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 Example 2 -0.8 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
LOGARITHMIC POTENTIALS − | z | 2 + 1 | z | ≤ 1 , 2 , 2 V µ ( z ) = log 1 | z | > 1 . | z | ,
LOGARITHMIC ENERGY
EQUILIBRIUM
CLASSICAL EQUILIBRIUM For K ⊂ C compact, the Robin constant is κ = min { I ( µ ) : µ unit measure on K } 1 0.8 0.6 dx Example 1: dµ ( x ) = 1 − x 2 on [ − 1 , 1] 0.4 √ 0.2 π 0 -0.2 Example 2: dµ ( x ) = 1 2 dx on [ − 1 , 1] -0.4 Example 1 -0.6 Example 2 -0.8 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
CLASSICAL EQUILIBRIUM For K ⊂ C compact, the Robin constant is κ = min { I ( µ ) : µ unit measure on K } Classical application: asymptotic distribution of zeros of standard orthogonal polynomials 1 0.8 0.6 dx Example 1: dµ ( x ) = 1 − x 2 on [ − 1 , 1] 0.4 √ 0.2 π 0 -0.2 Example 2: dµ ( x ) = 1 2 dx on [ − 1 , 1] -0.4 Example 1 -0.6 Example 2 -0.8 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
WEIGHTED EQUILIBRIUM weighted logarithmic energy Z Z I Q ( µ ) := k µ k 2 + 2 Q dµ = I ( µ ) + 2 Q dµ For K ⊂ C compact and Q and admissible external field, κ Q = min { I Q ( µ ) : µ unit measure on K } The unique minimizer, λ Q , such that I Q ( λ Q ) = κ Q , is the weighted equilibrium measure on K. Characterization : Potential Density V λ Q + Q ≥ c Q q.e. on K, on supp( λ Q ) ≤ c Q (so, basically, V λ Q + Q = c Q q.e. on supp( λ Q )). New feature: we ignore a priori what is supp( λ Q )!
WEIGHTED EQUILIBRIUM weighted logarithmic energy Z Z I Q ( µ ) := k µ k 2 + 2 Q dµ = I ( µ ) + 2 Q dµ For K ⊂ C compact and Q and admissible external field, κ Q = min { I Q ( µ ) : µ unit measure on K } The unique minimizer, λ Q , such that I Q ( λ Q ) = κ Q , is the Classical application: asymptotic distribution weighted equilibrium measure on K. of zeros of polynomials of varying orthogonality Characterization : Potential Density V λ Q + Q ≥ c Q q.e. on K, on supp( λ Q ) ≤ c Q (so, basically, V λ Q + Q = c Q q.e. on supp( λ Q )). New feature: we ignore a priori what is supp( λ Q )!
CONSTRAINED WEIGHTED EQUILIBRIUM weighted logarithmic energy Z Z I Q ( µ ) := k µ k 2 + 2 Q dµ = I ( µ ) + 2 Q dµ For K ⊂ C compact, Q an admissible external field, and τ a measure with finite energy, supp( τ ) = K , and τ ( K ) > 1, Q = min { I Q ( µ ) : µ unit measure on K with µ ≤ τ } κ τ Again, the unique minimizer, λ τ Q , with I Q ( λ τ Q ) = κ τ Q , is the con- strained equilibrium measure on K. Characterization : Q + Q ≥ c τ V λ τ q.e. on supp( τ − λ τ Constraint Q ) , Q on supp( λ τ Q ) ≤ c Q Potential Density New feature: Void Band Saturated region
CONSTRAINED WEIGHTED EQUILIBRIUM weighted logarithmic energy Z Z I Q ( µ ) := k µ k 2 + 2 Q dµ = I ( µ ) + 2 Q dµ For K ⊂ C compact, Q an admissible external field, and τ a measure with finite energy, supp( τ ) = K , and τ ( K ) > 1, Classical application: asymptotic distribution Q = min { I Q ( µ ) : µ unit measure on K with µ ≤ τ } κ τ of zeros of polynomials of discrete orthogonality Again, the unique minimizer, λ τ Q , with I Q ( λ τ Q ) = κ τ Q , is the con- strained equilibrium measure on K. Characterization : Q + Q ≥ c τ V λ τ q.e. on supp( τ − λ τ Constraint Q ) , Q on supp( λ τ Q ) ≤ c Q Potential Density New feature: Void Band Saturated region
BALAYAGE
MINIMAL ENERGY PARADIGM
CLASSICAL BALAYAGE ν µ Since for reasonable ε and σ 2 M , k µ � ν k k µ � (1 � ε ) ν � εσ k , we get Z ( V µ − V ν ) d ( ν − σ ) ≥ 0 , for all σ ∈ M In particular, supp( ν ) ⊂ ∂ D and ( “standard” definition q.e. in D c := C \ D of balayage V µ ( z ) = V ν ( z )
CLASSICAL BALAYAGE Recall: − | z | 2 + 1 | z | ≤ 1 , 2 , 2 V µ ( z ) = log 1 | z | > 1 . | z | , 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 ν − V µ is subharmonic in D , V ν ( z ) ≤ V µ ( z ) , z ∈ C (“balayage decreases the potential”). • ω a = Bal( δ a , ∂ D ) is the harmonic measure of ∂ D ν w.r.t. a . • If D is unbounded, we get instead µ V µ ( z ) = V ν ( z ) + c q.e. in D c • Extension: if supp( µ ) 6⇢ D , we we may take µ = µ out + µ in , � with µ out = µ D c , and define � ν = Bal( µ, D c ) := µ out + Bal( µ in , D c ) • Further extension: if µ is a signed measures, and µ = µ + − µ − is its Jordan decomposition, then Bal( µ, K ) := Bal( µ + , K ) − Bal( µ − , K )
PARTIAL BALAYAGE 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 ) A variational argument as before shows that • ν ≤ τ • V ν ≤ V µ on C • V ν = V µ on supp( τ − ν ) Moreover, min( µ, τ ) ≤ Bal( µ, τ ) ≤ τ
PARTIAL BALAYAGE Example 1: a 2D case. Let µ = βδ a ( β > 0), and τ = α mes 2 on p a su ffi ciently large disk | z | ≤ R . If R ≥ | a | + β / ( πα ), then ⇣ ⌘ p � � Bal βδ a , α mes 2 = α mes 2 | z − a | ≤ r , with r = β / ( πα ) . � � | z | ≤ R Notice: here Bal( µ, τ ) either = τ or = 0 (saturation). τ µ Bal( µ, τ )
PARTIAL BALAYAGE Example 1: a 2D case. Let µ = βδ a ( β > 0), and τ = α mes 2 on p a su ffi ciently large disk | z | ≤ R . If R ≥ | a | + β / ( πα ), then ⇣ ⌘ p � � Bal βδ a , α mes 2 = α mes 2 | z − a | ≤ r , with r = β / ( πα ) . � � | z | ≤ R Notice: here Bal( µ, τ ) either = τ 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 = P n k =1 c k δ a k , c k > 0, and τ = mes 2 on C . Then � Bal ( µ, τ ) = τ � S with ∂ S being an algebraic curve. Moreover, n d mes 2 ( t ) Z c k X = t − z a k − z S k =1 ( S is a quadrature domain). Example 3: a Hele-Shaw flow. Given a domain S 0 3 a , let � µ t = t δ a + mes 2 S 0 , and τ = mes 2 on C . Then � � Bal ( µ t , τ ) = τ t > 0 S t , � In particular, Z Z z k d mes 2 ( z ) = z k d mes 2 ( z ) , k ∈ N , S t S 0 Z Z d mes 2 ( z ) = t + d mes 2 ( z ) S 0 S t S 0
CONNECTIONS BETWEEN BALAYAGE AND EQUILIBRIUM
Weighted Constrained Equilibrium Energy minimization Balayage Classical Partial
CONSTRAINED & WEIGHTED EQUILIBRIUM Constraint Potential Density Potential Density Void Band Saturated region V λ Q + Q ≥ c Q Q + Q ≥ c τ V λ τ q.e. on supp( τ − λ τ Q ) , q.e. on K, Q on supp( λ τ on supp( λ Q ) ≤ c Q Q ) ≤ c Q
BALAYAGE & EQUILIBRIUM ν 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 Q ( z ) = α | z | 2 + β log α , β > 0 . | z − a | , Recall: − | z | 2 + 1 | z | ≤ 1 , 2 , 2 V µ ( z ) = log 1 | z | > 1 . | z | ,
BALAYAGE & EQUILIBRIUM Example: consider the weighted equilibrium on C in the external field 1 Q ( z ) = α | z | 2 + β log α , β > 0 . | z − a | , Hence, we can replace Q by V σ , with r σ = − 2 α β + 1 � | z | ≤ R + βδ a , R = π mes 2 � 2 α Thus, σ − = 2 α � σ + = βδ a , π mes 2 � | z | ≤ R Recall again: τ r Bal ( σ + , σ − ) = 2 α β � π mes 2 r = | z − a | ≤ r , 2 α , �
BALAYAGE & EQUILIBRIUM Example: consider the weighted equilibrium on C in the external field 1 Q ( z ) = α | z | 2 + β log α , β > 0 . | z − a | , Hence, we can replace Q by V σ , with r σ = − 2 α β + 1 � | z | ≤ R + βδ a , R = π mes 2 � 2 α Thus, σ − = 2 α � σ + = βδ a , π mes 2 � | z | ≤ R q q β +1 β We conclude: if | a | ≤ 2 α , 2 α − λ Q = 2 α | z | ≤ R − 2 α a � � π mes 2 π mes 2 � � | z − a | ≤ r
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