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 ≡ fA 2 Sym (R ): f (A) ≥ 0g: Then for u 2 C2 f (D2u) ≥ 0 () D2u 2 F (the subequation) and f (D2u) = 0 () D2u 2 @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 2 C2 f (D2u) ≥ 0 () D2u 2 F (the subequation) and f (D2u) = 0 () D2u 2 @F (the equation) A Geometric Approach – N. V. Krylov Consider instead the closed set 2 n F ≡ fA 2 Sym (R ): f (A) ≥ 0g: Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 8 / 53 and f (D2u) = 0 () D2u 2 @F (the equation) A Geometric Approach – N. V. Krylov Consider instead the closed set 2 n F ≡ fA 2 Sym (R ): f (A) ≥ 0g: Then for u 2 C2 f (D2u) ≥ 0 () D2u 2 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 ≡ fA 2 Sym (R ): f (A) ≥ 0g: Then for u 2 C2 f (D2u) ≥ 0 () D2u 2 F (the subequation) and f (D2u) = 0 () D2u 2 @F (the equation) Blaine Lawson Potential Theory for Nonlinear PDE’s April 13, 2014 8 / 53 1.