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Uniqueness of Mean field equations: Known results Open problems and Applications

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31 Pages
English

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Uniqueness of Mean field equations: Known results, Open problems and Applications. Chang-Shou Lin Department of Mathematics National Taiwan University Chang-Shou Lin Uniqueness of Mean field equations:Known results, Open problems and Applications.

  • mean field

  • inequality says

  • chang-shou lin

  • bounded domains

  • ?? inf

  • mathematics national


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Uniqueness of Mean field equations:
Known results, Open problems and Applications.
Chang-Shou Lin
Department of Mathematics
National Taiwan University
Chang-Shou Lin Uniqueness of Mean field equations:Known results, Open problems and Applications.1. Introduction
2. Uniqueness for bounded domains
3. Existence at 8π
24. Uniqueness for S
25. Uniqueness for T
Chang-Shou Lin Uniqueness of Mean field equations:Known results, Open problems and Applications.1.Introduction
2 1Let Ω be a bounded domain ofR , and H (Ω) = the completion0
1of C (Ω)under the norm0
Z
12
2k∇uk = ( |∇u| dx)
Ω
The Moser-Trudinger inequality says that there is a constant
c = c(Ω), such that
Z 2(u)
exp(4π )dx ≤ C.
2k∇ukΩ
Chang-Shou Lin Uniqueness of Mean field equations:Known results, Open problems and Applications.√
k∇uk 2 4πu 2√Since u≤ ( ) +( ) ,
k∇uk2 4π
Z Z
2 2k∇uk uue dx ≤ exp( +4π )dx
216π k∇ukΩ Ω
Z
2 2k∇u k u
≤ exp( )· exp(4π )dx
216π k∇ukΩ
2k∇uk
≤ cexp
16π
i.e.
Z Z
1 2 u 0|∇u| dx−8π e dx ≥ c (1)
2 Ω Ω
For ρ> 0, we consider
Z Z
1 2 uJ (u) = |∇u| −ρ e dxρ
2 Ω Ω
Chang-Shou Lin Uniqueness of Mean field equations:Known results, Open problems and Applications.Inequality (2) yields that J (u) is bounded from below providedρ
that ρ≤ 8π. If ρ< 8π, u be a minimizing sequence, i.e.k
Z Z
1 2 ukJ (u ) = |∇u | −ρ e dx −→ inf J (u)ρ k k ρ
12 H (Ω)Ω 0Z
1 ρ 2= ( − ) |∇u |k
2 16π
Z Z
1 2 uk+ρ( |∇u | − e dx)k16π ΩZ
1 ρ 2≥ ( − ) |∇u | dx +ck
2 16π
i.e. Z
2|∇u | dx ≤ ck 1
Ω
Chang-Shou Lin Uniqueness of Mean field equations:Known results, Open problems and Applications.2 1Then by the Moser-Trudinger inequality, expc u ∈ L (Ω). Thus,0 k
1a subsequence of{u } can be chosen such that u * u in H (Ω)k k 0
and Z Z
u (x) u(x) 1ke dx −→ e dx. Then u∈ H (Ω)0
Ω Ω
satisfies
J (u) = inf J(u).ρ
1H (Ω)0
Obviously, the Euler-Lagrange equation for J isρ
(
uρeΔu + = 0inΩue
u| = 0∂Ω
Chang-Shou Lin Uniqueness of Mean field equations:Known results, Open problems and Applications.RIn this lecture, I consider the equation:

Nu X h(x)e RΔu +ρ = α δ , in Ωk Pku (2)h(x)e
k=1
u = 0, on ∂Ω
1where h(x) is a positive C function, ρ is a constant positive.
α > 0 and δ is the Dirac measure at P ∈ Ω.k P kk
We are also interested in the situation when the space is a
compact Riemann surface:
u Xh(x)e 1 1
RΔu +ρ( − ) = α (δ − ), in M, (3)k Pkuh(x)e |M| |M|
M
where|M| is the area of M.
Chang-Shou Lin Uniqueness of Mean field equations:Known results, Open problems and Applications.Equation (2)or(3) is called mean field equation because it often
arises in the context of statistical mechanics of point vortices in
the mean field limit. See the works by
Caglioti-Lions-Marchioro-Pulvirenti, Kiessling, Polani-
Dritschel...etc.
Equation (2)or(3)have appeared in many different research field. In
2the conformal geometry, when M is S , and p = 8π, the equation
is related to the Nirenberg problem. For a given positive function
2 u 2h(x), we want to find a new metric ds = e kdxk such that h(x)
2is the Gaussian causative of ds . In physics, equations (2) and (3)
is also related to many physics models from Gauge field theory.
Chang-Shou Lin Uniqueness of Mean field equations:Known results, Open problems and Applications.Equation (2) and (3) has been extensively studied for the past
three decades, many results on existence has been shown.
For example, Chen and Lin have proved the following theorems.
Theorem 1 suppose ρ = 8πm, m is a positive integer. Then the
Leray-Schauder degree d for equation(1) is well-defined andρ

1 if ρ< 8π
d =ρ g(g+1)···(g+m−1) if 8πm <ρ< 8π(m+1)
m!
where g = the number of ”holes” of Ω.
Theorem 2 Suppose ρ = 8mπ, m is a positive integer. Then the
Leray-Schauder degree d for equation (3) is well-defined, andρ

1 if ρ< 8π
d =ρ (1−χ(M))(2−χ(M))···(m−χ(M))
if 8mπ <ρ< 8(m+1)π
m!
where χ(M) is the Euler characteritic of M.
Chang-Shou Lin Uniqueness of Mean field equations:Known results, Open problems and Applications.663.Uniqueness for equation
2We first consider equation in bounded domains ofR , without
singularity data, i.e. α = 0.k
2 2Theorem 3 suppose Ω is a bounded C domain inR , and ρ≤ 8π,
then equation (2) with α = 0 possesses at most one solution.k
2When Ω is a simply connected bounded domain inR and ρ< 8π,
Theorem 1 was proved by Nagasaki and Suzuki in 1988. Under the
same assumption of Ω and ρ = 8π, Theorem 1 was proved by
2Chen, Chang and Lin. For a general domain inR , i.e. without the
assumption of simple-connectedness, Theorem 1 was recently
proved by C.S. Lin.
Chang-Shou Lin Uniqueness of Mean field equations:Known results, Open problems and Applications.