Equilibrium and Balayage A mini-tutorial A. Martnez-Finkelshtein - - PowerPoint PPT Presentation
Equilibrium and Balayage A mini-tutorial A. Martnez-Finkelshtein (U. Almera) Optimal point configurations and orthogonal polynomials CIEM Castro Urdiales April 19, 2017 Equilibrium and Balayage A mini-tutorial A.
“Optimal point configurations and orthogonal polynomials” CIEM Castro Urdiales April 19, 2017
“Optimal point configurations and orthogonal polynomials” CIEM Castro Urdiales April 19, 2017
V µ(z) = −|z|2 2 + 1 2, |z| ≤ 1, log 1 |z|, |z| > 1.
For K ⊂ C compact, the Robin constant is κ = min {I(µ) : µ unit measure on K}
Example 1: dµ(x) =
dx π √ 1−x2 on [−1, 1]
Example 2: dµ(x) = 1
2dx on [−1, 1]
For K ⊂ C compact, the Robin constant is κ = min {I(µ) : µ unit measure on K}
Example 1: dµ(x) =
dx π √ 1−x2 on [−1, 1]
Example 2: dµ(x) = 1
2dx on [−1, 1]
Classical application: asymptotic distribution
IQ(µ) := kµk2 + 2 Z Q dµ = I(µ) + 2 Z Q dµ weighted logarithmic energy For K ⊂ C compact and Q and admissible external field, κQ = min {IQ(µ) : µ unit measure on K} The unique minimizer, λQ, such that IQ(λQ) = κQ, is the weighted equilibrium measure on K. New feature: we ignore a priori what is supp(λQ)! Characterization: V λQ + Q ≥ cQ q.e. on K, ≤ cQ
(so, basically, V λQ + Q = cQ q.e. on supp(λQ)).
Density Potential
IQ(µ) := kµk2 + 2 Z Q dµ = I(µ) + 2 Z Q dµ weighted logarithmic energy For K ⊂ C compact and Q and admissible external field, κQ = min {IQ(µ) : µ unit measure on K} The unique minimizer, λQ, such that IQ(λQ) = κQ, is the weighted equilibrium measure on K. New feature: we ignore a priori what is supp(λQ)! Characterization: V λQ + Q ≥ cQ q.e. on K, ≤ cQ
(so, basically, V λQ + Q = cQ q.e. on supp(λQ)).
Density Potential
Classical application: asymptotic distribution
IQ(µ) := kµk2 + 2 Z Q dµ = I(µ) + 2 Z Q dµ weighted logarithmic energy For K ⊂ C compact, Q an admissible external field, and τ a measure with finite energy, supp(τ) = K, and τ(K) > 1, κτ
Q = min {IQ(µ) : µ unit measure on K with µ ≤ τ}
Again, the unique minimizer, λτ
Q, with IQ(λτ Q) = κτ Q, is the con-
strained equilibrium measure on K. Characterization: V λτ
Q + Q ≥ cτ
Q
q.e. on supp(τ − λτ
Q),
≤ cQ
Q)
Density Potential Constraint
New feature:
Void Band Saturated region
IQ(µ) := kµk2 + 2 Z Q dµ = I(µ) + 2 Z Q dµ weighted logarithmic energy For K ⊂ C compact, Q an admissible external field, and τ a measure with finite energy, supp(τ) = K, and τ(K) > 1, κτ
Q = min {IQ(µ) : µ unit measure on K with µ ≤ τ}
Again, the unique minimizer, λτ
Q, with IQ(λτ Q) = κτ Q, is the con-
strained equilibrium measure on K. Characterization: V λτ
Q + Q ≥ cτ
Q
q.e. on supp(τ − λτ
Q),
≤ cQ
Q)
Density Potential Constraint
New feature:
Void Band Saturated region
Classical application: asymptotic distribution
(
“standard” definition
In particular, supp(ν) ⊂ ∂D and V µ(z) = V ν(z) q.e. in Dc := C \ D Since for reasonable ε and σ 2 M, kµ νk kµ (1 ε)ν εσk, we get Z (V µ − V ν) d(ν − σ) ≥ 0, for all σ ∈ M
µ ν
Recall: Hence, the unit Lebesgue measure on |z| = 1 is the balayage of the unit plane Lebesgue measure on |z| ≤ 1
V µ(z) = −|z|2 2 + 1 2, |z| ≤ 1, log 1 |z|, |z| > 1.
µ ν Observations:
V µ(z) = V ν(z) + c q.e. in Dc
V ν(z) ≤ V µ(z), z ∈ C (“balayage decreases the potential”).
with µout = µ
ν = Bal(µ, Dc) := µout + Bal(µin, Dc)
is its Jordan decomposition, then Bal(µ, K) := Bal(µ+, K) − Bal(µ−, K)
w.r.t. a.
Let µ be a positive measure, and τ a given measure on C such that µ(C) ≤ µ(τ). Take M := {ν : ν ≤ τ and ν(C) = µ(C)} is the partial balayage of µ under τ, denoted by ν = Bal(µ, τ) Moreover, min(µ, τ) ≤ Bal(µ, τ) ≤ τ A variational argument as before shows that
C
Notice: here Bal(µ, τ) either = τ or = 0 (saturation). Example 1: a 2D case. Let µ = βδa (β > 0), and τ = α mes2 on a sufficiently large disk |z| ≤ R. If R ≥ |a| + p β/(πα), then Bal ⇣ βδa, α mes2
⌘ = α mes2
with r = p β/(πα).
µ
Bal(µ, τ) τ
Notice: here Bal(µ, τ) either = τ or = 0 (saturation). Example 1: a 2D case. Let µ = βδa (β > 0), and τ = α mes2 on a sufficiently large disk |z| ≤ R. If R ≥ |a| + p β/(πα), then
Bal ⇣ βδa, α mes2
⌘ = α mes2
with r = p β/(πα).
Example 2: a generalization. Let µ = Pn
k=1 ckδak, ck > 0, and
τ = mes2 on C. Then Bal (µ, τ) = τ
with ∂S being an algebraic curve. Moreover, Z
S
dmes2(t) t − z =
n
X
k=1
ck ak − z (S is a quadrature domain). Example 3: a Hele-Shaw flow. Given a domain S0 3 a, let µt = tδa + mes2
Bal (µt, τ) = τ
t > 0 In particular, Z
St
zk dmes2(z) = Z
S0
zk dmes2(z), k ∈ N, Z
St
dmes2(z) = t + Z
S0
dmes2(z) S0
Constrained Weighted Classical Partial
V λQ + Q ≥ cQ q.e. on K, ≤ cQ
Density Potential
V λτ
Q + Q ≥ cτ
Q
q.e. on supp(τ − λτ
Q),
≤ cQ
Q)
Density Potential Constraint
Void Band Saturated region
µ ν
Example: consider the weighted equilibrium on C in the external field Q(z) = α|z|2 + β log 1 |z − a|, α, β > 0. Recall:
V µ(z) = −|z|2 2 + 1 2, |z| ≤ 1, log 1 |z|, |z| > 1.
Example: consider the weighted equilibrium on C in the external field Q(z) = α|z|2 + β log 1 |z − a|, α, β > 0. Hence, we can replace Q by V σ, with σ = −2α π mes2
R = r β + 1 2α Thus, σ+ = βδa, σ− = 2α π mes2
Recall again:
τ
Bal (σ+, σ−) = 2α π mes2
r = r β 2α,
Example: consider the weighted equilibrium on C in the external field Q(z) = α|z|2 + β log 1 |z − a|, α, β > 0. Hence, we can replace Q by V σ, with σ = −2α π mes2
R = r β + 1 2α Thus, σ+ = βδa, σ− = 2α π mes2
We conclude: if |a| ≤ q
β+1 2α −
q
β 2α,
λQ = 2α π mes2
π mes2
a