ ρ in Y , then F [ ρ ] ≥ F [ ¯ ρ ] , ∀ ρ ∈ Y with the equality cases given by dilations of ¯ ρ .Proof. For a given stationary state ¯ ρ ∈ Y and a given ρ ∈ Y , we denote by ψ the convexfunction whose gradient pushes forward the measure ¯ ρ ( a ) da onto ρ ( x ) dx : ψ (cid:48) ρ ( a ) da ) = ρ ( x ) dx . Using (4.1), the functional F [ ρ ] rewrites as follows: F [ ρ ] = 1 m − (cid:90) R (cid:18) ¯ ρ ( a ) ψ (cid:48)(cid:48) ac ( a ) (cid:19) m − ¯ ρ ( a ) da + χ k (cid:90) (cid:90) R × R (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:48) ( a ) − ψ (cid:48) ( b ) a − b (cid:12)(cid:12)(cid:12)(cid:12) k | a − b | k ¯ ρ ( a ) ¯ ρ ( b ) dadb = 1 m − (cid:90) R (cid:0) ψ (cid:48)(cid:48) ac ( a ) (cid:1) − m ¯ ρ ( a ) m da + χ k (cid:90) (cid:90) R × R (cid:10) ψ (cid:48)(cid:48) ([ a, b ]) (cid:11) k | a − b | k ¯ ρ ( a ) ¯ ρ ( b ) dadb , where (cid:10) u ([ a, b ]) (cid:11) = (cid:82) u ([ a, b ] s ) ds and [ a, b ] s = (1 − s ) a + bs for any a, b ∈ R and u : R → R + .By Lemma 24, we can write for any a ∈ R , ( ψ (cid:48)(cid:48) ac ( a )) − m ¯ ρ ( a ) m = χ (cid:90) R (cid:10) ψ (cid:48)(cid:48) ac ([ a, b ]) − m (cid:11) | a − b | k ¯ ρ ( a ) ¯ ρ ( b ) db , ROUND STATES IN THE DIFFUSION-DOMINATED REGIME 21 and hence F [ ρ ] = χ (cid:90) (cid:90) R × R (cid:26) m − (cid:10) ψ (cid:48)(cid:48) ac ([ a, b ]) − m (cid:11) + 1 k (cid:10) ψ (cid:48)(cid:48) ([ a, b ]) (cid:11) k (cid:27) | a − b | k ¯ ρ ( a ) ¯ ρ ( b ) dadb . Using the concavity of the power function ( · ) − m and and Lemma 23, we deduce F [ ρ ] ≥ χ (cid:90) (cid:90) R × R (cid:26) m − (cid:10) ψ (cid:48)(cid:48) ([ a, b ]) (cid:11) − m + 1 k (cid:10) ψ (cid:48)(cid:48) ([ a, b ]) (cid:11) k (cid:27) | a − b | k ¯ ρ ( a ) ¯ ρ ( b ) dadb . Applying characterisation (4.3) to the energy of the stationary state ¯ ρ , we obtain F [ ¯ ρ ] = χ (cid:90) (cid:90) R × R (cid:18) m −
1) + 1 k (cid:19) | a − b | k ¯ ρ ( a ) ¯ ρ ( b ) dadb . Since(4.4) z − m m − z k k ≥ m − k for any real z > and for m > m c = 1 − k , we conclude F [ ρ ] ≥ F [ ¯ ρ ] . Equality in Jensen’sinequality arises if and only if the derivative of the transport map ψ (cid:48)(cid:48) is a constant function,i.e. when ρ is a dilation of ¯ ρ . In agreement with this, equality in (4.4) is realised if and onlyif z = 1 . (cid:3) In fact, the result in Theorem 25 implies that all critical points of F in Y are globalminimisers. Further, we obtain the following uniqueness result: Corollary 26 (Uniqueness) . Let χ > and k ∈ ( − , . If m c < m , then there exists at mostone stationary state in Y to equation (1.1) . If m c < m < m ∗ , then there exists a unique globalminimiser for F in Y .Proof. Assume there are two stationary states to equation (1.1): ¯ ρ , ¯ ρ ∈ Y . Then Theorem25 implies that F [ ¯ ρ ] = F [ ¯ ρ ] , and so ¯ ρ is a dilation of ¯ ρ . By Theorem 11, there exists aminimiser of F in Y , which is a stationary state of equation (1.1) if m c < m < m ∗ by Theorem20, and so uniqueness follows. (cid:3) Theorem 25 and Corollary 26 complete the proof of the main result Theorem 2.
Appendix A. Properties of the Riesz potential
The estimates in Proposition 7 are mainly based on the fact that the Riesz potential of aradial function can be expressed in terms of the hypergeometric function F ( a, b ; c ; z ) := Γ( c )Γ( b )Γ( c − b ) (cid:90) (1 − zt ) − a (1 − t ) c − b − t b − dt, which we define for z ∈ ( − , , with the parameters a, b, c being positive. Notice that F ( a, b, c,
0) = 1 and F is increasing with respect to z ∈ ( − , . Moreover, if c > , b > and c > a + b , the limit as z ↑ is finite and it takes the value(A.1) Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) , see [34, §9.3]. We will also make use of some elementary relations. First of all, there holds(A.2) F ( a, b ; c ; z ) = (1 − z ) c − a − b F ( c − a, c − b ; c ; z ) , see [34, §9.5], and it is easily seen that ddz F ( a, b ; c ; z ) = abc F ( a + 1 , b + 1; c + 1; z ) . Inserting (A.2) we find(A.3) ddz F ( a, b ; c ; z ) = abc (1 − z ) c − a − b − F ( c − a, c − b ; c + 1; z ) . To simplify notation, let us define(A.4) H ( a, b ; c ; z ) := Γ( b )Γ( c − b )Γ( c ) F ( a, b ; c ; z ) = (cid:90) (1 − zt ) − a (1 − t ) c − b − t b − dt . Proof of Proposition 7.
For a given radial function ρ ∈ Y we use polar coordinates, still de-noting by ρ the radial profile of ρ , and compute as in [49, Theorem 5], see also [1], [25] or [26,§1.3],(A.5) | x | k ∗ ρ ( x ) = σ N − (cid:90) ∞ (cid:18)(cid:90) π (cid:0) | x | + η − | x | η cos θ (cid:1) k/ sin N − θ dθ (cid:19) ρ ( η ) η N − dη . Then we need to estimate the integral Θ k ( r, η ) := σ N − (cid:90) π (cid:0) r + η − rη cos( θ ) (cid:1) k/ sin N − ( θ ) dθ = (cid:40) r k ϑ k ( η/r ) , η < r ,η k ϑ k ( r/η ) , r < η , (A.6)where, for u ∈ [0 , , ϑ k ( u ) := σ N − (cid:90) π (cid:0) u − u cos( θ ) (cid:1) k/ sin N − ( θ ) dθ = σ N − (1 + u ) k (cid:90) π (cid:18) − u (1 + u ) cos (cid:18) θ (cid:19)(cid:19) k/ sin N − ( θ ) dθ . Using the change of variables t = cos (cid:0) θ (cid:1) , we get from the integral formulation (A.4), ϑ k ( u ) = 2 N − σ N − (1 + u ) k (cid:90) (cid:18) − u (1 + u ) t (cid:19) k/ t N − (1 − t ) N − dt = 2 N − σ N − (1 + u ) k H ( a, b ; c ; z ) (A.7)with a = − k , b = N − , c = N − , z = 4 u (1 + u ) . The function F ( a, b ; c ; z ) is increasing in z and then for any z ∈ (0 , there holds(A.8) F ( a, b ; c ; z ) ≤ lim z ↑ F ( a, b ; c ; z ) . Note that c − a − b = ( k + N − / changes sign at k = 1 − N , and the estimate of Θ k dependson the sign of c − a − b :Case k > − N The limit (A.8) is finite if c − a − b > and it is given by the expression(A.1). Therefore we get from (A.6)-(A.7) and (A.4) Θ k ( | x | , η ) ≤ C ( | x | + η ) k ≤ C | x | k if − N < k < with C := 2 N − σ N − Γ( b )Γ( c − a − b ) / Γ( c − a ) . Inserting this into (A.5) concludes the proofof (i).Case k < − N If c − a − b < we use (A.2) F ( a, b ; c ; z ) = (1 − z ) c − a − b F ( c − a, c − b ; c ; z ) , ROUND STATES IN THE DIFFUSION-DOMINATED REGIME 23 where now the right hand side, using (A.8) and (A.1), can be bounded from above by (1 − z ) c − a − b Γ( c )Γ( a + b − c ) / [Γ( a )Γ( b )] for z ∈ (0 , . This yields from (A.6)-(A.7) and (A.4) theestimate(A.9) Θ k ( | x | , η ) ≤ C | x | k (cid:18) | x | + η | x | − η (cid:19) − k − N if k < − N with C := 2 N − σ N − Γ( c − b )Γ( a + b − c ) / Γ( a ) .Case k = 1 − N If on the other hand c − a − b = 0 , we use (A.3) with c = 2 a = 2 b = N − ,integrating it and obtaining, since F = 1 for z = 0 , F ( a, b ; c ; z ) = 1 + N − (cid:90) z F ( c − a, c − b ; c + 1; t )1 − t dt, and the latter right hand side is bounded above, thanks to (A.8) and (A.1), by N − N )4(Γ( N/ / log (cid:18) − z (cid:19) for z ∈ (0 , . This leads from (A.6)-(A.7) to the new estimate(A.10) Θ k ( | x | , η ) ≤ C | x | k (cid:18) (cid:18) | x | + η | x | − η (cid:19)(cid:19) if k = 1 − N , with C := 2 N − σ N − N/ − / Γ( N − max (cid:110) , ( N − N )2Γ(( N +1) / (cid:111) .Now, if ρ is supported on a ball B R , the radial representation (A.5) reduces to(A.11) | x | k ∗ ρ ( x ) = (cid:90) R Θ k ( | x | , η ) ρ ( η ) η N − dη, x ∈ R N . If | x | > R , we have ( | x | + η )( | x | − η ) − ≤ ( | x | + R )( | x | − R ) − for any η ∈ (0 , R ) , thereforewe can put R in place of η in the right hand side of (A.9) and (A.10), insert into (A.11) andconclude. (cid:3) Acknowledgements
We thank Y. Yao and F. Brock for useful discussion about the continuous Steiner symmetrisa-tion. We thank X. Ros-Otón, P. R. Stinga and P. Mironescu for some fruitful explanations concerningthe regularity properties of fractional elliptic equations used in this work. JAC was partially sup-ported by the Royal Society via a Wolfson Research Merit Award and by the EPSRC grant numberEP/P031587/1. FH acknowledges support from the EPSRC grant number EP/H023348/1 for theCambridge Centre for Analysis. EM was partially supported by the FWF project M1733-N20. BVwas partially supported by GNAMPA of INdAM, "Programma triennale della Ricerca dell’Universitàdegli Studi di Napoli "Parthenope"- Sostegno alla ricerca individuale 2015-2017". EM and BV aremember of the GNAMPA group of the Istituto Nazionale di Alta Matematica (INdAM). The authorsare very grateful to the Mittag-Leffler Institute for providing a fruitful working environment duringthe special semester
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