with Reese Harvey
POTENTIAL THEORY FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 1 / 53 POTENTIAL THEORY FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
with Reese Harvey
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 1 / 53 To every differential equation on Rn of the form: f (D2u) = 0
there is an associated “Potential Theory” based on the functions which satisfy the condition:
f (D2u) ≥ 0
in a generalized sense.
The Idea
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 2 / 53 there is an associated “Potential Theory” based on the functions which satisfy the condition:
f (D2u) ≥ 0
in a generalized sense.
The Idea
To every differential equation on Rn of the form: f (D2u) = 0
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 2 / 53 based on the functions which satisfy the condition:
f (D2u) ≥ 0
in a generalized sense.
The Idea
To every differential equation on Rn of the form: f (D2u) = 0
there is an associated “Potential Theory”
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 2 / 53 The Idea
To every differential equation on Rn of the form: f (D2u) = 0
there is an associated “Potential Theory” based on the functions which satisfy the condition:
f (D2u) ≥ 0
in a generalized sense.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 2 / 53 1. The Laplace Equation:
∆u = tr(D2u) = 0
The Associated Potential Theory: The Theory of Subharmonic Functions
“∆u ≥ 0”.
The upper semi-continuous functions which are “sub the harmonics”: u ≤ h on ∂K ⇒ u ≤ h on K
Classical Examples
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 3 / 53 The Associated Potential Theory: The Theory of Subharmonic Functions
“∆u ≥ 0”.
The upper semi-continuous functions which are “sub the harmonics”: u ≤ h on ∂K ⇒ u ≤ h on K
Classical Examples
1. The Laplace Equation:
∆u = tr(D2u) = 0
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 3 / 53 The upper semi-continuous functions which are “sub the harmonics”: u ≤ h on ∂K ⇒ u ≤ h on K
Classical Examples
1. The Laplace Equation:
∆u = tr(D2u) = 0
The Associated Potential Theory: The Theory of Subharmonic Functions
“∆u ≥ 0”.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 3 / 53 u ≤ h on ∂K ⇒ u ≤ h on K
Classical Examples
1. The Laplace Equation:
∆u = tr(D2u) = 0
The Associated Potential Theory: The Theory of Subharmonic Functions
“∆u ≥ 0”.
The upper semi-continuous functions which are “sub the harmonics”:
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 3 / 53 Classical Examples
1. The Laplace Equation:
∆u = tr(D2u) = 0
The Associated Potential Theory: The Theory of Subharmonic Functions
“∆u ≥ 0”.
The upper semi-continuous functions which are “sub the harmonics”: u ≤ h on ∂K ⇒ u ≤ h on K
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 3 / 53 and D2u ≥ 0.
The Associated Potential Theory: The Theory of Convex Functions
“D2u ≥ 0”.
Classical Examples
2. The Real Monge-Ampere` Equation:
det(D2u) = 0
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 4 / 53 The Associated Potential Theory: The Theory of Convex Functions
“D2u ≥ 0”.
Classical Examples
2. The Real Monge-Ampere` Equation:
det(D2u) = 0 and D2u ≥ 0.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 4 / 53 Classical Examples
2. The Real Monge-Ampere` Equation:
det(D2u) = 0 and D2u ≥ 0.
The Associated Potential Theory: The Theory of Convex Functions
“D2u ≥ 0”.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 4 / 53 The Associated Potential Theory: The Theory of Plurisubharmonic Functions
2 “(D u)C ≥ 0”.
The upper semi-continuous functions on Cn whose restriction to every affine complex line is subharmonic
Poincare,´ Oka, Grauert, Lelong, Hormander,¨ etc.
Classical Examples
3. The Complex Monge-Ampere` Equation:
2 2 detC (D u)C = 0 and (D u)C ≥ 0.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 5 / 53 The upper semi-continuous functions on Cn whose restriction to every affine complex line is subharmonic
Poincare,´ Oka, Grauert, Lelong, Hormander,¨ etc.
Classical Examples
3. The Complex Monge-Ampere` Equation:
2 2 detC (D u)C = 0 and (D u)C ≥ 0.
The Associated Potential Theory: The Theory of Plurisubharmonic Functions
2 “(D u)C ≥ 0”.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 5 / 53 Poincare,´ Oka, Grauert, Lelong, Hormander,¨ etc.
Classical Examples
3. The Complex Monge-Ampere` Equation:
2 2 detC (D u)C = 0 and (D u)C ≥ 0.
The Associated Potential Theory: The Theory of Plurisubharmonic Functions
2 “(D u)C ≥ 0”.
The upper semi-continuous functions on Cn whose restriction to every affine complex line is subharmonic
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 5 / 53 Classical Examples
3. The Complex Monge-Ampere` Equation:
2 2 detC (D u)C = 0 and (D u)C ≥ 0.
The Associated Potential Theory: The Theory of Plurisubharmonic Functions
2 “(D u)C ≥ 0”.
The upper semi-continuous functions on Cn whose restriction to every affine complex line is subharmonic
Poincare,´ Oka, Grauert, Lelong, Hormander,¨ etc.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 5 / 53 Some Interesting Cases:
2 th σk D u = 0 k Hessian Equation
2 th σk D u = 0 k Complex Hessian Equation C
arctan D2u = 0 Special Lagrangian Potential Equation
Many of the Classical Constructions and Results Carry Over to General Equations
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 6 / 53 2 th σk D u = 0 k Hessian Equation
2 th σk D u = 0 k Complex Hessian Equation C
arctan D2u = 0 Special Lagrangian Potential Equation
Many of the Classical Constructions and Results Carry Over to General Equations
Some Interesting Cases:
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 6 / 53 2 th σk D u = 0 k Complex Hessian Equation C
arctan D2u = 0 Special Lagrangian Potential Equation
Many of the Classical Constructions and Results Carry Over to General Equations
Some Interesting Cases:
2 th σk D u = 0 k Hessian Equation
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 6 / 53 arctan D2u = 0 Special Lagrangian Potential Equation
Many of the Classical Constructions and Results Carry Over to General Equations
Some Interesting Cases:
2 th σk D u = 0 k Hessian Equation
2 th σk D u = 0 k Complex Hessian Equation C
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 6 / 53 Many of the Classical Constructions and Results Carry Over to General Equations
Some Interesting Cases:
2 th σk D u = 0 k Hessian Equation
2 th σk D u = 0 k Complex Hessian Equation C
arctan D2u = 0 Special Lagrangian Potential Equation
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 6 / 53 Many of the Classical Constructions and Results Carry Over to General Equations
Some Interesting Cases:
2 th σk D u = 0 k Hessian Equation
2 th σk D u = 0 k Complex Hessian Equation C
arctan D2u = 0 Special Lagrangian Potential Equation
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 6 / 53 One fixes a continuous function
2 n f : Sym (R ) → R, and associates to it the nonlinear differential equation
f (D2u) = 0
and differential subequation
f (D2u) ≥ 0.
The Usual Set-up
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 7 / 53 and associates to it the nonlinear differential equation
f (D2u) = 0
and differential subequation
f (D2u) ≥ 0.
The Usual Set-up
One fixes a continuous function
2 n f : Sym (R ) → R,
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 7 / 53 and differential subequation
f (D2u) ≥ 0.
The Usual Set-up
One fixes a continuous function
2 n f : Sym (R ) → R, and associates to it the nonlinear differential equation
f (D2u) = 0
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 7 / 53 The Usual Set-up
One fixes a continuous function
2 n f : Sym (R ) → R, and associates to it the nonlinear differential equation
f (D2u) = 0
and differential subequation
f (D2u) ≥ 0.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 7 / 53 Consider instead the closed set
2 n F ≡ {A ∈ Sym (R ): f (A) ≥ 0}. Then for u ∈ C2
f (D2u) ≥ 0 ⇐⇒ D2u ∈ F (the subequation) and f (D2u) = 0 ⇐⇒ D2u ∈ ∂F (the equation)
A Geometric Approach – N. V. Krylov
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 8 / 53 Then for u ∈ C2
f (D2u) ≥ 0 ⇐⇒ D2u ∈ F (the subequation) and f (D2u) = 0 ⇐⇒ D2u ∈ ∂F (the equation)
A Geometric Approach – N. V. Krylov
Consider instead the closed set
2 n F ≡ {A ∈ Sym (R ): f (A) ≥ 0}.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 8 / 53 and f (D2u) = 0 ⇐⇒ D2u ∈ ∂F (the equation)
A Geometric Approach – N. V. Krylov
Consider instead the closed set
2 n F ≡ {A ∈ Sym (R ): f (A) ≥ 0}. Then for u ∈ C2
f (D2u) ≥ 0 ⇐⇒ D2u ∈ F (the subequation)
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 8 / 53 A Geometric Approach – N. V. Krylov
Consider instead the closed set
2 n F ≡ {A ∈ Sym (R ): f (A) ≥ 0}. Then for u ∈ C2
f (D2u) ≥ 0 ⇐⇒ D2u ∈ F (the subequation) and f (D2u) = 0 ⇐⇒ D2u ∈ ∂F (the equation)
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 8 / 53 1. The Laplace Equation:
2 n F ≡ {A ∈ Sym (R ): tr(A) ≥ 0}.
2. The Real Monge-Ampere` Equation: det(D2u) = 0 and D2u ≥ 0.
2 n F ≡ P ≡ {A ∈ Sym (R ): A ≥ 0}.
Note: The additional condition is natural here.
3. The Complex Monge-Ampere` Equation:
F ≡ PC ≡ {A ∈ Sym2 ( n): A ≥ 0}. R C C
1 AC = 2 (A − JAJ)
Examples
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 9 / 53 2. The Real Monge-Ampere` Equation: det(D2u) = 0 and D2u ≥ 0.
2 n F ≡ P ≡ {A ∈ Sym (R ): A ≥ 0}.
Note: The additional condition is natural here.
3. The Complex Monge-Ampere` Equation:
F ≡ PC ≡ {A ∈ Sym2 ( n): A ≥ 0}. R C C
1 AC = 2 (A − JAJ)
Examples
1. The Laplace Equation:
2 n F ≡ {A ∈ Sym (R ): tr(A) ≥ 0}.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 9 / 53 Note: The additional condition is natural here.
3. The Complex Monge-Ampere` Equation:
F ≡ PC ≡ {A ∈ Sym2 ( n): A ≥ 0}. R C C
1 AC = 2 (A − JAJ)
Examples
1. The Laplace Equation:
2 n F ≡ {A ∈ Sym (R ): tr(A) ≥ 0}.
2. The Real Monge-Ampere` Equation: det(D2u) = 0 and D2u ≥ 0.
2 n F ≡ P ≡ {A ∈ Sym (R ): A ≥ 0}.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 9 / 53 3. The Complex Monge-Ampere` Equation:
F ≡ PC ≡ {A ∈ Sym2 ( n): A ≥ 0}. R C C
1 AC = 2 (A − JAJ)
Examples
1. The Laplace Equation:
2 n F ≡ {A ∈ Sym (R ): tr(A) ≥ 0}.
2. The Real Monge-Ampere` Equation: det(D2u) = 0 and D2u ≥ 0.
2 n F ≡ P ≡ {A ∈ Sym (R ): A ≥ 0}.
Note: The additional condition is natural here.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 9 / 53 Examples
1. The Laplace Equation:
2 n F ≡ {A ∈ Sym (R ): tr(A) ≥ 0}.
2. The Real Monge-Ampere` Equation: det(D2u) = 0 and D2u ≥ 0.
2 n F ≡ P ≡ {A ∈ Sym (R ): A ≥ 0}.
Note: The additional condition is natural here.
3. The Complex Monge-Ampere` Equation:
F ≡ PC ≡ {A ∈ Sym2 ( n): A ≥ 0}. R C C
1 AC = 2 (A − JAJ)
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 9 / 53 Start with a general closed constraint set 2 F ⊂ Sym (Rn) on second derivatives:
2 open n Dx u ∈ F for all x ∈ X ⊂ R .
for u ∈ C2(X).
Obective: To extend this to a larger class of functions u, called the F-subharmonic functions F(X), and to develop an accompanying potential theory for this class.
A General Geometric Approach
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 10 / 53 2 open n Dx u ∈ F for all x ∈ X ⊂ R .
for u ∈ C2(X).
Obective: To extend this to a larger class of functions u, called the F-subharmonic functions F(X), and to develop an accompanying potential theory for this class.
A General Geometric Approach
Start with a general closed constraint set 2 F ⊂ Sym (Rn) on second derivatives:
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 10 / 53 Obective: To extend this to a larger class of functions u, called the F-subharmonic functions F(X), and to develop an accompanying potential theory for this class.
A General Geometric Approach
Start with a general closed constraint set 2 F ⊂ Sym (Rn) on second derivatives:
2 open n Dx u ∈ F for all x ∈ X ⊂ R .
for u ∈ C2(X).
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 10 / 53 A General Geometric Approach
Start with a general closed constraint set 2 F ⊂ Sym (Rn) on second derivatives:
2 open n Dx u ∈ F for all x ∈ X ⊂ R .
for u ∈ C2(X).
Obective: To extend this to a larger class of functions u, called the F-subharmonic functions F(X), and to develop an accompanying potential theory for this class.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 10 / 53 Set
USC(X) ≡ {u : X → [−∞, ∞): u is upper semicontinuous}
Take u ∈ USC(X) and require that for any ϕ ∈ C2(X), the condition
u ≤ ϕ and u(x0) = ϕ(x0)
must imply
D2 ϕ ∈ F x0 .
Let’s Try the Viscosity Approach
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 11 / 53 Take u ∈ USC(X) and require that for any ϕ ∈ C2(X), the condition
u ≤ ϕ and u(x0) = ϕ(x0)
must imply
D2 ϕ ∈ F x0 .
Let’s Try the Viscosity Approach
Set
USC(X) ≡ {u : X → [−∞, ∞): u is upper semicontinuous}
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 11 / 53 must imply
D2 ϕ ∈ F x0 .
Let’s Try the Viscosity Approach
Set
USC(X) ≡ {u : X → [−∞, ∞): u is upper semicontinuous}
Take u ∈ USC(X) and require that for any ϕ ∈ C2(X), the condition
u ≤ ϕ and u(x0) = ϕ(x0)
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 11 / 53 Let’s Try the Viscosity Approach
Set
USC(X) ≡ {u : X → [−∞, ∞): u is upper semicontinuous}
Take u ∈ USC(X) and require that for any ϕ ∈ C2(X), the condition
u ≤ ϕ and u(x0) = ϕ(x0)
must imply
D2 ϕ ∈ F x0 .
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 11 / 53 As it stands. NO.
Note: If ϕ satisfies the conditions above for u at a point x0, then so does
ϕ( ) = ϕ( ) + 1 h ( − ), − i e x x 2 A x x0 x x0
where A ≥ 0.
2 Conclusion: We must require F ⊂ Sym (Rn) to satisfy the condition F + P ⊂ F
where P ≡ {A : A ≥ 0}.
Is This a Workable Concept?
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 12 / 53 Note: If ϕ satisfies the conditions above for u at a point x0, then so does
ϕ( ) = ϕ( ) + 1 h ( − ), − i e x x 2 A x x0 x x0
where A ≥ 0.
2 Conclusion: We must require F ⊂ Sym (Rn) to satisfy the condition F + P ⊂ F
where P ≡ {A : A ≥ 0}.
Is This a Workable Concept?
As it stands. NO.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 12 / 53 2 Conclusion: We must require F ⊂ Sym (Rn) to satisfy the condition F + P ⊂ F
where P ≡ {A : A ≥ 0}.
Is This a Workable Concept?
As it stands. NO.
Note: If ϕ satisfies the conditions above for u at a point x0, then so does
ϕ( ) = ϕ( ) + 1 h ( − ), − i e x x 2 A x x0 x x0
where A ≥ 0.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 12 / 53 Is This a Workable Concept?
As it stands. NO.
Note: If ϕ satisfies the conditions above for u at a point x0, then so does
ϕ( ) = ϕ( ) + 1 h ( − ), − i e x x 2 A x x0 x x0
where A ≥ 0.
2 Conclusion: We must require F ⊂ Sym (Rn) to satisfy the condition F + P ⊂ F
where P ≡ {A : A ≥ 0}.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 12 / 53 2 A closed subset F ⊂ Sym (Rn) is called a subequation if it satisfies the positivity condition
F + P ⊂ F
where P ≡ {A ≥ 0}.
A C2-function on a domain X ⊂ Rn is F-subharmonic if
2 Dx u ∈ F ∀ x ∈ X,
and it is F-harmonic if
2 Dx u ∈ ∂F ∀ x ∈ X,
We want to extend these notions to upper semi-continuous functions.
Definitions
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 13 / 53 A C2-function on a domain X ⊂ Rn is F-subharmonic if
2 Dx u ∈ F ∀ x ∈ X,
and it is F-harmonic if
2 Dx u ∈ ∂F ∀ x ∈ X,
We want to extend these notions to upper semi-continuous functions.
Definitions
2 A closed subset F ⊂ Sym (Rn) is called a subequation if it satisfies the positivity condition
F + P ⊂ F
where P ≡ {A ≥ 0}.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 13 / 53 We want to extend these notions to upper semi-continuous functions.
Definitions
2 A closed subset F ⊂ Sym (Rn) is called a subequation if it satisfies the positivity condition
F + P ⊂ F
where P ≡ {A ≥ 0}.
A C2-function on a domain X ⊂ Rn is F-subharmonic if
2 Dx u ∈ F ∀ x ∈ X,
and it is F-harmonic if
2 Dx u ∈ ∂F ∀ x ∈ X,
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 13 / 53 Definitions
2 A closed subset F ⊂ Sym (Rn) is called a subequation if it satisfies the positivity condition
F + P ⊂ F
where P ≡ {A ≥ 0}.
A C2-function on a domain X ⊂ Rn is F-subharmonic if
2 Dx u ∈ F ∀ x ∈ X,
and it is F-harmonic if
2 Dx u ∈ ∂F ∀ x ∈ X,
We want to extend these notions to upper semi-continuous functions.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 13 / 53 USC(X) ≡ {u : X → [−∞, ∞): u is upper semicontinuous}
Definition. Fix u ∈ USC(X).A test function for u at a point x ∈ X is a function ϕ, C2 near x, such that
u ≤ ϕ near x
u = ϕ at x
Definition. A function u ∈ USC(X) is F-subharmonic if for every x ∈ X and every test function ϕ for u at x
2 Dx ϕ ∈ F.
F(X) ≡ the set of these.
Viscosity Theory (Crandall, Ishii, Lions, Evans, et al.)
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 14 / 53 Definition. Fix u ∈ USC(X).A test function for u at a point x ∈ X is a function ϕ, C2 near x, such that
u ≤ ϕ near x
u = ϕ at x
Definition. A function u ∈ USC(X) is F-subharmonic if for every x ∈ X and every test function ϕ for u at x
2 Dx ϕ ∈ F.
F(X) ≡ the set of these.
Viscosity Theory (Crandall, Ishii, Lions, Evans, et al.)
USC(X) ≡ {u : X → [−∞, ∞): u is upper semicontinuous}
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 14 / 53 Definition. A function u ∈ USC(X) is F-subharmonic if for every x ∈ X and every test function ϕ for u at x
2 Dx ϕ ∈ F.
F(X) ≡ the set of these.
Viscosity Theory (Crandall, Ishii, Lions, Evans, et al.)
USC(X) ≡ {u : X → [−∞, ∞): u is upper semicontinuous}
Definition. Fix u ∈ USC(X).A test function for u at a point x ∈ X is a function ϕ, C2 near x, such that
u ≤ ϕ near x
u = ϕ at x
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 14 / 53 F(X) ≡ the set of these.
Viscosity Theory (Crandall, Ishii, Lions, Evans, et al.)
USC(X) ≡ {u : X → [−∞, ∞): u is upper semicontinuous}
Definition. Fix u ∈ USC(X).A test function for u at a point x ∈ X is a function ϕ, C2 near x, such that
u ≤ ϕ near x
u = ϕ at x
Definition. A function u ∈ USC(X) is F-subharmonic if for every x ∈ X and every test function ϕ for u at x
2 Dx ϕ ∈ F.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 14 / 53 Viscosity Theory (Crandall, Ishii, Lions, Evans, et al.)
USC(X) ≡ {u : X → [−∞, ∞): u is upper semicontinuous}
Definition. Fix u ∈ USC(X).A test function for u at a point x ∈ X is a function ϕ, C2 near x, such that
u ≤ ϕ near x
u = ϕ at x
Definition. A function u ∈ USC(X) is F-subharmonic if for every x ∈ X and every test function ϕ for u at x
2 Dx ϕ ∈ F.
F(X) ≡ the set of these.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 14 / 53 • u, v ∈ F(X) ⇒ max{u, v} ∈ F(X)
• F(X) is closed under decreasing limits.
• F(X) is closed under uniform limits.
• If F ⊂ F(X) is locally uniformly bounded above, then u∗ ∈ F(X) where
u(x) ≡ sup v(x) v∈F
• If u ∈ C2(X), then
2 u ∈ F(X) ⇐⇒ Dx u ∈ F ∀ x ∈ X.
Remarkable Properties
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 15 / 53 • F(X) is closed under decreasing limits.
• F(X) is closed under uniform limits.
• If F ⊂ F(X) is locally uniformly bounded above, then u∗ ∈ F(X) where
u(x) ≡ sup v(x) v∈F
• If u ∈ C2(X), then
2 u ∈ F(X) ⇐⇒ Dx u ∈ F ∀ x ∈ X.
Remarkable Properties
• u, v ∈ F(X) ⇒ max{u, v} ∈ F(X)
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 15 / 53 • F(X) is closed under uniform limits.
• If F ⊂ F(X) is locally uniformly bounded above, then u∗ ∈ F(X) where
u(x) ≡ sup v(x) v∈F
• If u ∈ C2(X), then
2 u ∈ F(X) ⇐⇒ Dx u ∈ F ∀ x ∈ X.
Remarkable Properties
• u, v ∈ F(X) ⇒ max{u, v} ∈ F(X)
• F(X) is closed under decreasing limits.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 15 / 53 • If F ⊂ F(X) is locally uniformly bounded above, then u∗ ∈ F(X) where
u(x) ≡ sup v(x) v∈F
• If u ∈ C2(X), then
2 u ∈ F(X) ⇐⇒ Dx u ∈ F ∀ x ∈ X.
Remarkable Properties
• u, v ∈ F(X) ⇒ max{u, v} ∈ F(X)
• F(X) is closed under decreasing limits.
• F(X) is closed under uniform limits.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 15 / 53 • If u ∈ C2(X), then
2 u ∈ F(X) ⇐⇒ Dx u ∈ F ∀ x ∈ X.
Remarkable Properties
• u, v ∈ F(X) ⇒ max{u, v} ∈ F(X)
• F(X) is closed under decreasing limits.
• F(X) is closed under uniform limits.
• If F ⊂ F(X) is locally uniformly bounded above, then u∗ ∈ F(X) where
u(x) ≡ sup v(x) v∈F
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 15 / 53 Remarkable Properties
• u, v ∈ F(X) ⇒ max{u, v} ∈ F(X)
• F(X) is closed under decreasing limits.
• F(X) is closed under uniform limits.
• If F ⊂ F(X) is locally uniformly bounded above, then u∗ ∈ F(X) where
u(x) ≡ sup v(x) v∈F
• If u ∈ C2(X), then
2 u ∈ F(X) ⇐⇒ Dx u ∈ F ∀ x ∈ X.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 15 / 53 2 Define the dual of F ⊂ Sym (Rn) by
Fe ≡ ∼ (−IntF) = −(∼ IntF)
• F is a subequation ⇐⇒ Fe is a subequation.
• For each subequation Fe = F • In our analysis
The roles of F and Fe are often interchangeable.
• Note that F ∩ −Fe = ∂F
Duality and F-Harmonics
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 16 / 53 • F is a subequation ⇐⇒ Fe is a subequation.
• For each subequation Fe = F • In our analysis
The roles of F and Fe are often interchangeable.
• Note that F ∩ −Fe = ∂F
Duality and F-Harmonics 2 Define the dual of F ⊂ Sym (Rn) by
Fe ≡ ∼ (−IntF) = −(∼ IntF)
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 16 / 53 • For each subequation Fe = F • In our analysis
The roles of F and Fe are often interchangeable.
• Note that F ∩ −Fe = ∂F
Duality and F-Harmonics 2 Define the dual of F ⊂ Sym (Rn) by
Fe ≡ ∼ (−IntF) = −(∼ IntF)
• F is a subequation ⇐⇒ Fe is a subequation.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 16 / 53 • In our analysis
The roles of F and Fe are often interchangeable.
• Note that F ∩ −Fe = ∂F
Duality and F-Harmonics 2 Define the dual of F ⊂ Sym (Rn) by
Fe ≡ ∼ (−IntF) = −(∼ IntF)
• F is a subequation ⇐⇒ Fe is a subequation.
• For each subequation Fe = F
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 16 / 53 • Note that F ∩ −Fe = ∂F
Duality and F-Harmonics 2 Define the dual of F ⊂ Sym (Rn) by
Fe ≡ ∼ (−IntF) = −(∼ IntF)
• F is a subequation ⇐⇒ Fe is a subequation.
• For each subequation Fe = F • In our analysis
The roles of F and Fe are often interchangeable.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 16 / 53 Duality and F-Harmonics 2 Define the dual of F ⊂ Sym (Rn) by
Fe ≡ ∼ (−IntF) = −(∼ IntF)
• F is a subequation ⇐⇒ Fe is a subequation.
• For each subequation Fe = F • In our analysis
The roles of F and Fe are often interchangeable.
• Note that F ∩ −Fe = ∂F
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 16 / 53 2 Let F ⊂ Sym (Rn) be a subequation.
Definition. A function u on X is F-harmonic if
u ∈ F(X) and − u ∈ Fe(X)
These are our solutions to the equation.
2 2 u ∈ C (X) is F-harmonic ⇐⇒ Dx u ∈ ∂F ∀ x ∈ X.
Duality and F-Harmonics
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 17 / 53 Definition. A function u on X is F-harmonic if
u ∈ F(X) and − u ∈ Fe(X)
These are our solutions to the equation.
2 2 u ∈ C (X) is F-harmonic ⇐⇒ Dx u ∈ ∂F ∀ x ∈ X.
Duality and F-Harmonics
2 Let F ⊂ Sym (Rn) be a subequation.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 17 / 53 These are our solutions to the equation.
2 2 u ∈ C (X) is F-harmonic ⇐⇒ Dx u ∈ ∂F ∀ x ∈ X.
Duality and F-Harmonics
2 Let F ⊂ Sym (Rn) be a subequation.
Definition. A function u on X is F-harmonic if
u ∈ F(X) and − u ∈ Fe(X)
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 17 / 53 2 2 u ∈ C (X) is F-harmonic ⇐⇒ Dx u ∈ ∂F ∀ x ∈ X.
Duality and F-Harmonics
2 Let F ⊂ Sym (Rn) be a subequation.
Definition. A function u on X is F-harmonic if
u ∈ F(X) and − u ∈ Fe(X)
These are our solutions to the equation.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 17 / 53 Duality and F-Harmonics
2 Let F ⊂ Sym (Rn) be a subequation.
Definition. A function u on X is F-harmonic if
u ∈ F(X) and − u ∈ Fe(X)
These are our solutions to the equation.
2 2 u ∈ C (X) is F-harmonic ⇐⇒ Dx u ∈ ∂F ∀ x ∈ X.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 17 / 53 P ≡ {A : A ≥ 0}
Pe = {A : A has at least one eigenvalue ≥ 0}.
Proposition. For an open set X ⊂ Rn P(X) = the convex functions on X.
Pe(X) = the subaffine functions on X.
The homogeneous real Monge-Ampere` Equation D2u ≥ 0 and det(D2u) = 0.
Examples P and Pe
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 18 / 53 Pe = {A : A has at least one eigenvalue ≥ 0}.
Proposition. For an open set X ⊂ Rn P(X) = the convex functions on X.
Pe(X) = the subaffine functions on X.
The homogeneous real Monge-Ampere` Equation D2u ≥ 0 and det(D2u) = 0.
Examples P and Pe
P ≡ {A : A ≥ 0}
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 18 / 53 Proposition. For an open set X ⊂ Rn P(X) = the convex functions on X.
Pe(X) = the subaffine functions on X.
The homogeneous real Monge-Ampere` Equation D2u ≥ 0 and det(D2u) = 0.
Examples P and Pe
P ≡ {A : A ≥ 0}
Pe = {A : A has at least one eigenvalue ≥ 0}.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 18 / 53 Pe(X) = the subaffine functions on X.
The homogeneous real Monge-Ampere` Equation D2u ≥ 0 and det(D2u) = 0.
Examples P and Pe
P ≡ {A : A ≥ 0}
Pe = {A : A has at least one eigenvalue ≥ 0}.
Proposition. For an open set X ⊂ Rn P(X) = the convex functions on X.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 18 / 53 The homogeneous real Monge-Ampere` Equation D2u ≥ 0 and det(D2u) = 0.
Examples P and Pe
P ≡ {A : A ≥ 0}
Pe = {A : A has at least one eigenvalue ≥ 0}.
Proposition. For an open set X ⊂ Rn P(X) = the convex functions on X.
Pe(X) = the subaffine functions on X.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 18 / 53 Examples P and Pe
P ≡ {A : A ≥ 0}
Pe = {A : A has at least one eigenvalue ≥ 0}.
Proposition. For an open set X ⊂ Rn P(X) = the convex functions on X.
Pe(X) = the subaffine functions on X.
The homogeneous real Monge-Ampere` Equation D2u ≥ 0 and det(D2u) = 0.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 18 / 53 THEOREM. (2009)
Let Ω ⊂⊂ Rn be a domain whose boundary ∂Ω is strictly F and F-convex.e Then for each ϕ ∈ C(∂Ω), there exists a unique u ∈ C(Ω) such that
(1) u Ω is F-harmonic, and
(2) u ∂Ω = ϕ.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 19 / 53 Then for each ϕ ∈ C(∂Ω), there exists a unique u ∈ C(Ω) such that
(1) u Ω is F-harmonic, and
(2) u ∂Ω = ϕ.
The Dirichlet Problem
THEOREM. (2009)
Let Ω ⊂⊂ Rn be a domain whose boundary ∂Ω is strictly F and F-convex.e
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 19 / 53
(2) u ∂Ω = ϕ.
The Dirichlet Problem
THEOREM. (2009)
Let Ω ⊂⊂ Rn be a domain whose boundary ∂Ω is strictly F and F-convex.e Then for each ϕ ∈ C(∂Ω), there exists a unique u ∈ C(Ω) such that
(1) u Ω is F-harmonic, and
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 19 / 53 The Dirichlet Problem
THEOREM. (2009)
Let Ω ⊂⊂ Rn be a domain whose boundary ∂Ω is strictly F and F-convex.e Then for each ϕ ∈ C(∂Ω), there exists a unique u ∈ C(Ω) such that
(1) u Ω is F-harmonic, and
(2) u ∂Ω = ϕ.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 19 / 53 2 The subequation F ⊂ Sym (Rn) is a cone which is invariant under a subgroup G ⊂ O(n)
which acts transitively on the sphere
n−1 n S ⊂ R .
Standing Assumption from now on:
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 20 / 53 Standing Assumption from now on:
2 The subequation F ⊂ Sym (Rn) is a cone which is invariant under a subgroup G ⊂ O(n)
which acts transitively on the sphere
n−1 n S ⊂ R .
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 20 / 53 Eigenvalue Equations
2 If F ⊂ Sym (Rn) is O(n)-invariant, then it is completely determined by a condition 2 on the eigenvalues of A ∈ Sym (Rn).
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 21 / 53 Any such F has a complex and quaternionic counterpart denoted F C and F H
by applying the same conditions to the hermitian symmetric parts
AC and AH
These subequations are U(n) and Sp(n) invariant.
Complex and Quaternionic Counterparts
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 22 / 53 These subequations are U(n) and Sp(n) invariant.
Complex and Quaternionic Counterparts
Any such F has a complex and quaternionic counterpart denoted F C and F H
by applying the same conditions to the hermitian symmetric parts
AC and AH
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 22 / 53 Complex and Quaternionic Counterparts
Any such F has a complex and quaternionic counterpart denoted F C and F H
by applying the same conditions to the hermitian symmetric parts
AC and AH
These subequations are U(n) and Sp(n) invariant.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 22 / 53 The eigenspaces of AC are complex lines with eigenvalues λ1, ..., λn.
n 4n H = (R , I, J, K ) 2 The quaternionic hermitian symmetric part of A ∈ Sym (R4n) is
1 AH = 4 (A − IAI − JAJ − KAK ).
The eigenspaces of AH are quaternionic lines with eigenvalues λ1, ..., λn.
n 2n C = (R , J) 2 The complex hermitian symmetric part of A ∈ Sym (R2n) is
1 AC = 2 (A − JAJ).
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 23 / 53 n 4n H = (R , I, J, K ) 2 The quaternionic hermitian symmetric part of A ∈ Sym (R4n) is
1 AH = 4 (A − IAI − JAJ − KAK ).
The eigenspaces of AH are quaternionic lines with eigenvalues λ1, ..., λn.
n 2n C = (R , J) 2 The complex hermitian symmetric part of A ∈ Sym (R2n) is
1 AC = 2 (A − JAJ).
The eigenspaces of AC are complex lines with eigenvalues λ1, ..., λn.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 23 / 53 The eigenspaces of AH are quaternionic lines with eigenvalues λ1, ..., λn.
n 2n C = (R , J) 2 The complex hermitian symmetric part of A ∈ Sym (R2n) is
1 AC = 2 (A − JAJ).
The eigenspaces of AC are complex lines with eigenvalues λ1, ..., λn.
n 4n H = (R , I, J, K ) 2 The quaternionic hermitian symmetric part of A ∈ Sym (R4n) is
1 AH = 4 (A − IAI − JAJ − KAK ).
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 23 / 53 n 2n C = (R , J) 2 The complex hermitian symmetric part of A ∈ Sym (R2n) is
1 AC = 2 (A − JAJ).
The eigenspaces of AC are complex lines with eigenvalues λ1, ..., λn.
n 4n H = (R , I, J, K ) 2 The quaternionic hermitian symmetric part of A ∈ Sym (R4n) is
1 AH = 4 (A − IAI − JAJ − KAK ).
The eigenspaces of AH are quaternionic lines with eigenvalues λ1, ..., λn.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 23 / 53 P ⇒ PC, PH
The complex and quaternionic Monge-Ampere` Equations
Example: Monge-Ampere` Equations
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 24 / 53 Example: Monge-Ampere` Equations
P ⇒ PC, PH
The complex and quaternionic Monge-Ampere` Equations
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 24 / 53 2 For A ∈ Sym (Rn) let
λ1(A) ≤ λ2(A) ≤ · · · ≤ λn(A)
be the ordered eigenvalues of A.
P(k) ≡ {λk (A) ≥ 0}
P](k) = P(n − k + 1)
Examples: Other Branches of the MA Equation
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 25 / 53 P(k) ≡ {λk (A) ≥ 0}
P](k) = P(n − k + 1)
Examples: Other Branches of the MA Equation
2 For A ∈ Sym (Rn) let
λ1(A) ≤ λ2(A) ≤ · · · ≤ λn(A)
be the ordered eigenvalues of A.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 25 / 53 P](k) = P(n − k + 1)
Examples: Other Branches of the MA Equation
2 For A ∈ Sym (Rn) let
λ1(A) ≤ λ2(A) ≤ · · · ≤ λn(A)
be the ordered eigenvalues of A.
P(k) ≡ {λk (A) ≥ 0}
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 25 / 53 Examples: Other Branches of the MA Equation
2 For A ∈ Sym (Rn) let
λ1(A) ≤ λ2(A) ≤ · · · ≤ λn(A)
be the ordered eigenvalues of A.
P(k) ≡ {λk (A) ≥ 0}
P](k) = P(n − k + 1)
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 25 / 53 Σk ≡ {A : σ1(A) ≥ 0, ..., σk (A) ≥ 0}
σk (A) ≡ σk (λ1(A), ..., λn(A))
This is the principal branch of the equation
2 σk (D u) = 0.
The equation has (k − 1) other branches. It also has complex and quaternionic counterparts.
Examples: Other Elementary Symmetric Functions Hessian Equations – Trudinger-Wang-Labutin
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 26 / 53 σk (A) ≡ σk (λ1(A), ..., λn(A))
This is the principal branch of the equation
2 σk (D u) = 0.
The equation has (k − 1) other branches. It also has complex and quaternionic counterparts.
Examples: Other Elementary Symmetric Functions Hessian Equations – Trudinger-Wang-Labutin
Σk ≡ {A : σ1(A) ≥ 0, ..., σk (A) ≥ 0}
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 26 / 53 This is the principal branch of the equation
2 σk (D u) = 0.
The equation has (k − 1) other branches. It also has complex and quaternionic counterparts.
Examples: Other Elementary Symmetric Functions Hessian Equations – Trudinger-Wang-Labutin
Σk ≡ {A : σ1(A) ≥ 0, ..., σk (A) ≥ 0}
σk (A) ≡ σk (λ1(A), ..., λn(A))
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 26 / 53 The equation has (k − 1) other branches. It also has complex and quaternionic counterparts.
Examples: Other Elementary Symmetric Functions Hessian Equations – Trudinger-Wang-Labutin
Σk ≡ {A : σ1(A) ≥ 0, ..., σk (A) ≥ 0}
σk (A) ≡ σk (λ1(A), ..., λn(A))
This is the principal branch of the equation
2 σk (D u) = 0.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 26 / 53 It also has complex and quaternionic counterparts.
Examples: Other Elementary Symmetric Functions Hessian Equations – Trudinger-Wang-Labutin
Σk ≡ {A : σ1(A) ≥ 0, ..., σk (A) ≥ 0}
σk (A) ≡ σk (λ1(A), ..., λn(A))
This is the principal branch of the equation
2 σk (D u) = 0.
The equation has (k − 1) other branches.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 26 / 53 Examples: Other Elementary Symmetric Functions Hessian Equations – Trudinger-Wang-Labutin
Σk ≡ {A : σ1(A) ≥ 0, ..., σk (A) ≥ 0}
σk (A) ≡ σk (λ1(A), ..., λn(A))
This is the principal branch of the equation
2 σk (D u) = 0.
The equation has (k − 1) other branches. It also has complex and quaternionic counterparts.
Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 26 / 53 For each real number p ∈ [1, n], define Pp ≡ A : λ1(A) + ··· + λ[p](A) + (p − [p])λ[p]+1(A) ≥ 0
where λ1(A) ≤ · · · ≤ λn(A) are the ordered eigenvalues of A.
The Pp-subharmonics are p-plurisubharmonic functions.
THEOREM. (2012) For p an integer:
The restriction of a Pp-subharmonic to any minimal p-dimensional submanifold Y is subharmonic in the induced metric on Y .
Pp-harmonics are solutions of the polynomial equation Y MAp(A) = λi1 (A) + ··· + λip (A) = 0.
i1<··· An Important Example: p-Convexity Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 27 / 53 where λ1(A) ≤ · · · ≤ λn(A) are the ordered eigenvalues of A. The Pp-subharmonics are p-plurisubharmonic functions. THEOREM. (2012) For p an integer: The restriction of a Pp-subharmonic to any minimal p-dimensional submanifold Y is subharmonic in the induced metric on Y . Pp-harmonics are solutions of the polynomial equation Y MAp(A) = λi1 (A) + ··· + λip (A) = 0. i1<··· An Important Example: p-Convexity For each real number p ∈ [1, n], define Pp ≡ A : λ1(A) + ··· + λ[p](A) + (p − [p])λ[p]+1(A) ≥ 0 Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 27 / 53 The Pp-subharmonics are p-plurisubharmonic functions. THEOREM. (2012) For p an integer: The restriction of a Pp-subharmonic to any minimal p-dimensional submanifold Y is subharmonic in the induced metric on Y . Pp-harmonics are solutions of the polynomial equation Y MAp(A) = λi1 (A) + ··· + λip (A) = 0. i1<··· An Important Example: p-Convexity For each real number p ∈ [1, n], define Pp ≡ A : λ1(A) + ··· + λ[p](A) + (p − [p])λ[p]+1(A) ≥ 0 where λ1(A) ≤ · · · ≤ λn(A) are the ordered eigenvalues of A. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 27 / 53 THEOREM. (2012) For p an integer: The restriction of a Pp-subharmonic to any minimal p-dimensional submanifold Y is subharmonic in the induced metric on Y . Pp-harmonics are solutions of the polynomial equation Y MAp(A) = λi1 (A) + ··· + λip (A) = 0. i1<··· An Important Example: p-Convexity For each real number p ∈ [1, n], define Pp ≡ A : λ1(A) + ··· + λ[p](A) + (p − [p])λ[p]+1(A) ≥ 0 where λ1(A) ≤ · · · ≤ λn(A) are the ordered eigenvalues of A. The Pp-subharmonics are p-plurisubharmonic functions. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 27 / 53 The restriction of a Pp-subharmonic to any minimal p-dimensional submanifold Y is subharmonic in the induced metric on Y . Pp-harmonics are solutions of the polynomial equation Y MAp(A) = λi1 (A) + ··· + λip (A) = 0. i1<··· An Important Example: p-Convexity For each real number p ∈ [1, n], define Pp ≡ A : λ1(A) + ··· + λ[p](A) + (p − [p])λ[p]+1(A) ≥ 0 where λ1(A) ≤ · · · ≤ λn(A) are the ordered eigenvalues of A. The Pp-subharmonics are p-plurisubharmonic functions. THEOREM. (2012) For p an integer: Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 27 / 53 Pp-harmonics are solutions of the polynomial equation Y MAp(A) = λi1 (A) + ··· + λip (A) = 0. i1<··· An Important Example: p-Convexity For each real number p ∈ [1, n], define Pp ≡ A : λ1(A) + ··· + λ[p](A) + (p − [p])λ[p]+1(A) ≥ 0 where λ1(A) ≤ · · · ≤ λn(A) are the ordered eigenvalues of A. The Pp-subharmonics are p-plurisubharmonic functions. THEOREM. (2012) For p an integer: The restriction of a Pp-subharmonic to any minimal p-dimensional submanifold Y is subharmonic in the induced metric on Y . Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 27 / 53 An Important Example: p-Convexity For each real number p ∈ [1, n], define Pp ≡ A : λ1(A) + ··· + λ[p](A) + (p − [p])λ[p]+1(A) ≥ 0 where λ1(A) ≤ · · · ≤ λn(A) are the ordered eigenvalues of A. The Pp-subharmonics are p-plurisubharmonic functions. THEOREM. (2012) For p an integer: The restriction of a Pp-subharmonic to any minimal p-dimensional submanifold Y is subharmonic in the induced metric on Y . Pp-harmonics are solutions of the polynomial equation Y MAp(A) = λi1 (A) + ··· + λip (A) = 0. i1<··· Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 27 / 53 The Riesz kernel |x|2−p if 1 ≤ p < 2 Kp(x) ≡ log|x| if p = 2 1 − |x|p−2 if p > 2 n is Pp-harmonic in R − {0} and Pp-subharmonic across 0. The Riesz Kernel Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 28 / 53 log|x| if p = 2 1 − |x|p−2 if p > 2 n is Pp-harmonic in R − {0} and Pp-subharmonic across 0. The Riesz Kernel The Riesz kernel |x|2−p if 1 ≤ p < 2 Kp(x) ≡ Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 28 / 53 1 − |x|p−2 if p > 2 n is Pp-harmonic in R − {0} and Pp-subharmonic across 0. The Riesz Kernel The Riesz kernel |x|2−p if 1 ≤ p < 2 Kp(x) ≡ log|x| if p = 2 Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 28 / 53 n is Pp-harmonic in R − {0} and Pp-subharmonic across 0. The Riesz Kernel The Riesz kernel |x|2−p if 1 ≤ p < 2 Kp(x) ≡ log|x| if p = 2 1 − |x|p−2 if p > 2 Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 28 / 53 and Pp-subharmonic across 0. The Riesz Kernel The Riesz kernel |x|2−p if 1 ≤ p < 2 Kp(x) ≡ log|x| if p = 2 1 − |x|p−2 if p > 2 n is Pp-harmonic in R − {0} Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 28 / 53 The Riesz Kernel The Riesz kernel |x|2−p if 1 ≤ p < 2 Kp(x) ≡ log|x| if p = 2 1 − |x|p−2 if p > 2 n is Pp-harmonic in R − {0} and Pp-subharmonic across 0. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 28 / 53 A Particular Instance Valentino Tosatti and Ben Weinkove made use of C C Pn−1-subharmonic and Pn−1-harmonic functions in their recent work on the Monge-Ampere` Equations and the Calabi-Yau problem on general complex manifolds. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 29 / 53 Fix a compact set n Gl ⊂ G(p, R ) and define ( ) = : ≥ . F Gl A tr A W 0 This is always a subequation. An Important Family: Geometric Subequations Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 30 / 53 and define ( ) = : ≥ . F Gl A tr A W 0 This is always a subequation. An Important Family: Geometric Subequations Fix a compact set n Gl ⊂ G(p, R ) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 30 / 53 This is always a subequation. An Important Family: Geometric Subequations Fix a compact set n Gl ⊂ G(p, R ) and define ( ) = : ≥ . F Gl A tr A W 0 Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 30 / 53 An Important Family: Geometric Subequations Fix a compact set n Gl ⊂ G(p, R ) and define ( ) = : ≥ . F Gl A tr A W 0 This is always a subequation. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 30 / 53 If Gl = G(1, Rn), then F(Gl ) = P If Gl = GC(1, Cn), then F(Gl ) = PC If Gl = GH(1, Hn), then F(Gl ) = PH n If Gl = G(p, R ), then F(Gl ) = Pp Gl = LAG ⊂ GR(n, Cn) Gl = SLAG ⊂ GR(n, Cn) Gl = G(φ) where φ is a calibration. Geometric Subequations – Examples Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 31 / 53 If Gl = GC(1, Cn), then F(Gl ) = PC If Gl = GH(1, Hn), then F(Gl ) = PH n If Gl = G(p, R ), then F(Gl ) = Pp Gl = LAG ⊂ GR(n, Cn) Gl = SLAG ⊂ GR(n, Cn) Gl = G(φ) where φ is a calibration. Geometric Subequations – Examples If Gl = G(1, Rn), then F(Gl ) = P Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 31 / 53 If Gl = GH(1, Hn), then F(Gl ) = PH n If Gl = G(p, R ), then F(Gl ) = Pp Gl = LAG ⊂ GR(n, Cn) Gl = SLAG ⊂ GR(n, Cn) Gl = G(φ) where φ is a calibration. Geometric Subequations – Examples If Gl = G(1, Rn), then F(Gl ) = P If Gl = GC(1, Cn), then F(Gl ) = PC Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 31 / 53 n If Gl = G(p, R ), then F(Gl ) = Pp Gl = LAG ⊂ GR(n, Cn) Gl = SLAG ⊂ GR(n, Cn) Gl = G(φ) where φ is a calibration. Geometric Subequations – Examples If Gl = G(1, Rn), then F(Gl ) = P If Gl = GC(1, Cn), then F(Gl ) = PC If Gl = GH(1, Hn), then F(Gl ) = PH Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 31 / 53 Gl = LAG ⊂ GR(n, Cn) Gl = SLAG ⊂ GR(n, Cn) Gl = G(φ) where φ is a calibration. Geometric Subequations – Examples If Gl = G(1, Rn), then F(Gl ) = P If Gl = GC(1, Cn), then F(Gl ) = PC If Gl = GH(1, Hn), then F(Gl ) = PH n If Gl = G(p, R ), then F(Gl ) = Pp Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 31 / 53 Gl = SLAG ⊂ GR(n, Cn) Gl = G(φ) where φ is a calibration. Geometric Subequations – Examples If Gl = G(1, Rn), then F(Gl ) = P If Gl = GC(1, Cn), then F(Gl ) = PC If Gl = GH(1, Hn), then F(Gl ) = PH n If Gl = G(p, R ), then F(Gl ) = Pp Gl = LAG ⊂ GR(n, Cn) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 31 / 53 Gl = G(φ) where φ is a calibration. Geometric Subequations – Examples If Gl = G(1, Rn), then F(Gl ) = P If Gl = GC(1, Cn), then F(Gl ) = PC If Gl = GH(1, Hn), then F(Gl ) = PH n If Gl = G(p, R ), then F(Gl ) = Pp Gl = LAG ⊂ GR(n, Cn) Gl = SLAG ⊂ GR(n, Cn) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 31 / 53 Geometric Subequations – Examples If Gl = G(1, Rn), then F(Gl ) = P If Gl = GC(1, Cn), then F(Gl ) = PC If Gl = GH(1, Hn), then F(Gl ) = PH n If Gl = G(p, R ), then F(Gl ) = Pp Gl = LAG ⊂ GR(n, Cn) Gl = SLAG ⊂ GR(n, Cn) Gl = G(φ) where φ is a calibration. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 31 / 53 Fix an invariant cone subequation 2 n F ⊂ Sym (R ). Definition. The Riesz characteristic of F is the number pF = sup {p : Pe⊥ − (p − 1)Pe ∈ F} where Pe = orthogonal projection onto Re and Pe⊥ = orthogonal projection onto the complement. 1 1 −(p − 1) The Riesz Characteristic Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 32 / 53 Definition. The Riesz characteristic of F is the number pF = sup {p : Pe⊥ − (p − 1)Pe ∈ F} where Pe = orthogonal projection onto Re and Pe⊥ = orthogonal projection onto the complement. 1 1 −(p − 1) The Riesz Characteristic Fix an invariant cone subequation 2 n F ⊂ Sym (R ). Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 32 / 53 where Pe = orthogonal projection onto Re and Pe⊥ = orthogonal projection onto the complement. 1 1 −(p − 1) The Riesz Characteristic Fix an invariant cone subequation 2 n F ⊂ Sym (R ). Definition. The Riesz characteristic of F is the number pF = sup {p : Pe⊥ − (p − 1)Pe ∈ F} Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 32 / 53 and Pe⊥ = orthogonal projection onto the complement. 1 1 −(p − 1) The Riesz Characteristic Fix an invariant cone subequation 2 n F ⊂ Sym (R ). Definition. The Riesz characteristic of F is the number pF = sup {p : Pe⊥ − (p − 1)Pe ∈ F} where Pe = orthogonal projection onto Re Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 32 / 53 1 1 −(p − 1) The Riesz Characteristic Fix an invariant cone subequation 2 n F ⊂ Sym (R ). Definition. The Riesz characteristic of F is the number pF = sup {p : Pe⊥ − (p − 1)Pe ∈ F} where Pe = orthogonal projection onto Re and Pe⊥ = orthogonal projection onto the complement. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 32 / 53 The Riesz Characteristic Fix an invariant cone subequation 2 n F ⊂ Sym (R ). Definition. The Riesz characteristic of F is the number pF = sup {p : Pe⊥ − (p − 1)Pe ∈ F} where Pe = orthogonal projection onto Re and Pe⊥ = orthogonal projection onto the complement. 1 1 −(p − 1) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 32 / 53 F = P pF = 1 F = PC pF = 2 F = PH pF = 4 pF = p ⇒ pF C = 2p and pF H = 4p n F = Σk p = k F = Pp pF = p n Gl ⊂ G(p, R ) pF(lG) = p Examples Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 33 / 53 F = PC pF = 2 F = PH pF = 4 pF = p ⇒ pF C = 2p and pF H = 4p n F = Σk p = k F = Pp pF = p n Gl ⊂ G(p, R ) pF(lG) = p Examples F = P pF = 1 Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 33 / 53 F = PH pF = 4 pF = p ⇒ pF C = 2p and pF H = 4p n F = Σk p = k F = Pp pF = p n Gl ⊂ G(p, R ) pF(lG) = p Examples F = P pF = 1 F = PC pF = 2 Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 33 / 53 pF = p ⇒ pF C = 2p and pF H = 4p n F = Σk p = k F = Pp pF = p n Gl ⊂ G(p, R ) pF(lG) = p Examples F = P pF = 1 F = PC pF = 2 F = PH pF = 4 Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 33 / 53 n F = Σk p = k F = Pp pF = p n Gl ⊂ G(p, R ) pF(lG) = p Examples F = P pF = 1 F = PC pF = 2 F = PH pF = 4 pF = p ⇒ pF C = 2p and pF H = 4p Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 33 / 53 F = Pp pF = p n Gl ⊂ G(p, R ) pF(lG) = p Examples F = P pF = 1 F = PC pF = 2 F = PH pF = 4 pF = p ⇒ pF C = 2p and pF H = 4p n F = Σk p = k Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 33 / 53 n Gl ⊂ G(p, R ) pF(lG) = p Examples F = P pF = 1 F = PC pF = 2 F = PH pF = 4 pF = p ⇒ pF C = 2p and pF H = 4p n F = Σk p = k F = Pp pF = p Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 33 / 53 Examples F = P pF = 1 F = PC pF = 2 F = PH pF = 4 pF = p ⇒ pF C = 2p and pF H = 4p n F = Σk p = k F = Pp pF = p n Gl ⊂ G(p, R ) pF(lG) = p Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 33 / 53 Examples F = P pF = 1 F = PC pF = 2 F = PH pF = 4 pF = p ⇒ pF C = 2p and pF H = 4p n F = Σk p = k F = Pp pF = p n Gl ⊂ G(p, R ) pF(lG) = p Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 33 / 53 The δ-uniformly elliptic equation: n(1 + δ) P (δ) ≡ A : A + δ tr(A)I ≥ 0 p = n n + δ The trace power equation: q 1 F ≡ {A : tr(A ) ≥ 0} p = 1 + (n − 1) q The Pucci equation: For 0 < λ < Λ λ P ≡ {A ∈ Sym2( n): λtrA+ + ΛtrA− ≥ 0}, p = (n − 1) + 1. λ,Λ R Λ Examples Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 34 / 53 The trace power equation: q 1 F ≡ {A : tr(A ) ≥ 0} p = 1 + (n − 1) q The Pucci equation: For 0 < λ < Λ λ P ≡ {A ∈ Sym2( n): λtrA+ + ΛtrA− ≥ 0}, p = (n − 1) + 1. λ,Λ R Λ Examples The δ-uniformly elliptic equation: n(1 + δ) P (δ) ≡ A : A + δ tr(A)I ≥ 0 p = n n + δ Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 34 / 53 The Pucci equation: For 0 < λ < Λ λ P ≡ {A ∈ Sym2( n): λtrA+ + ΛtrA− ≥ 0}, p = (n − 1) + 1. λ,Λ R Λ Examples The δ-uniformly elliptic equation: n(1 + δ) P (δ) ≡ A : A + δ tr(A)I ≥ 0 p = n n + δ The trace power equation: q 1 F ≡ {A : tr(A ) ≥ 0} p = 1 + (n − 1) q Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 34 / 53 Examples The δ-uniformly elliptic equation: n(1 + δ) P (δ) ≡ A : A + δ tr(A)I ≥ 0 p = n n + δ The trace power equation: q 1 F ≡ {A : tr(A ) ≥ 0} p = 1 + (n − 1) q The Pucci equation: For 0 < λ < Λ λ P ≡ {A ∈ Sym2( n): λtrA+ + ΛtrA− ≥ 0}, p = (n − 1) + 1. λ,Λ R Λ Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 34 / 53 THEOREM Let M be a convex cone subequation with Riesz characteristic p, and suppose F is any subequation satisfying F + M ⊂ F. Then any closed set E of locally finite Hausdorff p-measure is removable for F-subharmonics and F-harmonics. That is, for Ωopen ⊂ Rn and u ∈ F(Ω − E): If u is locally bounded above at points of E, then u extends to u ∈ F(Ω) If u is continuous on Ω and F-harmonic on Ω − E, then u is F-harmonic on Ω Removable Singularities Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 35 / 53 and suppose F is any subequation satisfying F + M ⊂ F. Then any closed set E of locally finite Hausdorff p-measure is removable for F-subharmonics and F-harmonics. That is, for Ωopen ⊂ Rn and u ∈ F(Ω − E): If u is locally bounded above at points of E, then u extends to u ∈ F(Ω) If u is continuous on Ω and F-harmonic on Ω − E, then u is F-harmonic on Ω Removable Singularities THEOREM Let M be a convex cone subequation with Riesz characteristic p, Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 35 / 53 Then any closed set E of locally finite Hausdorff p-measure is removable for F-subharmonics and F-harmonics. That is, for Ωopen ⊂ Rn and u ∈ F(Ω − E): If u is locally bounded above at points of E, then u extends to u ∈ F(Ω) If u is continuous on Ω and F-harmonic on Ω − E, then u is F-harmonic on Ω Removable Singularities THEOREM Let M be a convex cone subequation with Riesz characteristic p, and suppose F is any subequation satisfying F + M ⊂ F. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 35 / 53 That is, for Ωopen ⊂ Rn and u ∈ F(Ω − E): If u is locally bounded above at points of E, then u extends to u ∈ F(Ω) If u is continuous on Ω and F-harmonic on Ω − E, then u is F-harmonic on Ω Removable Singularities THEOREM Let M be a convex cone subequation with Riesz characteristic p, and suppose F is any subequation satisfying F + M ⊂ F. Then any closed set E of locally finite Hausdorff p-measure is removable for F-subharmonics and F-harmonics. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 35 / 53 If u is locally bounded above at points of E, then u extends to u ∈ F(Ω) If u is continuous on Ω and F-harmonic on Ω − E, then u is F-harmonic on Ω Removable Singularities THEOREM Let M be a convex cone subequation with Riesz characteristic p, and suppose F is any subequation satisfying F + M ⊂ F. Then any closed set E of locally finite Hausdorff p-measure is removable for F-subharmonics and F-harmonics. That is, for Ωopen ⊂ Rn and u ∈ F(Ω − E): Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 35 / 53 then u extends to u ∈ F(Ω) If u is continuous on Ω and F-harmonic on Ω − E, then u is F-harmonic on Ω Removable Singularities THEOREM Let M be a convex cone subequation with Riesz characteristic p, and suppose F is any subequation satisfying F + M ⊂ F. Then any closed set E of locally finite Hausdorff p-measure is removable for F-subharmonics and F-harmonics. That is, for Ωopen ⊂ Rn and u ∈ F(Ω − E): If u is locally bounded above at points of E, Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 35 / 53 If u is continuous on Ω and F-harmonic on Ω − E, then u is F-harmonic on Ω Removable Singularities THEOREM Let M be a convex cone subequation with Riesz characteristic p, and suppose F is any subequation satisfying F + M ⊂ F. Then any closed set E of locally finite Hausdorff p-measure is removable for F-subharmonics and F-harmonics. That is, for Ωopen ⊂ Rn and u ∈ F(Ω − E): If u is locally bounded above at points of E, then u extends to u ∈ F(Ω) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 35 / 53 Removable Singularities THEOREM Let M be a convex cone subequation with Riesz characteristic p, and suppose F is any subequation satisfying F + M ⊂ F. Then any closed set E of locally finite Hausdorff p-measure is removable for F-subharmonics and F-harmonics. That is, for Ωopen ⊂ Rn and u ∈ F(Ω − E): If u is locally bounded above at points of E, then u extends to u ∈ F(Ω) If u is continuous on Ω and F-harmonic on Ω − E, then u is F-harmonic on Ω Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 35 / 53 Suppose pF = p. The Function u(x) = ψ(|x|) is an increasing radial F-subharmonic function iff ψ(t) satisfies the one-variable subequation (p − 1) ψ00(t) + ψ0(t) ≥ 0 and ψ0(t) ≥ 0. t Increasing Radial Subharmonics Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 36 / 53 The Function u(x) = ψ(|x|) is an increasing radial F-subharmonic function iff ψ(t) satisfies the one-variable subequation (p − 1) ψ00(t) + ψ0(t) ≥ 0 and ψ0(t) ≥ 0. t Increasing Radial Subharmonics Suppose pF = p. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 36 / 53 iff ψ(t) satisfies the one-variable subequation (p − 1) ψ00(t) + ψ0(t) ≥ 0 and ψ0(t) ≥ 0. t Increasing Radial Subharmonics Suppose pF = p. The Function u(x) = ψ(|x|) is an increasing radial F-subharmonic function Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 36 / 53 Increasing Radial Subharmonics Suppose pF = p. The Function u(x) = ψ(|x|) is an increasing radial F-subharmonic function iff ψ(t) satisfies the one-variable subequation (p − 1) ψ00(t) + ψ0(t) ≥ 0 and ψ0(t) ≥ 0. t Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 36 / 53 Suppose pF = p. The increasing radial harmonics for F are: CKp(|x|) + k th where C ≥ 0, k ∈ R, and Kp(t) is the p Riesz function defined on 0 < t < ∞ by t2−p if 1 ≤ p < 2 Kp(t) = log t if p = 2 1 − tp−2 if 2 < p < ∞. Increasing Radial Harmonics Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 37 / 53 The increasing radial harmonics for F are: CKp(|x|) + k th where C ≥ 0, k ∈ R, and Kp(t) is the p Riesz function defined on 0 < t < ∞ by t2−p if 1 ≤ p < 2 Kp(t) = log t if p = 2 1 − tp−2 if 2 < p < ∞. Increasing Radial Harmonics Suppose pF = p. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 37 / 53 th where C ≥ 0, k ∈ R, and Kp(t) is the p Riesz function defined on 0 < t < ∞ by t2−p if 1 ≤ p < 2 Kp(t) = log t if p = 2 1 − tp−2 if 2 < p < ∞. Increasing Radial Harmonics Suppose pF = p. The increasing radial harmonics for F are: CKp(|x|) + k Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 37 / 53 Increasing Radial Harmonics Suppose pF = p. The increasing radial harmonics for F are: CKp(|x|) + k th where C ≥ 0, k ∈ R, and Kp(t) is the p Riesz function defined on 0 < t < ∞ by t2−p if 1 ≤ p < 2 Kp(t) = log t if p = 2 1 − tp−2 if 2 < p < ∞. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 37 / 53 Suppose u is F-subharmonic on a neighborhood of 0 ∈ Rn. Let Br = {|x| ≤ r}. Then M(r) = M(u, r) ≡ sup u. Br is an increasing radial F-subharmonic. When F is convex, so also are the functions 1 Z V (r) ≡ u |Br | Br 1 Z S(r) ≡ u |∂Br | ∂Br Basic Functions Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 38 / 53 Let Br = {|x| ≤ r}. Then M(r) = M(u, r) ≡ sup u. Br is an increasing radial F-subharmonic. When F is convex, so also are the functions 1 Z V (r) ≡ u |Br | Br 1 Z S(r) ≡ u |∂Br | ∂Br Basic Functions Suppose u is F-subharmonic on a neighborhood of 0 ∈ Rn. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 38 / 53 Then M(r) = M(u, r) ≡ sup u. Br is an increasing radial F-subharmonic. When F is convex, so also are the functions 1 Z V (r) ≡ u |Br | Br 1 Z S(r) ≡ u |∂Br | ∂Br Basic Functions Suppose u is F-subharmonic on a neighborhood of 0 ∈ Rn. Let Br = {|x| ≤ r}. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 38 / 53 When F is convex, so also are the functions 1 Z V (r) ≡ u |Br | Br 1 Z S(r) ≡ u |∂Br | ∂Br Basic Functions Suppose u is F-subharmonic on a neighborhood of 0 ∈ Rn. Let Br = {|x| ≤ r}. Then M(r) = M(u, r) ≡ sup u. Br is an increasing radial F-subharmonic. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 38 / 53 1 Z S(r) ≡ u |∂Br | ∂Br Basic Functions Suppose u is F-subharmonic on a neighborhood of 0 ∈ Rn. Let Br = {|x| ≤ r}. Then M(r) = M(u, r) ≡ sup u. Br is an increasing radial F-subharmonic. When F is convex, so also are the functions 1 Z V (r) ≡ u |Br | Br Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 38 / 53 Basic Functions Suppose u is F-subharmonic on a neighborhood of 0 ∈ Rn. Let Br = {|x| ≤ r}. Then M(r) = M(u, r) ≡ sup u. Br is an increasing radial F-subharmonic. When F is convex, so also are the functions 1 Z V (r) ≡ u |Br | Br 1 Z S(r) ≡ u |∂Br | ∂Br Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 38 / 53 Let p = pF with 1 ≤ p < ∞. THEOREM (The Fundamental Monotonicity Property). Let Ψ(r) = M(r), S(r) or V (r) Then, for 0 < r < t < R, the non-negative quantity Ψ(t) − Ψ(r) is increasing in r and t, K (t) − K (r) th where K = Kp is the p Riesz function. Monotonicity Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 39 / 53 THEOREM (The Fundamental Monotonicity Property). Let Ψ(r) = M(r), S(r) or V (r) Then, for 0 < r < t < R, the non-negative quantity Ψ(t) − Ψ(r) is increasing in r and t, K (t) − K (r) th where K = Kp is the p Riesz function. Monotonicity Let p = pF with 1 ≤ p < ∞. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 39 / 53 Then, for 0 < r < t < R, the non-negative quantity Ψ(t) − Ψ(r) is increasing in r and t, K (t) − K (r) th where K = Kp is the p Riesz function. Monotonicity Let p = pF with 1 ≤ p < ∞. THEOREM (The Fundamental Monotonicity Property). Let Ψ(r) = M(r), S(r) or V (r) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 39 / 53 th where K = Kp is the p Riesz function. Monotonicity Let p = pF with 1 ≤ p < ∞. THEOREM (The Fundamental Monotonicity Property). Let Ψ(r) = M(r), S(r) or V (r) Then, for 0 < r < t < R, the non-negative quantity Ψ(t) − Ψ(r) is increasing in r and t, K (t) − K (r) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 39 / 53 Monotonicity Let p = pF with 1 ≤ p < ∞. THEOREM (The Fundamental Monotonicity Property). Let Ψ(r) = M(r), S(r) or V (r) Then, for 0 < r < t < R, the non-negative quantity Ψ(t) − Ψ(r) is increasing in r and t, K (t) − K (r) th where K = Kp is the p Riesz function. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 39 / 53 THEOREM. Suppose 1 ≤ p < 2. Then every F-subharmonic function is α-Holder¨ continuous, where α = 2 − p. False for all p ≥ 2. ( log |x| if p = 2 Kp(|x|) = 1 − |x|p−2 if 2 < p < ∞. is F-subhamonic. When F = PC u(z) = log|f (z)| with f holomorphic is plurisubharmonic. A First Application – Regularity Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 40 / 53 Then every F-subharmonic function is α-Holder¨ continuous, where α = 2 − p. False for all p ≥ 2. ( log |x| if p = 2 Kp(|x|) = 1 − |x|p−2 if 2 < p < ∞. is F-subhamonic. When F = PC u(z) = log|f (z)| with f holomorphic is plurisubharmonic. A First Application – Regularity THEOREM. Suppose 1 ≤ p < 2. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 40 / 53 where α = 2 − p. False for all p ≥ 2. ( log |x| if p = 2 Kp(|x|) = 1 − |x|p−2 if 2 < p < ∞. is F-subhamonic. When F = PC u(z) = log|f (z)| with f holomorphic is plurisubharmonic. A First Application – Regularity THEOREM. Suppose 1 ≤ p < 2. Then every F-subharmonic function is α-Holder¨ continuous, Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 40 / 53 False for all p ≥ 2. ( log |x| if p = 2 Kp(|x|) = 1 − |x|p−2 if 2 < p < ∞. is F-subhamonic. When F = PC u(z) = log|f (z)| with f holomorphic is plurisubharmonic. A First Application – Regularity THEOREM. Suppose 1 ≤ p < 2. Then every F-subharmonic function is α-Holder¨ continuous, where α = 2 − p. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 40 / 53 ( log |x| if p = 2 Kp(|x|) = 1 − |x|p−2 if 2 < p < ∞. is F-subhamonic. When F = PC u(z) = log|f (z)| with f holomorphic is plurisubharmonic. A First Application – Regularity THEOREM. Suppose 1 ≤ p < 2. Then every F-subharmonic function is α-Holder¨ continuous, where α = 2 − p. False for all p ≥ 2. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 40 / 53 When F = PC u(z) = log|f (z)| with f holomorphic is plurisubharmonic. A First Application – Regularity THEOREM. Suppose 1 ≤ p < 2. Then every F-subharmonic function is α-Holder¨ continuous, where α = 2 − p. False for all p ≥ 2. ( log |x| if p = 2 Kp(|x|) = 1 − |x|p−2 if 2 < p < ∞. is F-subhamonic. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 40 / 53 A First Application – Regularity THEOREM. Suppose 1 ≤ p < 2. Then every F-subharmonic function is α-Holder¨ continuous, where α = 2 − p. False for all p ≥ 2. ( log |x| if p = 2 Kp(|x|) = 1 − |x|p−2 if 2 < p < ∞. is F-subhamonic. When F = PC u(z) = log|f (z)| with f holomorphic is plurisubharmonic. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 40 / 53 COROLLARY of the Basic Monotonicity Property The decreasing limit Ψ(t) − Ψ(r) ΘΨ(u, 0) = lim r, t → 0 K (t) − K (r) t > r > 0 exists and defines the Ψ-density, 0 ≤ ΘΨ < ∞, of u at 0. Ψ = M, S or V Existence of Densities Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 41 / 53 The decreasing limit Ψ(t) − Ψ(r) ΘΨ(u, 0) = lim r, t → 0 K (t) − K (r) t > r > 0 exists and defines the Ψ-density, 0 ≤ ΘΨ < ∞, of u at 0. Ψ = M, S or V Existence of Densities COROLLARY of the Basic Monotonicity Property Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 41 / 53 and defines the Ψ-density, 0 ≤ ΘΨ < ∞, of u at 0. Ψ = M, S or V Existence of Densities COROLLARY of the Basic Monotonicity Property The decreasing limit Ψ(t) − Ψ(r) ΘΨ(u, 0) = lim r, t → 0 K (t) − K (r) t > r > 0 exists Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 41 / 53 Existence of Densities COROLLARY of the Basic Monotonicity Property The decreasing limit Ψ(t) − Ψ(r) ΘΨ(u, 0) = lim r, t → 0 K (t) − K (r) t > r > 0 exists and defines the Ψ-density, 0 ≤ ΘΨ < ∞, of u at 0. Ψ = M, S or V Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 41 / 53 (When 1 ≤ p < 2, we must normalize so that Ψ(0) = 0.) Densities In fact, Ψ(r) ΘΨ(u, 0) = lim . r↓0 K (r) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 42 / 53 Densities In fact, Ψ(r) ΘΨ(u, 0) = lim . r↓0 K (r) (When 1 ≤ p < 2, we must normalize so that Ψ(0) = 0.) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 42 / 53 ΘΨ(u, x) is defined at each x in domain of u. THEOREM. The function x 7→ ΘΨ(u, x) is upper semi-continuous. Corollary. For each c > 0 the set Ψ Ec ≡ {x :Θ (u, x) ≥ c} is closed. Densities Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 43 / 53 THEOREM. The function x 7→ ΘΨ(u, x) is upper semi-continuous. Corollary. For each c > 0 the set Ψ Ec ≡ {x :Θ (u, x) ≥ c} is closed. Densities ΘΨ(u, x) is defined at each x in domain of u. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 43 / 53 Corollary. For each c > 0 the set Ψ Ec ≡ {x :Θ (u, x) ≥ c} is closed. Densities ΘΨ(u, x) is defined at each x in domain of u. THEOREM. The function x 7→ ΘΨ(u, x) is upper semi-continuous. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 43 / 53 Densities ΘΨ(u, x) is defined at each x in domain of u. THEOREM. The function x 7→ ΘΨ(u, x) is upper semi-continuous. Corollary. For each c > 0 the set Ψ Ec ≡ {x :Θ (u, x) ≥ c} is closed. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 43 / 53 In the classical plurisubharmonic case F = PC. In this case all densities agree up to universal constants. There is the following deep result due to Hormander,¨ Bombieri, and in its final form Siu. THEOREM. Ec is a complex analytic subvariety. Question: Are there analogous results for other subequations? The Hormander-¨ Bombieri-Siu Theorem Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 44 / 53 In this case all densities agree up to universal constants. There is the following deep result due to Hormander,¨ Bombieri, and in its final form Siu. THEOREM. Ec is a complex analytic subvariety. Question: Are there analogous results for other subequations? The Hormander-¨ Bombieri-Siu Theorem In the classical plurisubharmonic case F = PC. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 44 / 53 There is the following deep result due to Hormander,¨ Bombieri, and in its final form Siu. THEOREM. Ec is a complex analytic subvariety. Question: Are there analogous results for other subequations? The Hormander-¨ Bombieri-Siu Theorem In the classical plurisubharmonic case F = PC. In this case all densities agree up to universal constants. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 44 / 53 THEOREM. Ec is a complex analytic subvariety. Question: Are there analogous results for other subequations? The Hormander-¨ Bombieri-Siu Theorem In the classical plurisubharmonic case F = PC. In this case all densities agree up to universal constants. There is the following deep result due to Hormander,¨ Bombieri, and in its final form Siu. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 44 / 53 Question: Are there analogous results for other subequations? The Hormander-¨ Bombieri-Siu Theorem In the classical plurisubharmonic case F = PC. In this case all densities agree up to universal constants. There is the following deep result due to Hormander,¨ Bombieri, and in its final form Siu. THEOREM. Ec is a complex analytic subvariety. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 44 / 53 The Hormander-¨ Bombieri-Siu Theorem In the classical plurisubharmonic case F = PC. In this case all densities agree up to universal constants. There is the following deep result due to Hormander,¨ Bombieri, and in its final form Siu. THEOREM. Ec is a complex analytic subvariety. Question: Are there analogous results for other subequations? Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 44 / 53 THEOREM. Suppose strong uniqueness of tangents holds for F. Then for any F-subharmonic function u and any c > 0, the set Ec(u) is discrete. This result is essentially sharp. Structure Theorem Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 45 / 53 Then for any F-subharmonic function u and any c > 0, the set Ec(u) is discrete. This result is essentially sharp. Structure Theorem THEOREM. Suppose strong uniqueness of tangents holds for F. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 45 / 53 This result is essentially sharp. Structure Theorem THEOREM. Suppose strong uniqueness of tangents holds for F. Then for any F-subharmonic function u and any c > 0, the set Ec(u) is discrete. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 45 / 53 Structure Theorem THEOREM. Suppose strong uniqueness of tangents holds for F. Then for any F-subharmonic function u and any c > 0, the set Ec(u) is discrete. This result is essentially sharp. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 45 / 53 THEOREM. Suppose Ω ⊂⊂ Rn is a domain with strictly F-convex boundary. Suppose given: + (xj , Θj ) ∈ Ω × R j = 1, ..., N ϕ ∈ C(∂Ω) S Then there exists a unique H ∈ C Ω − j {xj } such that: S (1) H is F-harmonic on Ω − j {xj }, = ϕ (2) H ∂B , (3) There exists constants c, C. so that for each j, Θj Kp(|x − xj |) + c ≤ H(x) ≤ Θj Kp(|x − xj |) + C The Dirichlet Problem with Prescribed Asymptotics Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 46 / 53 Suppose given: + (xj , Θj ) ∈ Ω × R j = 1, ..., N ϕ ∈ C(∂Ω) S Then there exists a unique H ∈ C Ω − j {xj } such that: S (1) H is F-harmonic on Ω − j {xj }, = ϕ (2) H ∂B , (3) There exists constants c, C. so that for each j, Θj Kp(|x − xj |) + c ≤ H(x) ≤ Θj Kp(|x − xj |) + C The Dirichlet Problem with Prescribed Asymptotics THEOREM. Suppose Ω ⊂⊂ Rn is a domain with strictly F-convex boundary. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 46 / 53 ϕ ∈ C(∂Ω) S Then there exists a unique H ∈ C Ω − j {xj } such that: S (1) H is F-harmonic on Ω − j {xj }, = ϕ (2) H ∂B , (3) There exists constants c, C. so that for each j, Θj Kp(|x − xj |) + c ≤ H(x) ≤ Θj Kp(|x − xj |) + C The Dirichlet Problem with Prescribed Asymptotics THEOREM. Suppose Ω ⊂⊂ Rn is a domain with strictly F-convex boundary. Suppose given: + (xj , Θj ) ∈ Ω × R j = 1, ..., N Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 46 / 53 S Then there exists a unique H ∈ C Ω − j {xj } such that: S (1) H is F-harmonic on Ω − j {xj }, = ϕ (2) H ∂B , (3) There exists constants c, C. so that for each j, Θj Kp(|x − xj |) + c ≤ H(x) ≤ Θj Kp(|x − xj |) + C The Dirichlet Problem with Prescribed Asymptotics THEOREM. Suppose Ω ⊂⊂ Rn is a domain with strictly F-convex boundary. Suppose given: + (xj , Θj ) ∈ Ω × R j = 1, ..., N ϕ ∈ C(∂Ω) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 46 / 53 S (1) H is F-harmonic on Ω − j {xj }, = ϕ (2) H ∂B , (3) There exists constants c, C. so that for each j, Θj Kp(|x − xj |) + c ≤ H(x) ≤ Θj Kp(|x − xj |) + C The Dirichlet Problem with Prescribed Asymptotics THEOREM. Suppose Ω ⊂⊂ Rn is a domain with strictly F-convex boundary. Suppose given: + (xj , Θj ) ∈ Ω × R j = 1, ..., N ϕ ∈ C(∂Ω) S Then there exists a unique H ∈ C Ω − j {xj } such that: Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 46 / 53 = ϕ (2) H ∂B , (3) There exists constants c, C. so that for each j, Θj Kp(|x − xj |) + c ≤ H(x) ≤ Θj Kp(|x − xj |) + C The Dirichlet Problem with Prescribed Asymptotics THEOREM. Suppose Ω ⊂⊂ Rn is a domain with strictly F-convex boundary. Suppose given: + (xj , Θj ) ∈ Ω × R j = 1, ..., N ϕ ∈ C(∂Ω) S Then there exists a unique H ∈ C Ω − j {xj } such that: S (1) H is F-harmonic on Ω − j {xj }, Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 46 / 53 (3) There exists constants c, C. so that for each j, Θj Kp(|x − xj |) + c ≤ H(x) ≤ Θj Kp(|x − xj |) + C The Dirichlet Problem with Prescribed Asymptotics THEOREM. Suppose Ω ⊂⊂ Rn is a domain with strictly F-convex boundary. Suppose given: + (xj , Θj ) ∈ Ω × R j = 1, ..., N ϕ ∈ C(∂Ω) S Then there exists a unique H ∈ C Ω − j {xj } such that: S (1) H is F-harmonic on Ω − j {xj }, = ϕ (2) H ∂B , Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 46 / 53 The Dirichlet Problem with Prescribed Asymptotics THEOREM. Suppose Ω ⊂⊂ Rn is a domain with strictly F-convex boundary. Suppose given: + (xj , Θj ) ∈ Ω × R j = 1, ..., N ϕ ∈ C(∂Ω) S Then there exists a unique H ∈ C Ω − j {xj } such that: S (1) H is F-harmonic on Ω − j {xj }, = ϕ (2) H ∂B , (3) There exists constants c, C. so that for each j, Θj Kp(|x − xj |) + c ≤ H(x) ≤ Θj Kp(|x − xj |) + C Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 46 / 53 We now assume F to be convex with Riesz characteristic p = pF < ∞. The homogeneity of the Riesz functions leads one to following. Def. Suppose u ∈ F(BR) and consider the family of functions {ur }r>0 defined by p−2 (a) ur (x) = r u(rx) if p > 2, (b) u (x) = u(rx) − sup u if p = 2, and r Br 1 (c) ur (x) = r 2−p [u(rx) − u(0)] if 1 ≤ p < 2, n Note: One has ur ∈ F(BR/r ) and BR/r expands to R as r ↓ 0. Tangents to Subsolutions Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 47 / 53 The homogeneity of the Riesz functions leads one to following. Def. Suppose u ∈ F(BR) and consider the family of functions {ur }r>0 defined by p−2 (a) ur (x) = r u(rx) if p > 2, (b) u (x) = u(rx) − sup u if p = 2, and r Br 1 (c) ur (x) = r 2−p [u(rx) − u(0)] if 1 ≤ p < 2, n Note: One has ur ∈ F(BR/r ) and BR/r expands to R as r ↓ 0. Tangents to Subsolutions We now assume F to be convex with Riesz characteristic p = pF < ∞. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 47 / 53 Def. Suppose u ∈ F(BR) and consider the family of functions {ur }r>0 defined by p−2 (a) ur (x) = r u(rx) if p > 2, (b) u (x) = u(rx) − sup u if p = 2, and r Br 1 (c) ur (x) = r 2−p [u(rx) − u(0)] if 1 ≤ p < 2, n Note: One has ur ∈ F(BR/r ) and BR/r expands to R as r ↓ 0. Tangents to Subsolutions We now assume F to be convex with Riesz characteristic p = pF < ∞. The homogeneity of the Riesz functions leads one to following. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 47 / 53 and consider the family of functions {ur }r>0 defined by p−2 (a) ur (x) = r u(rx) if p > 2, (b) u (x) = u(rx) − sup u if p = 2, and r Br 1 (c) ur (x) = r 2−p [u(rx) − u(0)] if 1 ≤ p < 2, n Note: One has ur ∈ F(BR/r ) and BR/r expands to R as r ↓ 0. Tangents to Subsolutions We now assume F to be convex with Riesz characteristic p = pF < ∞. The homogeneity of the Riesz functions leads one to following. Def. Suppose u ∈ F(BR) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 47 / 53 p−2 (a) ur (x) = r u(rx) if p > 2, (b) u (x) = u(rx) − sup u if p = 2, and r Br 1 (c) ur (x) = r 2−p [u(rx) − u(0)] if 1 ≤ p < 2, n Note: One has ur ∈ F(BR/r ) and BR/r expands to R as r ↓ 0. Tangents to Subsolutions We now assume F to be convex with Riesz characteristic p = pF < ∞. The homogeneity of the Riesz functions leads one to following. Def. Suppose u ∈ F(BR) and consider the family of functions {ur }r>0 defined by Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 47 / 53 (b) u (x) = u(rx) − sup u if p = 2, and r Br 1 (c) ur (x) = r 2−p [u(rx) − u(0)] if 1 ≤ p < 2, n Note: One has ur ∈ F(BR/r ) and BR/r expands to R as r ↓ 0. Tangents to Subsolutions We now assume F to be convex with Riesz characteristic p = pF < ∞. The homogeneity of the Riesz functions leads one to following. Def. Suppose u ∈ F(BR) and consider the family of functions {ur }r>0 defined by p−2 (a) ur (x) = r u(rx) if p > 2, Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 47 / 53 1 (c) ur (x) = r 2−p [u(rx) − u(0)] if 1 ≤ p < 2, n Note: One has ur ∈ F(BR/r ) and BR/r expands to R as r ↓ 0. Tangents to Subsolutions We now assume F to be convex with Riesz characteristic p = pF < ∞. The homogeneity of the Riesz functions leads one to following. Def. Suppose u ∈ F(BR) and consider the family of functions {ur }r>0 defined by p−2 (a) ur (x) = r u(rx) if p > 2, (b) u (x) = u(rx) − sup u if p = 2, and r Br Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 47 / 53 n Note: One has ur ∈ F(BR/r ) and BR/r expands to R as r ↓ 0. Tangents to Subsolutions We now assume F to be convex with Riesz characteristic p = pF < ∞. The homogeneity of the Riesz functions leads one to following. Def. Suppose u ∈ F(BR) and consider the family of functions {ur }r>0 defined by p−2 (a) ur (x) = r u(rx) if p > 2, (b) u (x) = u(rx) − sup u if p = 2, and r Br 1 (c) ur (x) = r 2−p [u(rx) − u(0)] if 1 ≤ p < 2, Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 47 / 53 Tangents to Subsolutions We now assume F to be convex with Riesz characteristic p = pF < ∞. The homogeneity of the Riesz functions leads one to following. Def. Suppose u ∈ F(BR) and consider the family of functions {ur }r>0 defined by p−2 (a) ur (x) = r u(rx) if p > 2, (b) u (x) = u(rx) − sup u if p = 2, and r Br 1 (c) ur (x) = r 2−p [u(rx) − u(0)] if 1 ≤ p < 2, n Note: One has ur ∈ F(BR/r ) and BR/r expands to R as r ↓ 0. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 47 / 53 Definition. A function U ∈ F(Rn) is a tangent to u at 0 if there exists a sequence rj ↓ 0 such that 1 n urj → U in Lloc(R ) THEOREM. When F is convex, tangents always exist. Let T0u = the set of tangents to u at 0. Tangents to Subsolutions – Existence Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 48 / 53 if there exists a sequence rj ↓ 0 such that 1 n urj → U in Lloc(R ) THEOREM. When F is convex, tangents always exist. Let T0u = the set of tangents to u at 0. Tangents to Subsolutions – Existence Definition. A function U ∈ F(Rn) is a tangent to u at 0 Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 48 / 53 THEOREM. When F is convex, tangents always exist. Let T0u = the set of tangents to u at 0. Tangents to Subsolutions – Existence Definition. A function U ∈ F(Rn) is a tangent to u at 0 if there exists a sequence rj ↓ 0 such that 1 n urj → U in Lloc(R ) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 48 / 53 Let T0u = the set of tangents to u at 0. Tangents to Subsolutions – Existence Definition. A function U ∈ F(Rn) is a tangent to u at 0 if there exists a sequence rj ↓ 0 such that 1 n urj → U in Lloc(R ) THEOREM. When F is convex, tangents always exist. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 48 / 53 Tangents to Subsolutions – Existence Definition. A function U ∈ F(Rn) is a tangent to u at 0 if there exists a sequence rj ↓ 0 such that 1 n urj → U in Lloc(R ) THEOREM. When F is convex, tangents always exist. Let T0u = the set of tangents to u at 0. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 48 / 53 Definition. If T0u = {U} is always a singleton for F-subharmonic functions u, we say that uniqueness of tangents holds for F Definition. If T0u = {ΘKp(|x|)} (Θ ≥ 0) for all F-subharmonic functions u, we say that strong uniqueness of tangents holds for F Tangents to Subsolutions – Uniqueness Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 49 / 53 we say that uniqueness of tangents holds for F Definition. If T0u = {ΘKp(|x|)} (Θ ≥ 0) for all F-subharmonic functions u, we say that strong uniqueness of tangents holds for F Tangents to Subsolutions – Uniqueness Definition. If T0u = {U} is always a singleton for F-subharmonic functions u, Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 49 / 53 Definition. If T0u = {ΘKp(|x|)} (Θ ≥ 0) for all F-subharmonic functions u, we say that strong uniqueness of tangents holds for F Tangents to Subsolutions – Uniqueness Definition. If T0u = {U} is always a singleton for F-subharmonic functions u, we say that uniqueness of tangents holds for F Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 49 / 53 we say that strong uniqueness of tangents holds for F Tangents to Subsolutions – Uniqueness Definition. If T0u = {U} is always a singleton for F-subharmonic functions u, we say that uniqueness of tangents holds for F Definition. If T0u = {ΘKp(|x|)} (Θ ≥ 0) for all F-subharmonic functions u, Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 49 / 53 Tangents to Subsolutions – Uniqueness Definition. If T0u = {U} is always a singleton for F-subharmonic functions u, we say that uniqueness of tangents holds for F Definition. If T0u = {ΘKp(|x|)} (Θ ≥ 0) for all F-subharmonic functions u, we say that strong uniqueness of tangents holds for F Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 49 / 53 Let F = F(Gl ) be defined by a subset Gl ⊂ G(p, Rn) We say Gl ⊂ G(p, Rn) has the transitivity property if for all x, y ∈ Rn there exist W1, ..., Wk ∈ Gl with x ∈ W1, y ∈ Wk and dim(Wi ∩ Wi+1) > 0 for all i = 1, ..., k − 1. THEOREM. If Gl has the transitivity property, then strong uniqueness holds for F(Gl ). Tangents to Subsolutions – Strong Uniqueness Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 50 / 53 We say Gl ⊂ G(p, Rn) has the transitivity property if for all x, y ∈ Rn there exist W1, ..., Wk ∈ Gl with x ∈ W1, y ∈ Wk and dim(Wi ∩ Wi+1) > 0 for all i = 1, ..., k − 1. THEOREM. If Gl has the transitivity property, then strong uniqueness holds for F(Gl ). Tangents to Subsolutions – Strong Uniqueness Let F = F(Gl ) be defined by a subset Gl ⊂ G(p, Rn) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 50 / 53 if for all x, y ∈ Rn there exist W1, ..., Wk ∈ Gl with x ∈ W1, y ∈ Wk and dim(Wi ∩ Wi+1) > 0 for all i = 1, ..., k − 1. THEOREM. If Gl has the transitivity property, then strong uniqueness holds for F(Gl ). Tangents to Subsolutions – Strong Uniqueness Let F = F(Gl ) be defined by a subset Gl ⊂ G(p, Rn) We say Gl ⊂ G(p, Rn) has the transitivity property Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 50 / 53 there exist W1, ..., Wk ∈ Gl with x ∈ W1, y ∈ Wk and dim(Wi ∩ Wi+1) > 0 for all i = 1, ..., k − 1. THEOREM. If Gl has the transitivity property, then strong uniqueness holds for F(Gl ). Tangents to Subsolutions – Strong Uniqueness Let F = F(Gl ) be defined by a subset Gl ⊂ G(p, Rn) We say Gl ⊂ G(p, Rn) has the transitivity property if for all x, y ∈ Rn Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 50 / 53 with x ∈ W1, y ∈ Wk and dim(Wi ∩ Wi+1) > 0 for all i = 1, ..., k − 1. THEOREM. If Gl has the transitivity property, then strong uniqueness holds for F(Gl ). Tangents to Subsolutions – Strong Uniqueness Let F = F(Gl ) be defined by a subset Gl ⊂ G(p, Rn) We say Gl ⊂ G(p, Rn) has the transitivity property if for all x, y ∈ Rn there exist W1, ..., Wk ∈ Gl Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 50 / 53 for all i = 1, ..., k − 1. THEOREM. If Gl has the transitivity property, then strong uniqueness holds for F(Gl ). Tangents to Subsolutions – Strong Uniqueness Let F = F(Gl ) be defined by a subset Gl ⊂ G(p, Rn) We say Gl ⊂ G(p, Rn) has the transitivity property if for all x, y ∈ Rn there exist W1, ..., Wk ∈ Gl with x ∈ W1, y ∈ Wk and dim(Wi ∩ Wi+1) > 0 Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 50 / 53 THEOREM. If Gl has the transitivity property, then strong uniqueness holds for F(Gl ). Tangents to Subsolutions – Strong Uniqueness Let F = F(Gl ) be defined by a subset Gl ⊂ G(p, Rn) We say Gl ⊂ G(p, Rn) has the transitivity property if for all x, y ∈ Rn there exist W1, ..., Wk ∈ Gl with x ∈ W1, y ∈ Wk and dim(Wi ∩ Wi+1) > 0 for all i = 1, ..., k − 1. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 50 / 53 Tangents to Subsolutions – Strong Uniqueness Let F = F(Gl ) be defined by a subset Gl ⊂ G(p, Rn) We say Gl ⊂ G(p, Rn) has the transitivity property if for all x, y ∈ Rn there exist W1, ..., Wk ∈ Gl with x ∈ W1, y ∈ Wk and dim(Wi ∩ Wi+1) > 0 for all i = 1, ..., k − 1. THEOREM. If Gl has the transitivity property, then strong uniqueness holds for F(Gl ). Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 50 / 53 This proves the strong uniqueness of tangent cones when: (a) Gl = G(p, Rn) (p-plurisubharmonic functions) for p > 1. n (b) Gl = GC(k, C ) (complex k-plurisubharmonic functions) for k > 1 (p = 2k). n (c) Gl = GH(k, H ) (quaternionic k-plurisubharmonic functions) for k > 1 (p = 4k). (d) Gl = ASSOC (Associative subharmonic functions in R7)(p = 3). (e) Gl = COASSOC (Coassociative subharmonic functions in R7)(p = 4). (f) Gl = CAYLEY (Cayley subharmonic functions in R8)(p = 4). (g) Gl = LAG (Lagrangian subharmonic functions in Cn)(p = n). Special Cases Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 51 / 53 n (b) Gl = GC(k, C ) (complex k-plurisubharmonic functions) for k > 1 (p = 2k). n (c) Gl = GH(k, H ) (quaternionic k-plurisubharmonic functions) for k > 1 (p = 4k). (d) Gl = ASSOC (Associative subharmonic functions in R7)(p = 3). (e) Gl = COASSOC (Coassociative subharmonic functions in R7)(p = 4). (f) Gl = CAYLEY (Cayley subharmonic functions in R8)(p = 4). (g) Gl = LAG (Lagrangian subharmonic functions in Cn)(p = n). Special Cases This proves the strong uniqueness of tangent cones when: (a) Gl = G(p, Rn) (p-plurisubharmonic functions) for p > 1. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 51 / 53 n (c) Gl = GH(k, H ) (quaternionic k-plurisubharmonic functions) for k > 1 (p = 4k). (d) Gl = ASSOC (Associative subharmonic functions in R7)(p = 3). (e) Gl = COASSOC (Coassociative subharmonic functions in R7)(p = 4). (f) Gl = CAYLEY (Cayley subharmonic functions in R8)(p = 4). (g) Gl = LAG (Lagrangian subharmonic functions in Cn)(p = n). Special Cases This proves the strong uniqueness of tangent cones when: (a) Gl = G(p, Rn) (p-plurisubharmonic functions) for p > 1. n (b) Gl = GC(k, C ) (complex k-plurisubharmonic functions) for k > 1 (p = 2k). Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 51 / 53 (d) Gl = ASSOC (Associative subharmonic functions in R7)(p = 3). (e) Gl = COASSOC (Coassociative subharmonic functions in R7)(p = 4). (f) Gl = CAYLEY (Cayley subharmonic functions in R8)(p = 4). (g) Gl = LAG (Lagrangian subharmonic functions in Cn)(p = n). Special Cases This proves the strong uniqueness of tangent cones when: (a) Gl = G(p, Rn) (p-plurisubharmonic functions) for p > 1. n (b) Gl = GC(k, C ) (complex k-plurisubharmonic functions) for k > 1 (p = 2k). n (c) Gl = GH(k, H ) (quaternionic k-plurisubharmonic functions) for k > 1 (p = 4k). Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 51 / 53 (e) Gl = COASSOC (Coassociative subharmonic functions in R7)(p = 4). (f) Gl = CAYLEY (Cayley subharmonic functions in R8)(p = 4). (g) Gl = LAG (Lagrangian subharmonic functions in Cn)(p = n). Special Cases This proves the strong uniqueness of tangent cones when: (a) Gl = G(p, Rn) (p-plurisubharmonic functions) for p > 1. n (b) Gl = GC(k, C ) (complex k-plurisubharmonic functions) for k > 1 (p = 2k). n (c) Gl = GH(k, H ) (quaternionic k-plurisubharmonic functions) for k > 1 (p = 4k). (d) Gl = ASSOC (Associative subharmonic functions in R7)(p = 3). Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 51 / 53 (f) Gl = CAYLEY (Cayley subharmonic functions in R8)(p = 4). (g) Gl = LAG (Lagrangian subharmonic functions in Cn)(p = n). Special Cases This proves the strong uniqueness of tangent cones when: (a) Gl = G(p, Rn) (p-plurisubharmonic functions) for p > 1. n (b) Gl = GC(k, C ) (complex k-plurisubharmonic functions) for k > 1 (p = 2k). n (c) Gl = GH(k, H ) (quaternionic k-plurisubharmonic functions) for k > 1 (p = 4k). (d) Gl = ASSOC (Associative subharmonic functions in R7)(p = 3). (e) Gl = COASSOC (Coassociative subharmonic functions in R7)(p = 4). Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 51 / 53 (g) Gl = LAG (Lagrangian subharmonic functions in Cn)(p = n). Special Cases This proves the strong uniqueness of tangent cones when: (a) Gl = G(p, Rn) (p-plurisubharmonic functions) for p > 1. n (b) Gl = GC(k, C ) (complex k-plurisubharmonic functions) for k > 1 (p = 2k). n (c) Gl = GH(k, H ) (quaternionic k-plurisubharmonic functions) for k > 1 (p = 4k). (d) Gl = ASSOC (Associative subharmonic functions in R7)(p = 3). (e) Gl = COASSOC (Coassociative subharmonic functions in R7)(p = 4). (f) Gl = CAYLEY (Cayley subharmonic functions in R8)(p = 4). Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 51 / 53 Special Cases This proves the strong uniqueness of tangent cones when: (a) Gl = G(p, Rn) (p-plurisubharmonic functions) for p > 1. n (b) Gl = GC(k, C ) (complex k-plurisubharmonic functions) for k > 1 (p = 2k). n (c) Gl = GH(k, H ) (quaternionic k-plurisubharmonic functions) for k > 1 (p = 4k). (d) Gl = ASSOC (Associative subharmonic functions in R7)(p = 3). (e) Gl = COASSOC (Coassociative subharmonic functions in R7)(p = 4). (f) Gl = CAYLEY (Cayley subharmonic functions in R8)(p = 4). (g) Gl = LAG (Lagrangian subharmonic functions in Cn)(p = n). Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 51 / 53 In the three cases: G(1, Rn) (i.e., F = P) G(1, Cn) (i.e., F = PC) G(1, Hn) (i.e., F = PH) strong uniqueness fails. The possible tangents in these cases are completely characterized. In the convex case uniqueness of tangents holds, which is classical. In the complex case, uniqueness fails. This is due to Kiselman. Strong Uniqueness Fails Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 52 / 53 G(1, Cn) (i.e., F = PC) G(1, Hn) (i.e., F = PH) strong uniqueness fails. The possible tangents in these cases are completely characterized. In the convex case uniqueness of tangents holds, which is classical. In the complex case, uniqueness fails. This is due to Kiselman. Strong Uniqueness Fails In the three cases: G(1, Rn) (i.e., F = P) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 52 / 53 G(1, Hn) (i.e., F = PH) strong uniqueness fails. The possible tangents in these cases are completely characterized. In the convex case uniqueness of tangents holds, which is classical. In the complex case, uniqueness fails. This is due to Kiselman. Strong Uniqueness Fails In the three cases: G(1, Rn) (i.e., F = P) G(1, Cn) (i.e., F = PC) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 52 / 53 strong uniqueness fails. The possible tangents in these cases are completely characterized. In the convex case uniqueness of tangents holds, which is classical. In the complex case, uniqueness fails. This is due to Kiselman. Strong Uniqueness Fails In the three cases: G(1, Rn) (i.e., F = P) G(1, Cn) (i.e., F = PC) G(1, Hn) (i.e., F = PH) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 52 / 53 The possible tangents in these cases are completely characterized. In the convex case uniqueness of tangents holds, which is classical. In the complex case, uniqueness fails. This is due to Kiselman. Strong Uniqueness Fails In the three cases: G(1, Rn) (i.e., F = P) G(1, Cn) (i.e., F = PC) G(1, Hn) (i.e., F = PH) strong uniqueness fails. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 52 / 53 In the convex case uniqueness of tangents holds, which is classical. In the complex case, uniqueness fails. This is due to Kiselman. Strong Uniqueness Fails In the three cases: G(1, Rn) (i.e., F = P) G(1, Cn) (i.e., F = PC) G(1, Hn) (i.e., F = PH) strong uniqueness fails. The possible tangents in these cases are completely characterized. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 52 / 53 In the complex case, uniqueness fails. This is due to Kiselman. Strong Uniqueness Fails In the three cases: G(1, Rn) (i.e., F = P) G(1, Cn) (i.e., F = PC) G(1, Hn) (i.e., F = PH) strong uniqueness fails. The possible tangents in these cases are completely characterized. In the convex case uniqueness of tangents holds, which is classical. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 52 / 53 Strong Uniqueness Fails In the three cases: G(1, Rn) (i.e., F = P) G(1, Cn) (i.e., F = PC) G(1, Hn) (i.e., F = PH) strong uniqueness fails. The possible tangents in these cases are completely characterized. In the convex case uniqueness of tangents holds, which is classical. In the complex case, uniqueness fails. This is due to Kiselman. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 52 / 53 MANY QUESTIONS REMAIN OPEN. Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 53 / 53