Equilibrium and Balayage
A mini-tutorial
A. Martínez-Finkelshtein (U. Almería)
“Optimal point configurations and orthogonal polynomials”
CIEM Castro Urdiales
April 19, 2017
Equilibrium and Balayage
A mini-tutorial
A. Martínez-Finkelshtein (U. Almería)
“Optimal point configurations and orthogonal polynomials”
CIEM Castro Urdiales
April 19, 2017
LOGARITHMIC POTENTIAL
AND
LOGARITHMIC ENERGY
POTENTIAL THEORY AND POLYNOMIALS
LOGARITHMIC POTENTIALS
LOGARITHMIC POTENTIALS
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
Example 1
-0.6
-0.8
-2
Example 2
-1.5
-1
-0.5
0
0.5
1
1.5
2
LOGARITHMIC POTENTIALS
8
2
|z|
1
>
>
<
+ , |z|  1,
2
2
V µ (z) =
1
>
>
|z| > 1.
:log ,
|z|
LOGARITHMIC ENERGY
EQUILIBRIUM
CLASSICAL EQUILIBRIUM
For K ⇢ C compact, the Robin constant is
 = min {I(µ) : µ unit measure on K}
1
0.8
0.6
Example 1: dµ(x) =
pdx
⇡ 1 x2
on [ 1, 1]
0.4
0.2
0
Example 2: dµ(x) =
1
2 dx
-0.2
on [ 1, 1]
-0.4
Example 1
-0.6
-0.8
-2
Example 2
-1.5
-1
-0.5
0
0.5
1
1.5
2
CLASSICAL EQUILIBRIUM
For K ⇢ C compact, the Robin constant is
 = min {I(µ) : µ unit measure on K}
Classical application: asymptotic distribution
of zeros of standard orthogonal polynomials
1
0.8
0.6
Example 1: dµ(x) =
pdx
⇡ 1 x2
on [ 1, 1]
0.4
0.2
0
Example 2: dµ(x) =
1
2 dx
-0.2
on [ 1, 1]
-0.4
Example 1
-0.6
-0.8
-2
Example 2
-1.5
-1
-0.5
0
0.5
1
1.5
2
WEIGHTED EQUILIBRIUM
weighted logarithmic energy
Z
Z
IQ (µ) := kµk2 + 2 Q dµ = I(µ) + 2 Q dµ
For K ⇢ C compact and Q and admissible external field,
Q = min {IQ (µ) : µ unit measure on K}
V
Q
+Q
cQ
q.e. on K,
 cQ
(so, basically, V
on supp(
Q
Q)
= Q , is the
+ Q = cQ q.e. on supp(
Density
Characterization:
Q)
Potential
The unique minimizer, Q , such that IQ (
weighted equilibrium measure on K.
Q )).
New feature: we ignore a priori what is supp(
Q )!
WEIGHTED EQUILIBRIUM
weighted logarithmic energy
Z
Z
IQ (µ) := kµk2 + 2 Q dµ = I(µ) + 2 Q dµ
For K ⇢ C compact and Q and admissible external field,
Q = min {IQ (µ) : µ unit measure on K}
V
Q
+Q
cQ
q.e. on K,
 cQ
(so, basically, V
on supp(
Q
Q)
+ Q = cQ q.e. on supp(
Density
Characterization:
Potential
Classical
application:
asymptotic
The unique
minimizer,
that IQ ( distribution
Q , such
Q ) = Q , is the
zeros of polynomials
varying orthogonality
weightedofequilibrium
measure onofK.
Q )).
New feature: we ignore a priori what is supp(
Q )!
CONSTRAINED WEIGHTED EQUILIBRIUM
weighted logarithmic energy
Z
Z
IQ (µ) := kµk2 + 2 Q dµ = I(µ) + 2 Q dµ
For K ⇢ C compact, Q an admissible external field, and ⌧ a
measure with finite energy, supp(⌧ ) = K, and ⌧ (K) > 1,
⌧Q = min {IQ (µ) : µ unit measure on K with µ  ⌧ }
Again, the unique minimizer, ⌧Q , with IQ (
strained equilibrium measure on K.
⌧
Q)
= ⌧Q , is the con-
Characterization:
c⌧Q
 cQ
q.e. on supp(⌧
on supp(
⌧
Q)
New feature:
⌧
Q ),
Constraint
Density
+Q
Potential
V
⌧
Q
Void
Band
Saturated
region
CONSTRAINED WEIGHTED EQUILIBRIUM
weighted logarithmic energy
Z
Z
IQ (µ) := kµk2 + 2 Q dµ = I(µ) + 2 Q dµ
For K ⇢ C compact, Q an admissible external field, and ⌧ a
measure with finite energy, supp(⌧ ) = K, and ⌧ (K) > 1,
asymptotic
⌧QClassical
= min {IQapplication:
(µ) : µ unit measure
on Kdistribution
with µ  ⌧ }
of zeros of polynomials ⌧of discrete orthogonality
Again, the unique minimizer, Q , with IQ ( ⌧Q ) = ⌧Q , is the constrained equilibrium measure on K.
Characterization:
c⌧Q
 cQ
q.e. on supp(⌧
on supp(
⌧
Q)
New feature:
⌧
Q ),
Constraint
Density
+Q
Potential
V
⌧
Q
Void
Band
Saturated
region
BALAYAGE
MINIMAL ENERGY PARADIGM
CLASSICAL BALAYAGE
⌫
µ
Since for reasonable " and 2 M, kµ ⌫k  kµ
we get
Z
µ
⌫
(V
V ) d(⌫
) 0,
for all
V µ (z) = V ⌫ (z) q.e. in Dc := C \ D
(
In particular, supp(⌫) ⇢ @D and
(1
")⌫
" k,
2M
“standard” definition
of balayage
CLASSICAL BALAYAGE
Recall:
8
2
|z|
1
>
>
<
+ ,
2
2
V µ (z) =
1
>
>
:log ,
|z|
|z|  1,
|z| > 1.
Hence, the unit Lebesgue measure on |z| = 1 is the balayage of
the unit plane Lebesgue measure on |z|  1
CLASSICAL BALAYAGE
Observations:
• Since V ⌫ V µ is subharmonic in D, V ⌫ (z)  V µ (z),
(“balayage decreases the potential”).
z2C
• !a = Bal( a , @D) is the harmonic measure of @D
w.r.t. a.
• If D is unbounded, we get instead
µ
⌫
V (z) = V (z) + c
⌫
µ
q.e. in D
c
• Extension: if supp(µ) 6⇢ D, we we may take µ = µout + µin ,
with µout = µ Dc , and define
⌫ = Bal(µ, Dc ) := µout + Bal(µin , Dc )
• Further extension: if µ is a signed measures, and µ = µ+
is its Jordan decomposition, then
Bal(µ, K) := Bal(µ+ , K)
Bal(µ , K)
µ
PARTIAL BALAYAGE
Let µ be a positive measure, and ⌧ a given measure on C such
that µ(C)  µ(⌧ ). Take
M := {⌫ : ⌫  ⌧ and ⌫(C) = µ(C)}
is the partial balayage of µ under ⌧ , denoted by
⌫ = Bal(µ, ⌧ )
A variational argument as before shows that
•⌫⌧
• V ⌫  V µ on C
• V ⌫ = V µ on supp(⌧
⌫)
Moreover,
min(µ, ⌧ )  Bal(µ, ⌧ )  ⌧
PARTIAL BALAYAGE
Example 1: a 2D case. Let µ = a ( > 0),pand ⌧ = ↵ mes2 on
a sufficiently large disk |z|  R. If R |a| +
/(⇡↵), then
⇣
⌘
p
Bal
with r =
/(⇡↵).
a , ↵ mes2 |z|R = ↵ mes2 |z a|r ,
Notice: here Bal(µ, ⌧ ) either = ⌧ or = 0 (saturation).
⌧
µ
Bal(µ, ⌧ )
PARTIAL BALAYAGE
Example 1: a 2D case. Let µ = a ( > 0),pand ⌧ = ↵ mes2 on
a sufficiently large disk |z|  R. If R |a| +
/(⇡↵), then
⇣
⌘
p
Bal
with r =
/(⇡↵).
a , ↵ mes2 |z|R = ↵ mes2 |z a|r ,
Notice: here Bal(µ, ⌧ ) either = ⌧ or = 0 (saturation).
1.4
Robin measure on [-1,1]
Its constrained balayage under =dx
1.2
1
0.8
0.6
0.4
0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
PARTIAL BALAYAGE
Example 2: a generalization. Let µ =
⌧ = mes2 on C. Then
Bal (µ, ⌧ ) = ⌧ S
Pn
k=1 ck ak ,
ck > 0, and
with @S being an algebraic curve. Moreover,
Z
n
X
dmes2 (t)
ck
=
t z
ak z
S
k=1
(S is a quadrature domain).
Example 3: a Hele-Shaw flow. Given a domain S0 3 a, let
µt = t a + mes2 S0 , and ⌧ = mes2 on C. Then
Bal (µt , ⌧ ) = ⌧ St , t > 0
In particular,
Z
Z
k
k
z dmes2 (z) =
z dmes2 (z), k 2 N,
St
S0
Z
Z
dmes2 (z) = t +
dmes2 (z)
St
S0
S0
CONNECTIONS
BETWEEN BALAYAGE
AND EQUILIBRIUM
Weighted
Constrained
Equilibrium
Energy minimization
Balayage
Classical
Partial
CONSTRAINED & WEIGHTED EQUILIBRIUM
Density
Potential
Density
Potential
Constraint
Void
V
Q
+Q
cQ
 cQ
V
q.e. on K,
on supp(
Q)
⌧
Q
+Q
c⌧Q
 cQ
Saturated
region
Band
q.e. on supp(⌧
on supp(
⌧
Q)
⌧
Q ),
BALAYAGE & EQUILIBRIUM
⌫
µ
BALAYAGE & EQUILIBRIUM
1.4
Robin measure on [-1,1]
Its constrained balayage under =dx
1.2
1
0.8
0.6
0.4
0.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
BALAYAGE & EQUILIBRIUM
BALAYAGE & EQUILIBRIUM
Example: consider the weighted equilibrium on C in the external
field
1
2
Q(z) = ↵|z| + log
, ↵, > 0.
|z a|
Recall:
8
2
|z|
1
>
>
<
+ ,
2
2
V µ (z) =
1
>
>
:log ,
|z|
|z|  1,
|z| > 1.
BALAYAGE & EQUILIBRIUM
Example: consider the weighted equilibrium on C in the external
field
1
2
Q(z) = ↵|z| + log
, ↵, > 0.
|z a|
Hence, we can replace Q by V , with
=
2↵
mes2
⇡
Thus,
+
=
a,
Recall again:
Bal (
+,
2↵
)=
mes2
⇡
|z|R
+
a,
R=
2↵
=
mes2
⇡
|z a|r
,
r=
r
r
+1
2↵
|z|R
⌧
2↵
,
BALAYAGE & EQUILIBRIUM
Example: consider the weighted equilibrium on C in the external
field
1
2
Q(z) = ↵|z| + log
, ↵, > 0.
|z a|
Hence, we can replace Q by V , with
=
2↵
mes2
⇡
|z|R
Thus,
+
=
We conclude: if |a| 
Q
2↵
=
mes2
⇡
a,
q
|z|R
+1
2↵
+
a,
2↵
=
mes2
⇡
q
2↵ ,
2↵
mes2
⇡
R=
r
+1
2↵
|z|R
a
|z a|r
Thank you