On nonlocal Monge–Amp` ere equations Pablo Ra´ ul Stinga Iowa State University Fractional PDEs: Theory, Algorithms and Applications ICERM June 21st, 2018 Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 1 / 23
The Monge–Amp` ere equation Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 2 / 23
Monge optimal transport problem We want to transport µ onto ν (probability measures) in an optimal way given that the cost of moving x onto y = S ( x ) is | x − y | 2 = | x − S ( x ) | 2 Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 3 / 23
Monge optimal transport problem We want to transport µ onto ν (probability measures) in an optimal way given that the cost of moving x onto y = S ( x ) is | x − y | 2 = | x − S ( x ) | 2 Mathematical formulation. Minimize the functional � R n | x − S ( x ) | 2 d µ ( x ) F ( S ) = among all maps S that transport µ onto ν : for any Borel function ψ : R n → R � � R n ψ ( y ) d ν ( y ) = R n ψ ( S ( x )) d µ ( x ) Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 3 / 23
Monge optimal transport problem We want to transport µ onto ν (probability measures) in an optimal way given that the cost of moving x onto y = S ( x ) is | x − y | 2 = | x − S ( x ) | 2 Mathematical formulation. Minimize the functional � R n | x − S ( x ) | 2 d µ ( x ) F ( S ) = among all maps S that transport µ onto ν : for any Borel function ψ : R n → R � � R n ψ ( y ) d ν ( y ) = R n ψ ( S ( x )) d µ ( x ) Theorem (Brenier, 1991) If µ ( x ) = f ( x ) dx and ν ( y ) have finite second moments then there exists a µ -a.e. unique optimal transport map T. Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 3 / 23
Monge optimal transport problem We want to transport µ onto ν (probability measures) in an optimal way given that the cost of moving x onto y = S ( x ) is | x − y | 2 = | x − S ( x ) | 2 Mathematical formulation. Minimize the functional � R n | x − S ( x ) | 2 d µ ( x ) F ( S ) = among all maps S that transport µ onto ν : for any Borel function ψ : R n → R � � R n ψ ( y ) d ν ( y ) = R n ψ ( S ( x )) d µ ( x ) Theorem (Brenier, 1991) If µ ( x ) = f ( x ) dx and ν ( y ) have finite second moments then there exists a µ -a.e. unique optimal transport map T. Moreover, there exists a l.s.c. convex function ϕ , differentiable µ -a.e. such that T = ∇ ϕ µ -a.e. Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 3 / 23
Optimal transport and Monge–Amp` ere equation Suppose µ = f ( x ) dx and ν = g ( y ) dy Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 4 / 23
Optimal transport and Monge–Amp` ere equation Suppose µ = f ( x ) dx and ν = g ( y ) dy If the optimal transport map T is a diffeomorphism then, by changing variables, � � � R n ψ ( T ( x )) f ( x ) dx = R n ψ ( y ) g ( y ) dy = R n ψ ( T ( x )) g ( T ( x )) | det ∇ T ( x ) | dx Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 4 / 23
Optimal transport and Monge–Amp` ere equation Suppose µ = f ( x ) dx and ν = g ( y ) dy If the optimal transport map T is a diffeomorphism then, by changing variables, � � � R n ψ ( T ( x )) f ( x ) dx = R n ψ ( y ) g ( y ) dy = R n ψ ( T ( x )) g ( T ( x )) | det ∇ T ( x ) | dx Since ψ was arbitrary, g ( T ( x )) | det ∇ T ( x ) | = f ( x ) Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 4 / 23
Optimal transport and Monge–Amp` ere equation Suppose µ = f ( x ) dx and ν = g ( y ) dy If the optimal transport map T is a diffeomorphism then, by changing variables, � � � R n ψ ( T ( x )) f ( x ) dx = R n ψ ( y ) g ( y ) dy = R n ψ ( T ( x )) g ( T ( x )) | det ∇ T ( x ) | dx Since ψ was arbitrary, g ( T ( x )) | det ∇ T ( x ) | = f ( x ) Recall from Brenier that T = ∇ ϕ for ϕ convex , so that ∇ T = D 2 ϕ > 0 and f det( D 2 ϕ ) = g ◦ ∇ ϕ Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 4 / 23
Optimal transport and Monge–Amp` ere equation Suppose µ = f ( x ) dx and ν = g ( y ) dy If the optimal transport map T is a diffeomorphism then, by changing variables, � � � R n ψ ( T ( x )) f ( x ) dx = R n ψ ( y ) g ( y ) dy = R n ψ ( T ( x )) g ( T ( x )) | det ∇ T ( x ) | dx Since ψ was arbitrary, g ( T ( x )) | det ∇ T ( x ) | = f ( x ) Recall from Brenier that T = ∇ ϕ for ϕ convex , so that ∇ T = D 2 ϕ > 0 and f det( D 2 ϕ ) = g ◦ ∇ ϕ ◮ The fully nonlinear equation det( D 2 ϕ ) = F is the Monge–Amp` ere (MA) equation Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 4 / 23
Convex solutions and ellipticity Let ϕ be a solution to det( D 2 ϕ ) = F Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 5 / 23
Convex solutions and ellipticity Let ϕ be a solution to det( D 2 ϕ ) = F Equation for a directional derivative ∂ e ϕ of the solution det( D 2 ϕ )( D 2 ϕ ) − 1 D 2 ( ∂ e ϕ ) � � trace = ∂ e F Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 5 / 23
Convex solutions and ellipticity Let ϕ be a solution to det( D 2 ϕ ) = F Equation for a directional derivative ∂ e ϕ of the solution det( D 2 ϕ )( D 2 ϕ ) − 1 D 2 ( ∂ e ϕ ) � � trace = ∂ e F Here A ϕ ( x ) = det( D 2 ϕ ( x ))( D 2 ϕ ( x )) − 1 is the matrix of cofactors of D 2 ϕ Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 5 / 23
Convex solutions and ellipticity Let ϕ be a solution to det( D 2 ϕ ) = F Equation for a directional derivative ∂ e ϕ of the solution det( D 2 ϕ )( D 2 ϕ ) − 1 D 2 ( ∂ e ϕ ) � � trace = ∂ e F Here A ϕ ( x ) = det( D 2 ϕ ( x ))( D 2 ϕ ( x )) − 1 is the matrix of cofactors of D 2 ϕ If we call u = ∂ e ϕ G = ∂ e F then u solves the linearized MA equation L ϕ ( u ) = trace( A ϕ ( x ) D 2 u ) = G Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 5 / 23
Convex solutions and ellipticity Let ϕ be a solution to det( D 2 ϕ ) = F Equation for a directional derivative ∂ e ϕ of the solution det( D 2 ϕ )( D 2 ϕ ) − 1 D 2 ( ∂ e ϕ ) � � trace = ∂ e F Here A ϕ ( x ) = det( D 2 ϕ ( x ))( D 2 ϕ ( x )) − 1 is the matrix of cofactors of D 2 ϕ If we call u = ∂ e ϕ G = ∂ e F then u solves the linearized MA equation L ϕ ( u ) = trace( A ϕ ( x ) D 2 u ) = G Linearized MA is an elliptic equation as soon as D 2 ϕ ( x ) > 0 (convex!) and F > 0 Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 5 / 23
Convex solutions and ellipticity Let ϕ be a solution to det( D 2 ϕ ) = F Equation for a directional derivative ∂ e ϕ of the solution det( D 2 ϕ )( D 2 ϕ ) − 1 D 2 ( ∂ e ϕ ) � � trace = ∂ e F Here A ϕ ( x ) = det( D 2 ϕ ( x ))( D 2 ϕ ( x )) − 1 is the matrix of cofactors of D 2 ϕ If we call u = ∂ e ϕ G = ∂ e F then u solves the linearized MA equation L ϕ ( u ) = trace( A ϕ ( x ) D 2 u ) = G Linearized MA is an elliptic equation as soon as D 2 ϕ ( x ) > 0 (convex!) and F > 0 ◮ MA equation is degenerate elliptic . Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 5 / 23
The MA geometry There is an intrinsic quasi-metric space associated with MA Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 6 / 23
The MA geometry There is an intrinsic quasi-metric space associated with MA ◮ Uniformly elliptic. L ( u ) = trace( A ( x ) D 2 u ) with A ( x ) ∼ I Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 6 / 23
The MA geometry There is an intrinsic quasi-metric space associated with MA ◮ Uniformly elliptic. L ( u ) = trace( A ( x ) D 2 u ) with A ( x ) ∼ I P quadratic polynomial, then L ( P ) ≈ 1 Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 6 / 23
The MA geometry There is an intrinsic quasi-metric space associated with MA ◮ Uniformly elliptic. L ( u ) = trace( A ( x ) D 2 u ) with A ( x ) ∼ I P quadratic polynomial, then L ( P ) ≈ 1 Sublevel sets of P are all the Euclidean balls. Pablo Ra´ ul Stinga (Iowa State University) On nonlocal Monge–Amp` ere equations Providence RI, June 21 2018 6 / 23
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