Chemotactic systems in the presence of conflicts: a new functional inequality
CChemotactic systems in the presence of conflicts: a newfunctional inequality
G. WolanskyDepartment of Mathematics, Technion, Haifa 32000, israel
Abstract
The evolution of a chemotactic system involving a population of cells attracted to self-produced chemicals is described by the Keller-Segel system. In dimension 2, this systemdemonstrates a balance between the spreading effect of diffusion and the concentration dueto self-attraction. As a result, there exists a critical ”mass” (i.e. total cell’s population)above which the solution of this system collapses in a finite time, while below this criticalmass there is global existence in time. In particular, sub critical mass leads under certainadditional conditions to the existence of steady states, corresponding to the solution ofan elliptic Liouville equation. The existence of this critical mass is related to a functionalinequality known as the Moser-Trudinger inequality.An extension of the Keller-Segel model to several cells populations was consideredbefore in the literature. Here we review some of these results and, in particular, con-sider the case of conflict between two populations, that is, when population one attractspopulation two, while, at the same time, population two repels population one. Thisassumption leads to a new functional inequality which generalizes the Moser-Trudingerinequality. As an application of this inequality we derive sufficient conditions for theexistence of steady states corresponding to solutions of an elliptic Liouville system.
In this paper we study a non-local elliptic Liouville system in Ω of the form∆ u + M e αu ± βw (cid:82) Ω e αu ± βw = 0 ; ∆ w + M e − γw + βu (cid:82) Ω e − γw + βu = 0 , u = w = 0 on ∂ Ω (1.1)where M i >
0, all constants α, β, γ are non-negative and Ω is a planar bounded domain. Wedenote the + β case above as the ”conflict free” case, while the − β is the ”conflict” case. Thereasoning behind this notation is explained below (see also Section 3 and [15]).Our motivation for studying this system is the non-local parabolic-elliptic system ∂ρ∂t = ∆ ρ + ∇ · [ ρ ( ∓ β ∇ w − α ∇ u )] ; ∆ u + ρ = 0 ; ∆ w + M e βu − γw (cid:82) e βu − γw = 0 (1.2)System (1.2) is defined on Ω × [0 , ∞ ). The no-flux boundary condition for ρ takes the form( ∇ ρ − αρ ∇ u ∓ βρ ∇ w ) · n = 0 on ∂ Ω × (0 , ∞ ) (1.3)where n is the normal to ∂ Ω. In addition, u = w = 0 on ∂ Ω × (0 , ∞ ).In addition, ρ, u, w satisfy the initial conditions at t = 0: u ( ,
0) = u ∈ H (Ω) := H , w ( ,
0) = w ∈ H and ρ ( ,
0) := ρ ∈ L (Ω) := L where ρ ≥ a r X i v : . [ m a t h . A P ] A p r o-flux boundary condition (1.3) implies, by a formal application of the divergence theorem,the conservation of mass: (cid:90) Ω ρ ( x, t ) dx = (cid:90) Ω ρ ( x ) dx := M > . (1.4)The steady states of (1.2, 1.3, 1.4) are solutions of (1.1) where ρ = M e αu − βw (cid:82) Ω e αu − βw .The function ρ corresponds, in the language of chemotaxis [8,7], to the density of apopulation of organisms (living cells, bacteria, slime molds or, perhaps, crowded humanbeings ...) which evolve in time without multiplication and mortality. The individuals of thispopulation are moving on the planar domain Ω under a combination of random walk anddeterministic drift force along the gradient of self produced chemicals u and w .We remark at this point that the sign of the off diagonal terms in (1.1) represents theinteraction force between the populations. A positive off diagonal term for a given componentrepresents that the population corresponding to this component is rejected from the otherpopulation. Thus, β > β in the first equation implies that the firstpopulation is rejected from the second one as well, so there is no conflict. On the otherhand, the choice − β implies that the first population is attracted to the second one (whilethe second one is still rejected by the first). This unhappy situation is the origin of conflictof interests between the two populations. In [6], the general version of Liouville system (1.1) was considered∆ u i + M i (cid:82) Ω e (cid:80) nj =1 a ij u j e (cid:80) nj =1 a ij u j = 0 (1.5)of n ≥ ⊂ R , where u i =0 on ∂ Ω and where M i > ≤ i ≤ n . The coefficients { a i,j } are assumes tobe non-negative and a i,j = a j,i .Let J ⊂ { , . . . n } , andΛ J ( M , . . . M n ) := 4 π (cid:88) i ∈ J M i − (cid:88) i,j ∈ J a i,j M i M j . Theorem 1.1 in [6] implies that a sufficient condition for the existence of a solution of (1.5)is the inequalities Λ J ( M , . . . M n ) > J ⊆ { , . . . n } , J (cid:54) = ∅ . (1.6)Theorem 1.2 of the same paper deals with radial solutions of (1.5) under the assumption thatΩ is a disk in R . It follows that in that case the same result holds even if we give up thecondition of non-negative off diagonal elements a i,j , i (cid:54) = j . The diagonal elements a i,i arestill assumed to be non negative. 2 a) (b) (c) Figure 1: The 3 cases of conflict free chemotaxis for β > β < α/ β > α/ γ = 0 (b)and β > α/ γ > M = 8 π/α .Note that the conflict free case + β in (1.1) corresponds to a symmetric matrix { a i,j } in (1.5) where n = 2. However, the presence of the negative coefficient − γ in the secondequation of (1.1) violates the non-negative diagonal assumption of Theorem 1.2 in [6]. Still,the proof in [6] can be extended to this case as well, provided condition (1.6 ) is replaced byΛ I ( m , . . . m n ) > , < m i ≤ M i for any i ∈ I := { , . . . n } . (1.7)Note that (1.6) implies (1.7) if the diagonal elements are non-negative ( a i,i ≥ < M < πα , and 4 π ( M + m ) − α M + γ m − βM m > ∀ m ∈ (0 , M ) . (1.8)If β < M < π/α and M < ∞ . If β > β < α ⇒ M ∈ (0 , π/α ), M > β ≥ α , γ = 0 ⇒ π ( M + M ) − α M + γ M − βM M > β ≥ α , γ > ⇒ either M ∈ (0 , M ) and M > M ∈ ( M , π/α ) and (1.9). Here M = M is the vertical asymptote to the hyperbolic branch of4 π ( M + M ) − α M + γ M − βM M = 0in the positive quadrate M , M > .2 The case of conflict The main result of this paper is referred to the case of conflict, i.e ( − β ) in (1.1). We alsoassume β > M , M ) := 2( M − M ) − αM π + βM M π − γM π . Theorem 1.
For any choice of α > , β ≥ , γ ≥ there exists a solution of (1.1) in theconflict case for any < M < π/α , ≤ M < ∞ .If, moreover, β > α/ and Ω is a disk in R , then a radial solution exists ifi) Λ( M , M ) > , (1.10) and M < M where M is determined by the larger root of Λ (cid:18) M , πγ (cid:18) βα − (cid:19)(cid:19) = 0 (1.11) if γ > , and M = ∞ if γ = 0 .ii) If ( M , M ) satisfies (i) then the solution exists also for all ( M , M ) where M > M . The proof of Theorem 1 follows from Proposition 3.1, Proposition 4.1 and Theorem 3in section 4.2. In Fig. 2 we sketch in gray the solvability domain of (1.1) in the ( M , M )parameters for β > α/ (a) (b) Figure 2: The 2 cases of conflict for β > α/ γ = 0 (a), and γ > βM − γM = 4 π , (3): M = 4 π/β , (4): M = 8 π/α , (5): M = M [a] γ = 0. In that case the domain of solvability coincide with Λ > γ >
0. Here the curve Λ = 0 is a quadratic curve (either an ellipse or an hyperbola) andthe solvability domain is bounded by from the right by M on the M axis. Remark 1.1.
We do not know if the conditions of Theorem 1 are optimal. However, The-orem 2 and the remark below Theorem 3 in Section 4.2 suggest that this may be the case, atleast if γ = 0 . .3 Structure of the paper In section 2 we review the Free energy method for chemotactic system of a single component,the connection with Moser-Trudinger inequality and its relation with the parabolic and ellipticLiouville equation.In section 3 we extend the discussion to chemotactic systems of two components, con-sider 3 limit cases and the associated Free Energies. The elliptic Liouville systems for twocomponents are derived in both conflict/noconclict cases.From section 4 forward we concentrate in the case of conflict for two component chemo-taxis. We discuss the solution of the Liouville system as steady states of the chemotacticsystem and its stability under the 3 limit cases. In sections 4.1, 4.2 we describe the mainobjectives of this paper and its main results summarized in Theorems 2 and 3. The mosttechnical part of this paper is the proof of Theorem 3, given in Section 5.
1. Ω ⊂ R is an open, bounded domain.2. ∂ Ω is the boundary of Ω. We assume that ∂ Ω is C regular.3. u ∈ H (Ω) iff (cid:82) Ω | u | + (cid:82) Ω |∇ u | < ∞ and admits zero trace on ∂ Ω.4. Γ M := { ρ ∈ L (Ω) , ρ ≥ , (cid:82) Ω ρ ln ρ < ∞ , (cid:82) Ω ρ = M } .5. ∆ − is the Green function of the Dirichlet Laplacian on Ω. Define F M on Γ M × H as F M ( ρ, w ) := (cid:90) Ω ρ ln ρ + α (cid:90) Ω ρ ∆ − ρ + (cid:20) γ (cid:90) Ω |∇ w | + M ln (cid:18)(cid:90) e − γw − β ∆ − ( ρ ) (cid:19)(cid:21) . Noting u = − ∆ − ρ , it follows that (1.2) can be written as ∂ρ∂t = ∇ · (cid:18) ρ ∇ δF M δρ (cid:19) , δF M δw = 0 on Ω × (0 , ∞ ) (2.1)while (1.3) is equivalent to ∇ δF M δρ · n = 0 on ∂ Ω. A formal integration by parts yields ddt F M ( ρ ( , t ) , w ( , t )) = − (cid:90) ρ (cid:12)(cid:12)(cid:12)(cid:12) ∇ δF M δρ (cid:12)(cid:12)(cid:12)(cid:12) (2.2)so F M is monotone non-increasing along solutions of (1.2).From the representation (2.1) it follows that any solution of (1.1) corresponds to a criticalpoint of F M on Γ M × H . In particular, the monotonicity (2.2) suggests that local minimizersof F M on this domain correspond to stable steady states of (1.2). Thus, the question5egarding the bound from below of F M on Γ M × H is interesting in that respect, as it isa necessary condition for the existence of a global minimizer on this domain. This globalminimizer is, evidently, a critical point, and thus a steady state of (1.2).If we substitute γ = β = 0 in F M we get, up to an irrelevant constant, the Free Energyfunctional ρ ∈ Γ M (cid:55)→ F ( ρ ) := (cid:90) Ω ρ ln ρ + α (cid:90) Ω ρ ∆ − ( ρ ) . (2.3)This functional is monotone non-increasing along solutions of the parabolic-elliptic Keller-Segel system for chemotaxis of a single component [13,15,16, 2...] (see also [17,18] for appli-cation to self-gravitating systems) ∂ρ∂t = ∆ ρ − ∇ · [ ρ ( α ∇ u )] ; ∆ u + ρ = 0 ; (cid:90) Ω ρ = M . (2.4)Note that (2.4) is obtained from the substitution γ = β = 0 in (1.2). This can be written as ∂ρ∂t = ∇ · (cid:18) ρ ∇ δFδρ (cid:19) on Γ M . (2.5)The bound from below of F on Γ M for M ≤ π/α follows from the logarithmic HLS inequality[1, 5]: ∀ ρ ∈ L ln L ( D ) , (cid:90) D | ρ | ln | ρ | + (4 π ) − (cid:90) D (cid:90) D ρ ( x ) ln | x − y | ρ ( y ) dxdy > − C ( D ) , (cid:107) ρ (cid:107) = 1for functions in a two dimensional bounded domain D . Using scaling and taking into accountthat ∆ − ( x, y ) ≈ (2 π ) − ln | x − y | , up to lower order terms, imply the bound from below onΓ M for M ≤ π/α .This is a key inequality for the proof of global existence of (2.4) for M < π/α as well asthe existence of solution to the nonlocal Liouville equation∆ u + M (cid:82) e αu e αu = 0 , M < π/α (2.6)in a bounded domain Ω [16, 9,10, 14].The parabolic elliptic Keller-Segel (2.4) is a limiting case of the parabolic parabolic system[4] δ ∂ρ∂t = ∆ ρ − ∇ · [ ρ ( α ∇ u )] ; ε ∂u∂t = ∆ u + ρ = 0 , (cid:90) Ω ρ = M, u ∈ H (2.7)where δ = 1 and ε = 0. Another, less known limit of (2.7) [16, 9] is ε = 1 , δ = 0: ∂u∂t = ∆ u + M (cid:82) e αu e αu . (2.8)We observe that (2.8) is itself a gradient descend system on H of the form ∂u∂t = − α − δH M δu u ∈ H (cid:55)→ H Mα ( u ) := α (cid:90) Ω |∇ u | − M ln (cid:18)(cid:90) Ω e αu (cid:19) . A simple scaling shows that the bound from below of H M on H where M ≤ π/α followsfrom the Moser-Trudinger inequality12 (cid:90) Ω |∇ u | − π ln (cid:18)(cid:90) Ω e u (cid:19) > − C (2.9)for any u ∈ H [11, 12...]. This gives an alternative proof for the existence of solution to (2.6)for M < π/α , as well as the global (in time) existence of (2.8) under the same condition [3].Motivated by the above, we consider in this paper the condition for bound from belowof the functional F M on Γ M × H . Note that for M = 0 and γ = 0, F M is just the FreeEnergy F (2.3).It follows, then, that a new inequality for F M > − C on Γ M × H is a generalization ofthe Logarithmic HLS inequality for the case M = 0. Note also that if γ = 0, M > F M ( − ∆ w ) is, by integration by parts, just − α (cid:90) |∇ w | + M ln (cid:18)(cid:90) e βw (cid:19) which is related to the Moser-Trudinger inequality (with opposite sign, however). In fact, itis known that the Moser-Trudinger and Logarithmic HLS inequalities are equivalent. To seethis, consider( ρ, u ) ∈ Γ M × H (cid:55)→ H ( ρ, u ) := (cid:90) Ω ρ ln ρ + α (cid:90) Ω |∇ u | − α (cid:90) Ω ρu . and note that F ( ρ ) = inf u ∈ H H ( ρ, u ) ; H Mα ( u ) := inf ρ ∈ Γ M H ( ρ, u ) , so both logarithmic HLS and Moser-Trudinger inequalities follow from the bound H ( ρ, u ) > − C for ( ρ, u ) ∈ Γ π/α × H .It is also interesting to note that H induces a gradient descend flow for the parabolic-parabolic Keller-Segel equation (2.7) via δ ∂ρ∂t = ∇ · (cid:18) ρ ∇ δHδρ (cid:19) ; ε ∂u∂t = − α − δHδu and that (2.4) (resp. (2.8)) are singular limits of (2.7) for ε = 0 (resp. δ = 0). The general system of Chemotaxis for two components is a special case of the system of n populations [15]: δ i ∂ρ i ∂t = σ i ∆ ρ i + ∇ · [ ρ i ( a ii ∇ u i + a ij ∇ u j )] (3.1)7here ( i, j ) ∈ { , } i (cid:54) = j , and ε ∂u i ∂t = b i ∆ u i + ρ i , i = 1 , σ i > , b i > δ i , ε > a i,j are constants. Eq. (3.1,3.2) are defined on Ω × [0 , ∞ ) where Ω ⊂ R , u i are subjected to Dirichlet boundary condition u i = 0 on ∂ Ω × [0 , ∞ ) and initial data u i ( ,
0) = u i ∈ H (Ω) . (3.3) ρ i satisfy the no-flux boundary conditions n · { σ i ∇ ρ i + [ ρ ( a ii ∇ u i + a ij ∇ u j )] } = 0 (3.4)on ∂ Ω × [0 , ∞ ), where n is the normal to ∂ Ω. In addition, ρ i satisfy the initial conditions at t = 0: ρ i ( ,
0) = ρ i , where ρ i ∈ L (Ω) and ρ i ≥ (cid:90) Ω ρ i ( x, t ) dx = (cid:90) Ω ρ i ( x ) dx := M i . (3.5)The functions ρ i correspond to the densities of the two populations of organisms whichevolve with time without multiplication and mortality. The individuals of these populationsare moving on the planar domain Ω by a combination of random walk (corresponding to thediffusion coefficients σ i ), and deterministic drift forces along the gradient of self producedchemicals u i .Five of the constants in (3.1) can be eliminated by scaling ρ , ρ , u , u and the time t .In particular, we can assume, without loosing generality, that σ = σ = b = b = 1 andthat a = ± a := β . Let a := − α, a = γ . Assumption 3.1. α ≥ (self-attractive first population), γ ≥ (self repulsive second popu-lation) as well as β > (first population is rejected by the second one). We get δ ∂ρ ∂t = ∆ ρ + ∇ · [ ρ ( β ∇ u − α ∇ u )] ; δ ∂ρ ∂t = ∆ ρ + θ ∇ · [ ρ ( β ∇ u + θγ ∇ u )] (3.6) ε ∂u i ∂t = ∆ u i + ρ i , i = 1 , . (3.7)Here θ ∈ {− , } corresponds to the choice of sign in a = ± β . The case θ = 1 is the conflict free case studied in [15]. In that case the second population is rejected by the firstone, so both population has the same attitude to each other (mutual rejection, in that case).Let us define H θ : Γ M × Γ M × ( H ) → R ∪ {∞} : H θ ( ρ , ρ , u , u ) := (cid:90) ρ ln ρ + θ (cid:90) ρ ln ρ + α (cid:18)(cid:90) Ω |∇ u | − (cid:90) Ω u ρ (cid:19) − θγ (cid:18)(cid:90) Ω |∇ u | − (cid:90) Ω u ρ (cid:19) − β (cid:18)(cid:90) Ω ∇ u · ∇ u − (cid:90) Ω ρ u − ρ u (cid:19) (3.8)8he system (3.6, 3.7) subject to initial data (3.3, 3.5) takes the form δ ∂ρ ∂t = ∇ · (cid:18) ρ ∇ δH θ δρ (cid:19) ; δ ∂ρ ∂t = θ ∇ · (cid:18) ρ ∇ δH θ δρ (cid:19) . (3.9) ε ∂u ∂t = 1 β + αγθ (cid:18) β δH θ δu − θγ δH θ δu (cid:19) , ε ∂u ∂t = 1 β + αγθ (cid:18) β δH θ δu + α δH θ δu (cid:19) . (3.10) i) The limit ε = 0 of system (3.9, 3.10) is reduced into the parabolic-elliptic system (3.6)where (3.7) is replaced by ∆ u i + ρ i = 0 , i = 1 , . (3.11)If we substitute u i = − ∆ − ( ρ i ) in H θ we get H θ ( ρ , ρ ) := (cid:90) ρ ln ρ + θ (cid:90) ρ ln ρ + α (cid:90) ρ ∆ − ( ρ ) − θγ (cid:90) ρ ∆ − ( ρ ) − β (cid:90) ρ ∆ − ( ρ ) . (3.12)Then, (3.6,3.11) takes the form δ ∂ρ ∂t = ∇ · (cid:18) ρ ∇ δH θ δρ (cid:19) ; δ ∂ρ ∂t = θ ∇ · (cid:18) ρ ∇ δH θ δρ (cid:19) . (3.13)ii) The limit ε = δ = 0, δ = 1. Substitute δ = 0 in (3.6) and integrate to obtain ρ = M e − θβu − γu / (cid:82) e − θβu + γu . Hence ∂ρ ∂t = ∆ ρ + ∇ · [ ρ ( β ∇ u − α ∇ u )] ; ∆ u + ρ = 0 ; ∆ u + M e − θβu − γu (cid:82) e − θβu − γu = 0 . (3.14)Let us define now F Mθ on Γ M × H as F Mθ ( ρ, w ) := (cid:90) ρ ln ρ + α (cid:90) ρ ∆ − ( ρ ) − θ (cid:20) γ (cid:90) |∇ w | + M ln (cid:18)(cid:90) e − γw + θβ ∆ − ( ρ ) (cid:19)(cid:21) Then, with ρ := ρ , u := w , (3.14) can be written as ∂ρ∂t = ∇ · ρ ∇ (cid:32) δF M θ δρ (cid:33) ; δF M θ δw = 0 . (3.15)iii) The limit δ = δ = 0 for (3.9, 3.10) is reduced into ∂u ∂t = ∆ u + M e αu − βu (cid:82) Ω e αu − βu ; ∂u ∂t = ∆ u + M e − γu − θβu (cid:82) Ω e − γu − θβu . (3.16)If we substitute (3.19) in H θ and apply integration by parts, we get H M ,M θ ( u , u ) + M ln M + θM ln M where H M ,M θ ( u , u ) := α (cid:90) Ω |∇ u | − θγ (cid:90) Ω |∇ u | − β (cid:90) Ω ∇ u ·∇ u − M ln (cid:18)(cid:90) e αu − βu (cid:19) − θM ln (cid:18)(cid:90) e − γu − θβu (cid:19) . (3.17)9hen (3.16) takes the form ε ∂u ∂t = 1 β + αγθ (cid:18) β δH θ δu − θγ δH θ δu (cid:19) , ε ∂u ∂t = 1 β + αγθ (cid:18) β δH θ δu + α δH θ δu (cid:19) . (3.18) Any critical point of H θ in Γ M × Γ M × ( H ) is also an equilibrium solution of (3.6, 3.7).The variation of H θ with respect to ρ i yieldsln ρ − αu + βu = λ ; θ (ln ρ + γu ) + βu = λ where λ i are the Lagrange multipliers associated with the constraints (cid:82) ρ i = M i . Hence ρ = M e αu − βu (cid:82) Ω e αu − βu ; ρ = M e − γu − θβu (cid:82) Ω e − γu − θβu . (3.19)The variation of H θ with respect to ( u , u ) ∈ ( H ) yields ρ i = − ∆ u i . (3.20)Combining (3.19, 3.20) together we obtain the Liouville type system∆ u + M e αu − βu (cid:82) Ω e αu − βu = 0 ; ∆ u + M e − γu − θβu (cid:82) Ω e − γu − θβu = 0 , u = u = 0 on ∂ Ω . (3.21)It can be verified directly that a solution ( ρ i , u i ) of (3.20, 3.21) yields a steady state solutionof (3.6, 3.7). Proposition 3.1. ( u , u ) is a solution of the Liouville system (3.21) iff either ( u , u ) is acritical point of H M ,M θ in ( H ) or ( ρ , ρ ) , ρ i = − ∆ u i is a critical point of H θ in Γ M × Γ M or ( ρ , u ) is a critical point of F M θ in Γ M × H . From now on we assume the case of conflict θ = −
1. The Liouville system (3.21) takes theform∆ u + M e αu − βu (cid:82) Ω e αu − βu = 0 ; ∆ u + M e − γu + βu (cid:82) Ω e − γu + βu = 0 , u = u = 0 on ∂ Ω . (4.1)Here and thereafter we omit the index θ from H θ , H θ and F Mθ . In particular F M ( ρ, w ) := (cid:90) ρ ln ρ + α (cid:90) ρ ∆ − ( ρ ) + (cid:20) γ (cid:90) |∇ w | + M ln (cid:18)(cid:90) e − γw − β ∆ − ( ρ ) (cid:19)(cid:21) (4.2)10 emma 4.1. sup ρ ∈ Γ M H ( ρ , ρ ) = inf w ∈ H F M ( ρ, w ) − M ln M . Proof.
First note that 12 (cid:90) ρ ∆ − ( ρ ) = inf w ∈ H (cid:90) |∇ w | + (cid:90) ρw .H ( ρ , ρ ) := (cid:90) ρ ln ρ − (cid:90) ρ ln ρ + α (cid:90) ρ ∆ − ( ρ ) + γ (cid:90) ρ ∆ − ( ρ ) − β (cid:90) ρ ∆ − ( ρ ) . ≤ (cid:90) ρ ln ρ − (cid:90) ρ ln ρ + α (cid:90) ρ ∆ − ( ρ )+ γ (cid:18) (cid:90) |∇ w | − (cid:90) ρ w (cid:19) − β (cid:90) ρ ∆ − ( ρ ) := H ( ρ , ρ , w ) , and inf w ∈ H H ( ρ , ρ , w ) = H ( ρ , ρ ). A direct calculation shows thatsup ρ ∈ Γ M (cid:26) − (cid:90) ρ ln ρ − γ (cid:90) ρ w − β (cid:90) ρ ∆ − ( ρ ) (cid:27) = M ln (cid:18)(cid:90) e − γw − β ∆ − ( ρ ) (cid:19) − M ln M . In particular, sup ρ ∈ Γ M H ( ρ, ρ , w ) = F M ( ρ, w ) − M ln M . Hence inf w ∈ H F M ( ρ, w ) − M ln M =inf w ∈ H sup ρ ∈ Γ M H ( ρ, ρ , w ) = sup ρ ∈ Γ M inf w ∈ H H ( ρ, ρ , w ) = sup ρ ∈ Γ M H ( ρ, ρ ) Lemma 4.2.
For any ( u , u ) ∈ ( H ) H M ,M ( u , u ) + M ln M − M ln M =inf ρ ∈ Γ M sup ρ ∈ Γ M H ( ρ , ρ , u , u ) ≡ sup ρ ∈ Γ M inf ρ ∈ Γ M H ( ρ , ρ , u , u ) . (4.3) If, in addition, αγ ≥ β then for any ( ρ , ρ ) ∈ Γ M ,M H ( ρ , ρ ) = inf u ,u ∈ H H ( ρ , ρ , u , u ) . (4.4) Proof.
The equalities in (4.3) follow from the definition of H (3.17) which use (3.19). Indeed,(3.19) are the unique minimizer (maximizer) of H as a function of ρ ( ρ ) where u , u arefixed, since H is strictly convex in ρ and strictly concave in ρ .To get (4.4) note that αγ > β implies that H is strictly convex, jointly in ( u , u ), andthe only minimizer is ∆ u i + ρ i = 0, namely u i = − ∆ − ( ρ i ), i = 1 ,
2. Then (4.4) followsdirectly from definition (3.12). The case of equality αγ = β follows from a simple limitargument. 11 emma 4.3. If αγ ≥ β then inf u ,u ∈ H H M ,M ( u , u ) = inf ρ ∈ Γ M ; w ∈ H F M ( ρ, w ) − M ln M . Proof.
Since H is jointly convex in ( u , u ) and concave in ρ it follows, by the minmaxTheorem, that sup ρ ∈ Γ M inf u ,u ∈ H H ( ρ , ρ , u , u ) = inf u ,u ∈ H sup ρ ∈ Γ M H ( ρ , ρ , u , u ) . (4.5)So inf ρ ∈ Γ M sup ρ ∈ Γ M H ( ρ , ρ ) =inf ρ ∈ Γ M sup ρ ∈ Γ M inf u ,u ∈ H H ( ρ , ρ , u , u ) = inf ρ ∈ Γ M inf u ,u ∈ H sup ρ ∈ Γ M H ( ρ , ρ , u , u )= inf u ,u ∈ H inf ρ ∈ Γ M sup ρ ∈ Γ M H ( ρ , ρ , u , u ) = inf u ,u ∈ H H M ,M ( u , u ) , (4.6)where the first equality from (4.4), the second one from (4.5), the third one is trivial and thelast one follows from (4.3). Lemma 4.4. If αγ > β then H M ,M is a Lyapunov functional for (3.18), that is ddt H M ,M ( u ( · , t ) , u ( · , t )) ≤ where ( u , u ) is a solution of (3.18) in C (cid:16) R + ; (cid:0) H (Ω) (cid:1) (cid:17) . The above equality is strictunless ( u , u ) is a steady state of this system.Proof. of Lemma4.4From (3.10) we get ddt H M ,M = δ u H M ,M ∂u ∂t + δ u H M ,M ∂u ∂t = − εαγ − β (cid:104)(cid:16) βδ u H M ,M + γδ u H M ,M (cid:17) δ u H M ,M + (cid:16) βδ u H M ,M + αδ u H M ,M (cid:17) δ u H M ,M (cid:105) = − εαγ − β (cid:104) γ (cid:107) δ u H M ,M (cid:107) + α (cid:107) δ u H M ,M (cid:107) + 2 β (cid:68) δ u H M ,M , δ u H M ,M (cid:69)(cid:105) ≤ − εαγ − β (cid:104) γ (cid:107) δ u H M ,M (cid:107) + α (cid:107) δ u H M ,M (cid:107) − β (cid:107) δ u H M ,M (cid:107) (cid:107) δ u H M ,M (cid:107) (cid:105) (4.7)where we used Cauchy-Schwartz inequality. Since the quadratic form in δ u i H M ,M is positivedefinite, the last term is non-positive and, in fact, negative unless δ u i H M ,M = 0 for i = 1 ,
2. 12et ¯ F M ( ρ ) := inf w ∈ H F M ( ρ, w ) . (4.8) Definition 4.1.
Let M > , M ≥ . ( M , M ) ∈ Λ if and only if ¯ F M is unbounded frombelow on Γ M . The set where ¯ F M is bounded from below on Γ M is Λ .In the case where Ω is a disc D R := {| x | ≤ R } we denote Γ RM ⊂ Γ M ( D R ) the set of allradial functions in Γ M ( D R ) . Then Λ R (resp. Λ R ) is defined as above for ¯ F M restricted to Γ RM . Proposition 4.1. If ( M , M ) is an interior point of Λ then there exists a minimizer of F M on Γ M × H . If, moreover, αγ > β then this minimizer induces a minimizer of H M ,M on ( H ) as well.Proof. Let q ∈ (0 , γ := γq , ˜ β := β/q . Let ˜Γ defined according to Definition 4.1 withrespect to ˜ F , where˜ F M ( ρ, w ) := (cid:90) ρ ln ρ + α (cid:90) ρ ∆ − ( ρ ) + ˜ γ (cid:90) |∇ w | + M ln (cid:18)(cid:90) e − ˜ γw − ˜ β ∆ − ( ρ ) (cid:19) . We can find such q for which ( ˜ M , ˜ M ) := ( qM , q − M ) ∈ ˜Γ. Set ρ := q ˜ ρ , w = q ˜ w . Then˜ F ˜ M ( ˜ ρ, ˜ w ) := q − (cid:90) ρ ln ρ + α q (cid:90) ρ ∆ − ( ρ ) + ˜ γ q (cid:90) |∇ w | + ˜ M ln (cid:18)(cid:90) e − (˜ γ/q ) w − ˜ βq − ∆ − ( ρ ) (cid:19) + q − M ln q = q − (cid:20)(cid:90) ρ ln ρ + α q (cid:90) ρ ∆ − ( ρ ) + γ (cid:90) |∇ w | + M ln (cid:18)(cid:90) e − γw − β ∆ − ( ρ ) (cid:19)(cid:21) + q − M ln q = q − (cid:20) F M ( ρ, w ) − α − q − ) (cid:90) ρ ∆ − ( ρ ) (cid:21) + q − M ln q . Since ˜ F ˜ M ( ˜ ρ, ˜ w ) > C for some C ∈ R independent of ˜ ρ, ˜ w ∈ Γ ˜ M × H by assumption, itfollows F M ( ρ, w ) ≥ qC − M ln q + α − q − ) (cid:90) ρ ∆ − ( ρ ) (4.9)for any ( ρ, w ) ∈ Γ M × H . Let now ( ρ n , w n ) be a minimizing sequence for F M in Γ M × H .From (4.9), and since q ∈ (0 ,
1) we conclude that (cid:82) ρ n ∆ − ( ρ n ) is bounded uniformly frombelow. Since γ (cid:90) |∇ w n | + M ln (cid:18)(cid:90) e − γw n − β ∆ − ( ρ n ) (cid:19) ≥ γ (cid:90) |∇ w n | + M ln (cid:18)(cid:90) e − γw n (cid:19) is bounded from below as well, we obtain that (cid:82) ρ n ln ρ n is bounded from above. Let ¯ ρ be aweak limit of ρ n in the Zygmund space L ln L . Then (cid:90) ¯ ρ ln ¯ ρ ≤ lim inf n →∞ (cid:90) ρ n ln ρ n .
13n the other hand, ∆ − is a compact operator from L ln L to its dual space L exp , composedof all functions w for which e λ | w | is integrable for some λ ( w ) >
0. Hence v n := − ∆ − ( ρ n )admits a strongly convergent subsequence in L exp , whose limit is ¯ v = − ∆ − ( ¯ ρ ). Hencelim n →∞ (cid:90) ρ n ∆ − ( ρ n ) = (cid:90) ¯ ρ ∆ − ( ¯ ρ ) . Observe that w n are uniformly bounded in the H norm. Let ¯ w be its weak limit. Byembedding of H in L exp we also obtain that e − γw n is a strongly convergent sequence in L p for any p < ∞ , and its limit is e − γ ¯ w . This, and the strong convergence of v n in L exp implythat lim n →∞ (cid:90) e − γw n − β ∆ − ( ρ n ) = (cid:90) e − γ ¯ w − β ∆ − (¯ ρ ) . as well as lim inf n →∞ (cid:90) |∇ w n | ≥ (cid:90) |∇ ¯ w | . In particular it follows that inf ρ ∈ Γ M ,u ∈ H F M ( ρ, u ) = F M ( ¯ ρ, ¯ w ) . The proof for H M ,M in case αγ > β is easier, and is left to the reader. Our object is to characterize the sets Λ and Λ.Let H Mγ ( ρ, w ) := γ (cid:90) |∇ w | + M ln (cid:18)(cid:90) e − γw − β ∆ − ( ρ ) (cid:19) , ¯ H Mγ ( ρ ) = inf w ∈ H H Mγ ( ρ, w ) (4.10)and recall F given by (2.3). By (4.2, 4.10) we get F M ( ρ, w ) = F ( ρ ) + H γ ( ρ, w )and by (4.8, 4.10) ¯ F M ( ρ ) = F ( ρ ) + ¯ H Mγ ( ρ ) (4.11)Note that ¯ H Mγ is bounded from below uniformly in ρ ∈ Γ M . Indeed, since ∆ − ( ρ ) ≤ H Mγ ( ρ, w ) ≥ H Mγ (0 , w ) = γ (cid:90) |∇ w | + M ln (cid:18)(cid:90) e − γw (cid:19) for any w ∈ H . The last expression is bounded from below on H for any M > F M ( ρ ) is bounded from below whenever F is. The Free energy F is bounded frombelow on Γ M for M ≤ π/α . Hence, we expect that Λ contains M ≤ π/α for any M ≥ ρ j ∈ Γ M for which ¯ F M ( ρ j ) → −∞ . Itis enough to establish such a sequence of radial functions in the disc, i.e in Λ R . The evaluationof Λ is more subtle. At this stage we can only investigate Λ R .14 .2 Main results Let Λ( M , M ) := 2( M − M ) − αM π + βM M π − γ M π . (4.12)Note thatΛ( M , M ) = M (cid:18) − βM − γM π (cid:19) + (cid:18) M − αM π + γM π (cid:19) := Λ ( M , M )+Λ ( M , M ) . Theorem 2.
If both Λ( M , M ) < and Λ ( M , M ) < then ( M , M ) ∈ Λ . Theorem 3. If M ≤ π/α then ( M , M ) ∈ Λ for any M > . Let the disc D R := {| z | ≤ R } be our domain, for some R > , and Λ R as in Definition 4.1. Assumea) Λ( M , M ) > and β/α > γM / π + 1 .b) ( M , M ) satisfies (a) and M ≥ M then ( M , M ) ∈ Λ R . We now show that Theorem 3 implies Theorem 1.Note that Λ (8 π/α, M ) > β/α > γM / π + 1 and M >
0. In thatcase, Λ ( M , M ) = 0 intersects M = 8 π/α at M := πγ (cid:16) βα − (cid:17) > β/α > γ = 0 then there is no intersection. In any case, the domain where bothΛ( M , M ) > β/α > γM / π + 1 is contained in the strip M < M as defined in(1.11). By part (b) of the Theorem 3 we observe that, indeed, Λ R contains the domain abovethe lower branch of Λ = 0 in that strip.In the case γ = 0, Theorems 2 and 3 give an (almost) complete description. Indeed,in that case Λ ( M , M ) = 0 iff M = 8 π/α , so the conditions of Theorems 2 and 3 arecomplementary. Without any limitation of generality we may assume that Ω := D is the unit disk {| x | ≤ } .Denote ( f, g ) := (cid:82) D f ( x ) g ( x ) dx for a pair of integrable functions f, g on D . Proof. of Theorem 2:Let ρ = ρ ( | z | ). For ψ ≥ ρ ψ ( r ) = ψ ρ ( ψr ) if r ∈ [0 , /ψ ], ρ ψ = 0 if r ∈ (1 /ψ, w ∈ H and M > w ψM ( r ) := (cid:26) w ( ψr ) − (2 π ) − M ln(1 /ψ ) if 0 ≤ r ≤ /ψ − (2 π ) − M ln( r ) if 1 ≥ r ≥ /ψ Note that under this scaling ∆ − ρ ψ = − w ψM if ∆ w = − ρ and ρ ∈ Γ M . Also w ψM ∈ H forany M > w ∈ H . We obtain for ρ ∈ Γ M (cid:90) D ρ ψ ln ρ ψ = 2 M ln ψ + (cid:90) D ρ ln ρ . (5.1)15∆ − ρ ψ , ρ ψ ) = (∆ − ρ, ρ ) − M π ln ψ , (5.2)and for w ∈ H : (cid:90) D |∇ w ψM | = (cid:90) D |∇ w | + M π ln ψ . (5.3)In addition (cid:90) D e βu ψM − γw ψM = 2 π (cid:32) e ( γM − βM ) ln(1 /ψ ) / π (cid:90) /ψ re βu ( ψr ) − γw ( ψr ) + (cid:90) /ψ r γM − βM ) / π dr (cid:33) = (cid:20) π (cid:90) re βu ( r ) − γw ( r ) + O (1) (cid:21) ψ − βM − γM ) / π + O (1) (5.4)It follows from (4.2, 5.1-5.4) that if − βM − γM ) / (2 π ) > ( M , M ) >
0) then F M ( ρ ψ , w ψM ) = F M ( ρ, w ) + O (1)+ (cid:20) M − M ) − αM π + βM M π − γM π (cid:21) ln ψ ≡ F M ( ρ, w ) + O (1) + Λ( M , M ) ln ψ (5.5)Letting ψ → ∞ we obtain a blow-down sequence for F M ( ρ ψ , w ψ ) where ( ρ ψ , w ψ ) ∈ Γ M × H ,provided Λ( M , M ) < ( M , M ) < F M ( ρ ψ , w ψM ) = F M ( ρ, w )+ (cid:18) M − αM π + γM π (cid:19) ln ψ + O (1) ≡ F M ( ρ, w )+Λ ( M , M ) ln ψ + O (1) (5.6)and the same holds if Λ ( M , M ) < γ > Lemma 5.1.
For ψ ∈ (0 , , γ > , let v = v ( r ) be a solution of r − ( rv r ) r + r − βM/ π e − γv = 0 ; ψ ≤ r ≤ satisfying v r ≤ on the interval [ ψ, and v r (1) = − M / π . If βM − γM π − > then lim ψ → ln − (cid:18) √ ψ (cid:19) (cid:90) √ ψ r | v r | dr = (cid:18) M π (cid:19) . Remark 5.1.
Note that we cannot give up the condition γ > . Indeed, if γ = 0 then thesolution of (5.7) does not satisfy v r ≤ on [ ψ, under the stated condition βM > π , if ψ > is small enough. roof. Under the change of variables: r → e − t we get that (5.7) is transformed toˆ v tt + e ( βM/ π − t − γ ˆ v = 0 , ≤ t ≤ ln(1 /ψ )for ˆ v ( t ) := v ( e − t ). The end point r = 1 are transformed into t = 0 andˆ v t (0) = M / π Setting now ¯ v ( t ) := ˆ v ( t ) − γ − ( βM/ (2 π ) − t (5.8)we get ¯ v tt + e − γ ¯ v = 0 (5.9)and ¯ v t (0) = − γ − (cid:20) βM − γM π − (cid:21) . (5.10)From (5.9) it follows that | ¯ v t | / − γ − e − γ ¯ v := E is an invariant, so¯ v t = ± (cid:112) E + γ − e − γ ¯ v ) . for some constant E . The assumption βM − γM π − > − sign above,so ¯ v t = − (cid:112) E + γ − e − γ ¯ v ) (5.11)The exact solution of (5.11) which is defined on the interval [0 , ln(1 /ψ )) and blows down at t = ln(1 /ψ ) is¯ v ( t ) = − γ − ln(4 Eγ ) − √ E ( t + ln( ψ )) + 2 γ ln (cid:16) − e √ Eγ ( t +ln( ψ )) (cid:17) . (5.12)Substitute condition (5.10) in this solution implies √ E coth (cid:32)(cid:114) E γ ln(1 /ψ ) (cid:33) = γ − (cid:20) βM − γM π − (cid:21) . In the limit ψ → √ E → γ − (cid:104) βM − γM π − (cid:105) . From (5.12) we obtain that ¯ v t converges uniformly on [0 , ln(1 / √ ψ )] to − γ − (cid:104) βM − γM π − (cid:105) , as ψ → v t converges uniformly on [0 , ln(1 / √ ψ )] to M / π .Returning to the variable r = e − t , recalling v ( r ) := ˆ v ( − ln( r )) we get that for any solution v of (5.7) which satisfies the conditions of the lemma, the function rv r converges uniformlyon [ √ ψ,
1] to − M / π , as ψ →
0. Hence (cid:90) √ ψ r | v r | dr = (cid:90) √ ψ r − ( r | v r | ) dr = ( M / π ) ln(1 / (cid:112) ψ ) + o (ln(1 /ψ ))at ψ <<
1. 17 roof. of Theorem 3The bound of ¯ F M from below on Γ M for M ≤ π/α follows from the bound from below of F on Γ π/α . See (4.11) and the discussion below (2.5). This concludes the first alternative ofthe Theorem. We assume, from now on, that M ≥ π/α .Assume alternative (a), i.e. Λ( M , M ) >
0, 2 β/α > γM / π + 1 (so Λ (8 π/α, M ) > γ >
0. For the case γ = 0 see Remark 5.2 at the end of this proof.Note that Λ (8 π/α, M ) ≥
0. Hence Λ(8 π/α, M ) := Λ (8 π/α, M ) + Λ (8 π/α, M ) > M , M ) > s → Λ ( s, M ) is linear and s → Λ( s, M ) isconcave, then both Λ( s, M ) > ( s, M ) > π/α ≤ s ≤ M :Λ ( s, M ) > ∀ s ∈ [8 π/α, M ] . (5.13)Let δ := ( M − π/α ) /n where n is so large, for whichmin π/α ≤ s ≤ M Λ( s, M ) > αδM π . (5.14)We shall prove that if ¯ F M is unbounded from below on Γ Rs where s ∈ [8 π/α + δ, M ] then itis still unbounded from below on Γ Rs − δ . Iterating this argument n times we obtain that ¯ F M is unbounded from below on Γ R π/α and get a contradiction.So let s in this interval and { ρ j } ∈ Γ s a blow down sequence (e.g. ¯ F M ( ρ j ) < − j ).Choose ψ j ∈ (0 ,
1) such that (cid:82) D ψj ρ j = s − δ . Set ρ j ∈ Γ s − δ which is the restriction of ρ j to D ψ j := {| x | ≤ ψ } , that is: ρ j ( r ) = ρ j ( r ) for r ∈ [0 , ψ j ] , ρ j ( r ) = 0 for r ∈ ( ψ j , . Our first step is to show ¯ F M ( ρ j ) ≤ ¯ F M ( ρ j ) + πe − − αδs π ln ψ j . (5.15)Since the function s → − s ln s is bounded from above by e − it follows that (cid:90) D ρ j ln ρ j − (cid:90) D ρ j ln ρ j = − (cid:90) D − D ψj ρ j ln ρ j ≤ πe − . (5.16)Since ∆ − ≤ D and ρ j ≤ ρ j then ∆ − ρ j ≥ ∆ − ρ j (same reasoning can be appliedvia the maximum principle). Then, since β > H M γ ( ρ j ) ≤ ¯ H M γ ( ρ j ) . (5.17)Finally, to estimate the difference ( ρ j , ∆ − ρ j ) − ( ρ j , ∆ − ρ j ) we observe( ρ j , ∆ − ρ j ) − ( ρ j , ∆ − ρ j ) = 2( ρ j − ρ j , ∆ − ρ j ) + ( ρ j − ρ j , ∆ − ρ j − ∆ − ρ j ) . (5.18)18ince ρ j − ρ j is supported in the ring 1 ≥ r ≥ ψ j we obtain from (5.21)2( ρ j − ρ j , ∆ − ρ j ) ≥ s ( s − δ )2 π ln ψ j . (5.19)The second term is the (negative) energy due to a mass concentrated in the ring r ∈ [ ψ j , r = ψ j . The potential inducedby the mass δ concentrated on this circle is just δ/ (2 π ) ln ψ j , so the energy is bounded frombelow by δ / (2 π ) ln ψ j . Hence( ρ j , ∆ − ρ j ) − ( ρ j , ∆ − ρ j ) ≥ δs π ln ψ j . (5.20)Summarizing (5.16-5.20) and using (4.11) we obtain (5.15).Note that potential u j = − ∆ − ρ j satisfies u j ( r ) = s − δ π ln (cid:18) r (cid:19) for 1 ≥ r ≥ ψ j . (5.21)Set now ρ ψ j ( r ) = ψ j ρ ( (cid:112) ψ j r ) for r ∈ [0 , ρ ψ j j is supported on the disc or radius √ ψ . Evidently, ρ ψ j ∈ Γ s − δ as well.For ρ ψ j j as above, (5.1, 5.2) imply (cid:90) D ρ ψ j j ln ρ ψ j j = 2 M ln (cid:112) ψ j + (cid:90) D ρ j ln ρ j (5.22)(∆ − ρ ψ j j , ρ ψ j j ) = (∆ − ρ j , ρ j ) − M π ln (cid:112) ψ j . (5.23)We obtained F ( ρ ψ j j ) = F ( ρ j ) + (cid:18) M − αM π (cid:19) ln (cid:112) ψ j . (5.24)Next we estimate ¯ H M γ ( ρ ψ j j ) in terms of ¯ H M γ ( ρ j ).Set w j ∈ H ( D ) to be the solution of∆ w j + M (cid:82) D e γw j + βu j e − γw j + βu j = 0 . (5.25)Recall that w j is the minimizer of H M γ ( ρ j , w ) (see 4.10)). In particular¯ H M γ ( ρ j ) = H M γ ( ρ j , w j ) . (5.26)The function w j is a radial function on the interval [0 , r r − ( rw j,r ) r + λe − γw j + βu j = 0 , (5.27)where λ > w j,r (1) = − M / (2 π ). Since w j,r (0) = 0it follows w j,r ≤ , w j is19dentified, up to an additive constant, with a solution v of (5.7) on the annulus ψ j ≤ r ≤ M = s − δ . Recalling (5.13), we obtain from Lemma 5.1 γ (cid:90) √ ψ ≤| x |≤ |∇ w j | = γM π ln(1 / (cid:112) ψ ) + o ( | ln ψ | ) . (5.28)The potential corresponding to ρ ψ j j is u ψ j j ( r ) = u j ( (cid:112) ψ j r ) + (( s − δ ) / π ) ln (cid:112) ψ j . Define also w ψ j j ( r ) = w j ( (cid:112) ψ j r ) − w j ( (cid:112) ψ j ) ∈ H ( D ). As in (5.4 ) (with (cid:112) ψ j replacing ψ j ) we obtain (cid:90) D e βu ψjj − γw ψjj = 2 πe ( β ( s − δ ) ln √ ψ j / π + γw j ( √ ψ j )) (cid:90) re βu j ( √ ψ j r ) − γw j ( √ ψ j r ) dr = 2 πe ( β/ (2 π )( s − δ ) ln √ ψ j + γw j ( √ ψ j ) − √ ψ j ) (cid:90) √ ψ j re βu j ( r ) − γw j ( r ) dr ≤ πe ( β ( s − δ ) / (2 π ) ln √ ψ j + γw j ( √ ψ j ) − √ ψ j ) (cid:90) re βu j ( r ) − γw j ( r ) dr (5.29)It follows that M ln (cid:18)(cid:90) D e βu ψjj − γw ψjj (cid:19) ≤ M ln (cid:18)(cid:90) D e βu j − γw j (cid:19) + M (cid:110) [ β ( s − δ ) / (2 π ) −
2] ln (cid:112) ψ j + γw j ( (cid:112) ψ j )) (cid:111) . (5.30)Recall that w j is the solution of (5.27) satisfying w j (1) = 0 , w j,r (1) = − M / (2 π ). In par-ticular ( rw j,r ) r < , w j ( r ) ≤ − ( M / π ) ln( r ) for any r ∈ [0 , β ( s − δ ) / (2 π ) −
2] ln (cid:112) ψ j + γw j ( (cid:112) ψ j )) ≤ (cid:18) β ( s − δ )2 π − γM π − (cid:19) ln (cid:112) ψ j , so M ln (cid:18)(cid:90) D e βu ψjj − γw ψjj (cid:19) ≤ M ln (cid:18)(cid:90) D e βu j − γw j (cid:19) + M (cid:18) β ( s − δ )2 π − γM π − (cid:19) ln (cid:112) ψ j . (5.31)Next, γ (cid:90) D (cid:12)(cid:12)(cid:12) ∇ w ψ j j (cid:12)(cid:12)(cid:12) = πγ (cid:90) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dw ψ j j dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) rdr = πγ (cid:90) (cid:12)(cid:12)(cid:12) w j (cid:48) ( (cid:112) ψ j r ) (cid:12)(cid:12)(cid:12) ψ j rdr = πγ (cid:90) √ ψ j (cid:12)(cid:12) w j (cid:48) ( r ) (cid:12)(cid:12) rdr = πγ (cid:90) (cid:12)(cid:12) w j (cid:48) ( r ) (cid:12)(cid:12) rdr − γπ (cid:90) √ ψ j (cid:12)(cid:12) w j (cid:48) ( r ) (cid:12)(cid:12) rdr = γ (cid:90) D (cid:12)(cid:12) ∇ w j (cid:12)(cid:12) + γM π ln (cid:112) ψ j + o ( | ln ψ j | ) (5.32)20here we used (5.28) in the last equality.From (4.10, 5.26, 5.31, 5.32) we obtain¯ H M γ ( ρ ψ j j ) ≤ H M γ ( ρ ψ j j , w ψ j j ) ≤ H M γ ( ρ j , w j )+ M (cid:18) β ( s − δ )2 π − γM π − (cid:19) ln( (cid:112) ψ j )+ o (ln( ψ j ))= ¯ H M γ ( ρ j ) + M (cid:18) β ( s − δ )2 π − γM π − (cid:19) ln( (cid:112) ψ j ) + o (ln( ψ j )) (5.33)This and (5.24), together with the definition (4.12) of Λ, imply¯ F M ( ρ ψ j j ) ≤ ¯ F M ( ρ j ) + [Λ( s − δ, M ) + o (1)] ln (cid:112) ψ j . Using this in (5.15):¯ F M ( ρ ψ j j ) ≤ ¯ F M ( ρ j ) + (cid:18) Λ( s − δ, M ) − αδsπ + o (1) (cid:19) ln (cid:112) ψ j + πe − . Since ψ j ∈ (0 ,
1) it follows by (5.14) that F M ( ρ ψ j j ) ≤ F M ( ρ j ) + πe − , as long as s − δ ≤ M .In particular lim j →∞ F M ( ρ ψ j j ) = −∞ if lim j →∞ F M ( ρ j ) = −∞ . Recalling ρ j ∈ Γ s while ρ ψ j j ∈ Γ s − δ we obtain the desired result by n iteration, as explained below (5.14). Remark 5.2.
In the case γ = 0 , (5.32) is reduced to the trivial identity , while H M ( ρ, w ) is independent on w , so we may take w j = 0 and ¯ H M ( ρ j ) = H M ( ρ j , . Theinequality (5.33) holds with this substitution and the result follows as well. Note that we donot apply Lemma 5.1 in that case. We finally turn to the proof of part (b): By (4.10) H Mγ ( ρ, w ) − H M γ ( ρ, w ) = ( M − M ) ln (cid:18)(cid:90) e − γw − β ∆ − ( ρ ) (cid:19) ≥ ( M − M ) ln (cid:18)(cid:90) e − γw (cid:19) since ∆ − ρ ≤ M ≥ M . If γ = 0 then (4.10, 4.11) imply that F M is bounded frombelow on Γ M is F M is. Otherwise, Jensen, Poincare and Caushy-Schwartz inequalities implythe existence of a constant C ( ε ) > M − M ) ln (cid:18)(cid:90) e − γw (cid:19) ≥ − ε (cid:107)∇ w (cid:107) − C ( ε )for any ε > w ∈ H . Hence H Mγ ( ρ, w ) ≥ H M γ ( ρ, w ) − ε (cid:107)∇ w (cid:107) − C ( ε )for any ( ρ, w ) ∈ Γ M × H . Scaling w (cid:55)→ γw we obtain from (4.10)¯ H Mγ ( ρ ) ≥ ¯ H M ˆ γ ( ρ ) − C ( ε )where ˆ γ := γ − ε/γ . From this and (4.11) we obtain that ¯ F Mγ := F + ¯ H Mγ is bounded frombelow on Γ M if ¯ F M ˆ γ := F + ¯ H M ˆ γ is. Since the conditions of (a), determined by stronginequalities, are preserved under the change γ (cid:55)→ ˆ γ for ε > Sharp Sobolev inequalities on the sphere and the MoserTrudinger inequal-ity . Ann. of Math. (2) 138, 213-242 (1993)2. Blanchet, A., Carlen, E.A. and Carrillo, J.A.:
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