The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Existence of solutions for the Toda System via variational methods. David Ruiz Departamento de Análisis Matemático (Universidad de Granada, Spain)

joint work with Andrea Malchiodi (SISSA).

The problem

The scalar case

Center of mass and rate of concentration

Outline

1

The problem

2

The scalar case

3

Center of mass and rate of concentration

4

A Moser-Trudinger inequality

5

End of the proof

A Moser-Trudinger inequality

End of the proof

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

The Toda System In this talk we study existence of solutions for the Toda system:  −∆u1 = 2ρ1 (h1 eu1 − 1) − ρ2 (h2 eu2 − 1) in Σ, −∆u2 = 2ρ2 (h2 eu2 − 1) − ρ1 (h1 eu1 − 1) in Σ, where ∆ is the Laplace-Beltrami operator, ρi > 0, hi (x) > 0, and Σ is ´ a compact surface with Σ 1dVg = 1. This problem has a close relationship with geometry. Moreover, it arises in the study of the non-abelian Chern-Simons theory. M. A. Guest, Harmonic maps, loops groups, and integrable systems. Cambridge Univ. Press, 1997. G. Dunne, Self-dual Chern-Simons Theories, Lecture Notes in Physics, Springer-Verlag, 1995. G. Tarantello, Self-Dual Gauge Field Vortices: An Analytical Approach, PNLDE 72, Birkhäuser 2007.

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

By integrating on Σ both equations, we get that Hence, our problem is equivalent to:

´ Σ

End of the proof

hi eui dVg = 1.

      −∆u1 = 2ρ1 ´ h1 euu1 − 1 − ρ2 ´ h2 euu2 − 1 in Σ, 1 2 h e dV h e dV g g  Σ 1   Σ 2   −∆u2 = 2ρ2 ´ h2 euu2 − 1 − ρ1 ´ h1 euu1 − 1 in Σ. h2 e 2 dVg h1 e 1 dVg Σ

(1)

Σ

The solutions of (1) correspond to critical points of the functional: ˆ Jρ (u1 , u2 ) =

Q(u1 , u2 ) dVg + Σ

2 X i=1

ˆ

ˆ

ui

ui dVg − log

ρi Σ

hi e dVg , Σ

where ρ = (ρ1 , ρ2 ) and Q(u1 , u2 ) is defined as: Q(u1 , u2 ) =




1 |∇u1 |2 + |∇u2 |2 + ∇u1 · ∇u2 . 3

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

It is known that if both ρi ≤ 4π, then Jρ is bounded from below. In particular, if ρi < 4π, Jρ is coercive and achieves its minimum. J. Jost and G. Wang, Analytic aspects of the Toda system I. A Moser-Trudinger inequality, CPAM 2001. If we make now ρ1 → 4π, the solution (u1 , u2 ) could exhibit a blow-up behavior. In such case, ! 4λ 1 u1 ∼ Uλ,x (y ) = log , u2 ∼ − u1 . 2 2 2 (1 + λ d(x, y ) ) where y ∈ Σ, d(x, y ) stands for the geodesic distance and λ is a large parameter. Those functions Uλ,x are the unique entire solutions of the Liouville equation in R2 : ˆ U −∆U = 2e , eU dx < +∞. R2

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Under certain conditions on h1 and K (x), blow-up does not appear, and then a solution is found for ρ1 = 4π, ρ2 < 4π. J. Jost, C. S. Lin and G. Wang, Analytic aspects of the Toda system II. Bubbling behavior and existence of solutions, CPAM 2006. J. Li and Y. Li, Solutions for Toda systems on Riemann surfaces, Ann. Sc. Norm. Super. Pisa 2005. Moreover, in [1] it is proved that the set of solutions is compact for ρi ∈ / 4πN. The Leray-Schauder degree is not yet known, though.

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Under certain conditions on h1 and K (x), blow-up does not appear, and then a solution is found for ρ1 = 4π, ρ2 < 4π. J. Jost, C. S. Lin and G. Wang, Analytic aspects of the Toda system II. Bubbling behavior and existence of solutions, CPAM 2006. J. Li and Y. Li, Solutions for Toda systems on Riemann surfaces, Ann. Sc. Norm. Super. Pisa 2005. Moreover, in [1] it is proved that the set of solutions is compact for ρi ∈ / 4πN. The Leray-Schauder degree is not yet known, though.

Theorem Assume ρi ∈ (4π, 8π). Then Jρ has a critical point. The only previous result in this direction considers ρ1 < 4π, ρ2 ∈ / 4πN: A. Malchiodi and C. B. Ndiaye, Some existence results for the Toda system on closed surfaces, Atti Accad. Naz. Lincei 2007.

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

A glance to the scalar case The scalar counterpart of (1) is a Liouville-type problem in the form:   h(x)eu − ∆u = 2ρ ´ − 1 in Σ, (2) h(x)eu dVg Σ with ρ ∈ R. This equation has been very much studied. Solutions of (2) correspond to critical points of the functional: ˆ  ˆ ˆ 1 2 u Iρ (u) = |∇g u| dVg + 2ρ udVg − log h(x)e dVg . 2 Σ Σ Σ

(3)

The Moser-Trudinger inequality implies that Iρ is bounded from below for ρ ≤ 4π. Existence for any ρ ∈ / 4πN has been proved in: Z. Djadli and A. Malchiodi, Existence of conformal metrics with constant Q-curvature, Annals of Math. 2008.

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Let us describe the proof if ρ ∈ (4π, 8π). The first tool is a local version of the Moser-Trudinger inequality:

Proposition Let Ω ⊂ Ω0 two open domains in Σ, with dist(Ω, ∂Ω0 ) = δ > 0. Then, for any ε > 0 there exists a constant C = C(C0 , ε, δ) such that ˆ ˆ 1 log e2u dVg ≤ |∇u|2 dVg + 2 udVg + C. 4π − ε 0 Ω Ω Ω0 As a consequence, we have:

Proposition Let Ω1 , Ω2 be subsets of Σ with dist(Ω1 , Ω2 ) ≥ δ > 0, and assume that ˆ ˆ e2u dVg ≥ γ e2u dVg , i = 1, 2, Ωi

Σ

for γ > 0. Then, given ρ < 8π − ε, there exists C = C(ε, δ, γ) such that Iρ (u) ≥ −C.

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

Last proposition implies that if Iρ (un ) → −∞, then ´

e un * δx , x ∈ Σ. eun dVg Σ

This allows us to define a continuous map (for L >> 1): Ψ : Iρ−L = {u ∈ H 1 (Σ) : Iρ (u) < −L} → Σ. Moreover, one can define the map Φ : Σ → H 1 (Σ), Ψ(y ) = uy ∼ log

!

4λ 2

(1 + λ d(x, y )2 )

.

One can show that if λ is chosen large enough, Φ(y ) ∈ Iρ−L .

End of the proof

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Finally, one shows that Ψ ◦ Φ is homotopically equivalent to the identity. Since Σ is not contractible, then Φ(Σ) is not contractible in Iρ−L . This allows us to use a min-max scheme. Let us define the cone C = {tu : u ∈ Φ(Σ)}. Moreover, we define: Γ = {η : C → H 1 (Σ) continuous : η|∂Φ(Σ) = Id}. For any η ∈ Γ, Φ(Σ) is contractible in η(C), and hence η(C) * Iρ−L . This implies that: m = infη∈Γ maxu∈C Iρ ◦ η(u) ≥ −L. However, we cannot conclude directly since the (PS) property of Iρ is not known. But the existence of bounded (PS) sequences for almost all values of ρ, together with the compactness of solutions, allow us to conclude.

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Proposition Let Ω ⊂ Ω0 two open domains in Σ, with dist(Ω, ∂Ω0 ) = δ > 0. Then, for any ε > 0 there exists a constant C = C(C0 , ε, δ) such that 2 X i=1

log

ˆ

ui

e dVg

Ω



(1 + ε) − ui ≤ 4π Ω0

ˆ Q(u1 , u2 )dVg + C. Ω0

As in the scalar case, this implies that if Jρ (u1,n , u2,n ) → −∞, euin * δx , x ∈ Σ, for some i. euin dVg Σ

´

In other words, at least one component concentrates around a point. Moreover, one can construct examples in which both components concentrate around different points. Even more, one can construct sequences with both components concentrating at the same point, but at different rates.

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

We define:  A=

ˆ 1

f ∈ L (Σ) : f > 0 a. e. and

 f dVg = 1 ,

Σ

Moreover, given δ > 0, we define the cone over Σ: Σδ = (Σ × (0, +∞)) |(Σ×[δ,+∞)) .

(4)

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

We define:  A=

ˆ 1

f ∈ L (Σ) : f > 0 a. e. and

 f dVg = 1 ,

Σ

Moreover, given δ > 0, we define the cone over Σ: Σδ = (Σ × (0, +∞)) |(Σ×[δ,+∞)) .

Proposition Given R > 1, there exists δ = δ(R) > 0 and a continuous map: ψ : A → Σδ ,

ψ(f ) = (β, σ),

satisfying that for any f ∈ A there exists p ∈ Σ: a) If σ < δ, d(p, β) ≤ Cσ, C = C(R, Σ). ˆ ˆ b) f dVg > τ , f dVg > τ, with τ = τ (R, Σ) > 0. Bp (σ)

Bp (Rσ)c

(4)

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

Define R0 = 3R and σ : A × Σ → (0, +∞) such that: ˆ ˆ f dVg = f dVg . Bx (R0 σ(x,f ))c

Bx (σ(x,f ))

Moreover, we define: ˆ T : A × Σ → R,

T (x, f ) =

fdVg . Bx (σ(x,f ))

We now give the definition of σ: σ : A → R,

σ(f ) = 3min{σ(x, f ) : x ∈ Σ}.

End of the proof

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Define R0 = 3R and σ : A × Σ → (0, +∞) such that: ˆ ˆ f dVg = f dVg . Bx (R0 σ(x,f ))c

Bx (σ(x,f ))

Moreover, we define: ˆ T : A × Σ → R,

T (x, f ) =

fdVg . Bx (σ(x,f ))

We now give the definition of σ: σ : A → R,

σ(f ) = 3min{σ(x, f ) : x ∈ Σ}.

A geometrical argument provides that there exists τ > 0 such that S(f ) = {x ∈ Σ : T (x, f ) > τ, σ(x, f ) < σ(f )} , is always non-empty. Furthermore, diam(S(f )) ≤ Cσ(f ). We take p ∈ S(f ). However, such a choice could be discontinuous.

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Assume that Σ ⊂ RN isometrically. Take a tubular neighborhood of Σ and denote by Π the projection onto Σ. If σ is small, we can define: ˆ  + + (T (x, f ) − τ ) (σ(f ) − σ(x, f )) x dV g  Σ . ˆ β(f ) = Π    + + (T (x, f ) − τ ) (σ(f ) − σ(x, f )) dVg Σ

The important thing is that β is continuous and that dist(β(f ), S(f )) ≤ Cσ(f ).

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Assume that Σ ⊂ RN isometrically. Take a tubular neighborhood of Σ and denote by Π the projection onto Σ. If σ is small, we can define: ˆ  + + (T (x, f ) − τ ) (σ(f ) − σ(x, f )) x dV g  Σ . ˆ β(f ) = Π    + + (T (x, f ) − τ ) (σ(f ) − σ(x, f )) dVg Σ

The important thing is that β is continuous and that dist(β(f ), S(f )) ≤ Cσ(f ). Observe that β cannot be defined for larger σ: this is the reason of the identification to a cone. This, together with the definition of σ as a continuous map, is the main novelty with respect to the definitions of: A. Malchiodi and D. Ruiz, New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces, to appear in GAFA.

The problem

The scalar case

Center of mass and rate of concentration

Given u ∈ H 1 (Σ), we define f = ´

A Moser-Trudinger inequality

End of the proof

eu . eu dVg Σ

Proposition If ρi < 8π − ε, there exist R = R(ε) > 1 and ψ : A → Σδ as defined before such that: ψ(f1 ) = ψ(f2 ) ⇒ Jρ (u1 , u2 ) ≥ −C for some C = C(ε).

The problem

The scalar case

Center of mass and rate of concentration

Given u ∈ H 1 (Σ), we define f = ´

A Moser-Trudinger inequality

End of the proof

eu . eu dVg Σ

Proposition If ρi < 8π − ε, there exist R = R(ε) > 1 and ψ : A → Σδ as defined before such that: ψ(f1 ) = ψ(f2 ) ⇒ Jρ (u1 , u2 ) ≥ −C for some C = C(ε). There exists pi such that dist(pi , β) ≤ Cσ and: ˆ ˆ ui e dVg ≥ τ eui dVg , i = 1, 2; ˆ

Bp1 (σ)

Σ

ˆ eui dVg ≥ τ

Bp2 (Rσ)c

eui dVg , i = 1, 2. Σ

If σ ≥ δ, we are done. If not, assume for simplicity that p1 = p2 := p.

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Lemma (via dilation) For any ε > 0 there exists C = C(ε) > 0 such that ˆ Q(u1 , u2 ) dVg + C

(1 + ε)

≥

Bp (s)

4π

2 X i=1

−

ˆ

! ui

log

e dVg Bp (s/2)

¯1 (s) + u ¯2 (s) + 4 log s), 4π(u

¯i (s) = for any ui ∈ H 1 (Σ), p ∈ Σ, s > 0 small and for u

ui dVg . Bp (s)

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Lemma (via dilation) For any ε > 0 there exists C = C(ε) > 0 such that ˆ Q(u1 , u2 ) dVg + C

(1 + ε)

≥

4π

Bp (s)

2 X

ˆ log

i=1

−

! ui

e dVg Bp (s/2)

¯1 (s) + u ¯2 (s) + 4 log s), 4π(u

¯i (s) = for any ui ∈ H 1 (Σ), p ∈ Σ, s > 0 small and for u

ui dVg . Bp (s)

Lemma (via Kelvin Transform) Given ε > 0, r > 0, ∃ C = C(ε, r ) > 0, such that if ui = 0 in ∂Bp (2r ), ˆ Q(u1 , u2 ) dVg + C

(1 + ε) Ap (s/2,2r )

≥ 4π

2 X i=1

ˆ

! eui dVg

log Ap (s,r )

¯1 (s) + u ¯2 (s) + 4 log s). + 4π(u

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Therefore, we have: ˆ Q(u1 , u2 ) dVg + C

(1 + ε)

≥ 4π

Bp (2σ)

2 X

ˆ

! eui dVg

log Bp (σ)

i=1

¯1 (2σ) + u ¯2 (σ) + 4 log 2σ), − 4π(u

ˆ Q(u1 , u2 ) dVg + C

(1 + ε) Ap (Rσ/2,2r )

≥ 4π

2 X i=1

ˆ

! eui dVg

log Ap (Rσ,r )

¯1 (Rσ) + u ¯2 (Rσ) + 4 log(Rσ)) + 4π(u

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Therefore, we have: ˆ Q(u1 , u2 ) dVg + C

(1 + ε)

≥ 4π

Bp (2σ)

2 X

ˆ

! eui dVg

log Σ

i=1

¯1 (2σ) + u ¯2 (σ) + 4 log 2σ), − 4π(u

ˆ Q(u1 , u2 ) dVg + C

(1 + ε) Ap (Rσ/2,2r )

≥ 4π

2 X i=1

ˆ

! eui dVg

log Σ

¯1 (Rσ) + u ¯2 (Rσ) + 4 log(Rσ)) + 4π(u

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Therefore, we have: ˆ ≥ 4π

Q(u1 , u2 ) dVg + C

(1 + ε) Bp (2σ)

2 X

ˆ

! eui dVg

log Σ

i=1

¯1 (2σ) + u ¯2 (σ) + 4 log 2σ), − 4π(u

ˆ Q(u1 , u2 ) dVg + C

(1 + ε)

≥ 4π

Ap (Rσ/2,2r )

2 X

ˆ eui dVg

log

i=1

!

Σ

¯1 (Rσ) + u ¯2 (Rσ) + 4 log(Rσ)) + 4π(u By adding the above expressions, we conclude the proof, taking into account that: ˆ ¯1 (2σ) − u ¯1 (Rσ)| ≤ C |u

|∇u| Σ

2

ˆ

1/2

|∇u|2 + C.

≤ε Σ

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

Therefore, if L >> 1, we have a map:   Ψ : Jρ−L → Σδ × Σδ \ D, where D is the diagonal set and Ψ is defined as: Ψ(u1 , u2 ) = (ψ(f1 ), ψ(f2 )).

End of the proof

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Therefore, if L >> 1, we have a map:   Ψ : Jρ−L → Σδ × Σδ \ D, where D is the diagonal set and Ψ is defined as: Ψ(u1 , u2 ) = (ψ(f1 ), ψ(f2 )).  Let us call X = Σδ × Σδ \ D. Given ν > 0, we can define Xν ⊂ X satisfying the following properties: 

1

Xν is a retraction of X .

2

dist(Xν , D) > δ 2 .

3

If (x1 , t1 , x2 , t2 ) ∈ Xν , then min{t1 , t2 } ≤ ν.

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Then we define the map Φ : Xν → Jρ−L , Φ(x1 , t1 , x2 , t2 ) = (ϕ1 , ϕ2 ),

ϕ1 (y ) = log

1 + λ22 d(x2 , y )2
2 , 1 + λ21 d(x1 , y )2

with  λi = λi (ti ) =

ϕ2 (y ) = log

1 ti ,

− δ42 (ti

− δ)

1 + λ21 d(x1 , y )2
2 , 1 + λ22 d(x2 , y )2

for ti ≤ 2δ , for ti ≥ 2δ .

One can check that (ϕ1 , ϕ2 ) ∈ Jρ−L for a convenient choice of ν > 0.

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

Then we define the map Φ : Xν → Jρ−L , Φ(x1 , t1 , x2 , t2 ) = (ϕ1 , ϕ2 ),

ϕ1 (y ) = log

1 + λ22 d(x2 , y )2
2 , 1 + λ21 d(x1 , y )2

with  λi = λi (ti ) =

ϕ2 (y ) = log

1 ti ,

− δ42 (ti

− δ)

1 + λ21 d(x1 , y )2
2 , 1 + λ22 d(x2 , y )2

for ti ≤ 2δ , for ti ≥ 2δ .

One can check that (ϕ1 , ϕ2 ) ∈ Jρ−L for a convenient choice of ν > 0. Moreover, let us set (β1 , σ1 , β2 , σ2 ) = Ψ((ϕ1 , ϕ2 )). Then one can deform continuously (βi , σi ) on (xi , ti ) in X .

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

The min-max scheme works as in the scalar case, provided that:

Proposition   The set X = Σδ × Σδ \ D is not contractible. If Σ = S2 , then it is easy to show that X ∼ S2 .

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

End of the proof

The min-max scheme works as in the scalar case, provided that:

Proposition   The set X = Σδ × Σδ \ D is not contractible. If Σ = S2 , then it is easy to show that X ∼ S2 . For other surfaces we have a less precise description on X . However, there holds:

Proposition If the genus of Σ is bigger than one, then H 4 (X , R) is nontrivial.

The problem

The scalar case

Center of mass and rate of concentration

A Moser-Trudinger inequality

Thank you for your attention!

End of the proof