Self-dual Variational Calculus Nassif Ghoussoub, University of - PowerPoint PPT Presentation

Self-dual Variational Calculus Nassif Ghoussoub, University of British Columbia Vienna, July 2010

Convex analysis making a come back ◮ Monge’s Mass transport: Convexity in the optimal map. Geodesic convexiy in Wasserstein space. c -convexity on general manifolds. Nonlinear PDEs as gradient flows of geodesically convex energies on infinite dimensional manifolds. ◮ Mather’s Hamiltonian dynamics theory and its connection to mass transport. ◮ Weak KAM theory: Homogenization, effective Hamiltonians and their connection to the above.

My focus today ◮ Selfduality or How to solve PDEs by completing squares. Much simpler things on flat space, but with potential extensions to manifolds. ◮ Besides existence and uniqueness issues, “Self-dual Variational Calculus" provides a natural and unifying approach for dealing with – nonlinear inverse and control problems and – the homogenization of monotone vector fields.

Certain advantages of functional/convex analysis (of knowing Schachermayer) φ : X → R ∪ { + ∞} convex lower semi-continuous on a Banach space X . ◮ The availability of a subdifferential ∂φ ( x ) in non-smooth situations. ◮ The global definition of a Legendre transform φ ∗ defined on X ∗ by φ ∗ ( p ) = sup {� x , p � − φ ( x ); x ∈ X } , with the important inequality φ ( x ) + φ ∗ ( p ) ≥ � x , p � with equality. φ ( x ) + φ ∗ ( p ) = � x , p � x ∈ ∂φ ∗ ( p ) . iff p ∈ ∂φ ( x ) iff ◮ Inverting the vector field ∂φ (i.e., solving p ∈ ∂φ ( x ) ) by minimizing I ( x ) = φ ( x ) − � p , x � .

Very first course in variational approach to PDEs To solve � − ∆ u + | u | p − 2 u = f on Ω , u = 0 on ∂ Ω . It suffices to minimize, on H 1 0 (Ω) , the convex functional � � � φ ( u ) = 1 |∇ u | 2 dx + 1 | u | p dx + fudx 2 p Ω Ω Ω But what about the following non-selfadjoint equation ?? � − ∆ u + | u | p − 2 u + Σ n i = 1 a i ( x ) ∂ u = f on Ω , ∂ x i u = 0 on ∂ Ω . Is there a variational resolution?

Variational approach to gradient flows (Brezis-Ekeland,1976) � − ˙ v ( t ) ∈ ∂φ ( v ( t )) a . e . on [ 0 , T ] , v ( 0 ) = v 0 . � φ convex l.s.c on Hilbert space H . (e.g., φ ( u ) = 1 Ω |∇ u ( x ) | 2 dx . ) 2 Minimize the “selfdual functional" � T dt + 1 � � φ ( u ( t )) + φ ∗ ( − ˙ 2 | u ( T ) | 2 I ( u ) = u ( t )) 0 over all path u ∈ L 2 H , with ˙ u ∈ L 2 H and u ( 0 ) = v 0 . � T 2 | u ( T ) | 2 − 1 u ( t ) � dt = 1 2 | u ( 0 ) | 2 , 0 � u ( t ) , ˙ Using that � T dt + 1 � � 2 | v 0 | 2 ≥ 0 , φ ( u ( t )) + φ ∗ ( − ˙ I ( u ) = u ( t )) + � u ( t ) , ˙ u ( t ) � 0 u ) = 1 2 | v 0 | 2 , then we are Now if (BIG IF) we can prove inf I ( u ) = I (¯ done by Legendre duality.

Bogomolnyi’s trick of completing the square (Yang-Mills, Chern-Simon, Seiberg-Witten, Ginsburg-Landau) Associate to any connection A ∈ Ω 1 ( Ad E ) where E is a vector bundle over an oriented closed 4-manifold M , the curvature F A = dA + 1 2 [ A ∧ A ] ∈ Ω 2 ( Ad E ) , and the exterior differential on k -forms d A w = dw + [ A ∧ w ] . � � 1 2 ( � F A � 2 + � ∗ F A � 2 ) � F A � 2 I ( A ) := = M M 1 � � � F A + ∗ F A � 2 − = � F A ∧ ∗ F A � 2 M M 1 � � F A + ∗ F A � 2 + 8 π 2 c 2 ( P ) ≥ 8 π 2 c 2 ( P ) . = 2 M c 2 ( P ) is 2d Chern class (topological invariant). ∗ is Hodge operator. Inner product is negative trace of the product of the matrices. ◮ IF the infimum I = 8 π 2 c 2 ( P ) and if it is attained for some connection A , then F A = − ∗ F A , the anti-selfdual Yang-Mills. ◮ They are first order equations obtained variationally but not via Euler-Lagrange. ◮ In particular, YM equations d ∗ A F A = 0 via Bianchi identities.

Self-dualize the problem: Ghoussoub-Tzou (2004) Minimize the “selfdual functional" � T dt + 1 2 | u ( 0 ) | 2 − 2 � u ( 0 ) , v 0 � + | v 0 | 2 + 1 � � φ ( u ( t )) + φ ∗ ( − ˙ 2 | u ( T ) | 2 I ( u ) = u ( t )) 0 � T � � dt + | u ( 0 ) − v 0 | 2 ≥ 0 , φ ( u ( t )) + φ ∗ ( − ˙ u ( t )) + � u ( t ) , ˙ I ( u ) = u ( t ) � 0 The Lagrangian on L 2 X × L 2 X ∗ defined by � � T 0 φ ( u ( t )) + φ ∗ ( − ˙ if ˙ u ∈ L 2 u ( t ) + p ( t )) dt + ℓ ( u ( 0 ) , u ( T )) L ( u , p ) = X + ∞ otherwise is selfdual! that is for every pair ( u ( t ) , p ( t )) ∈ L 2 X × L 2 X ∗ , L ∗ ( p , u ) = L ( u , p ) . Why is inf I ( u ) = inf L ( u , 0 ) = 0 ? Because by basic convex duality: −L ∗ ( 0 , v ) ≤ inf α = sup u ∈X L ( u , 0 ) = β v ∈X In our setting β = − α , and since the minimum is attained if and only if there is no duality gap, then α = β = 0 .

Key concept: Selfdual Lagrangians 1. Selfdual Lagrangians: L : X × X ∗ → R ∪ { + ∞} is convex lsc in both variables. L ∗ ( p , x ) = L ( x , p ) for all ( p , x ) ∈ X ∗ × X . In this case, L ( x , p ) − � x , p � ≥ 0 for every ( x , p ) ∈ X × X ∗ , and L ( x , p ) − � x , p � = 0 if and only if ( p , x ) ∈ ∂ L ( x , p ) 2. Selfdual Vector Field: F : X → X ∗ such that there is L selfdual Lagrangian with F = ¯ ∂ L , i.e., F ( x ) = ¯ ∂ L ( x ) : = { p ∈ X ∗ ; L ( x , p ) − � x , p � = 0 } = { p ∈ X ∗ ; ( p , x ) ∈ ∂ L ( x , p ) } .

3. The Completely Selfdual Equations. p = ¯ ∂ L ( x ) ( p , x ) = ∂ L ( x , p ) . or Important: ◮ ¯ ∂ L is NOT necessarily a differential, yet it is derived from a potential in the sense that a solution can be obtained by minimizing I p ( x ) = L ( x , p ) − � x , p � and by showing that inf x ∈ X I p ( x ) = 0. ◮ inf x ∈ X L ( x , p ) − � x , p � is equal to zero! ◮ Many equations (evolutions) are completely selfdual.

Basic examples of selfdual Lagrangians: 1. If φ is any convex lsc functional on X , then L ( x , p ) = φ ( x ) + φ ∗ ( p ) is a selfdual Lagrangian on X × X ∗ and ∂ L ( x ) = ∂φ ( x ) . 2. If L is selfdual and Γ : X → X ∗ is skew-symmetric (i.e., Γ ∗ = − Γ ), then L Γ ( x , p ) = L ( x , − Γ x + p ) is also selfdual. Vector fields F = Γ + ∂φ (i.e., superposition of a dissipative and conservative vector fields) are also derived from a selfdual potential: Γ x + ∂φ ( x ) = ∂ L ( x ) where L ( x , p ) = ϕ ( x ) + ϕ ∗ ( − Γ x − p ) . On can then solve Γ x + ∂φ ( x ) = p by minimizing I p ( x ) = L ( x , p ) − � x , p � = ϕ ( x ) + ϕ ∗ ( − Γ x + p ) − � x , p � � − ∆ u + | u | p − 2 u + Σ n i = 1 a i ( x ) ∂ u = f on Ω ∂ x i Simple example u = 0 on ∂ Ω . 3. T : D ( T ) ⊂ X → 2 X ∗ is a maximal monotone operator with a non-empty domain, then there exists a selfdual Lagrangian L on X × X ∗ such that T = ¯ ∂ L . (Analogue of Rockafellar’s theorem for cyclically monotone maps)

New variational calculus vs.old � − ∆ u + | u | p − 2 u + Σ n i = 1 a i ( x ) ∂ u = f on Ω , ∂ x i u = 0 on ∂ Ω . Assuming div ( a ) ≥ 0 on Ω , then it suffices to minimize, on the same H 1 0 (Ω) , the new convex functional I ( u ) = Ψ( u ) + Ψ ∗ ( a · ∇ u + 1 2 div ( a ) u ) , where ψ is the convex functional � � � � Ψ( u ) = 1 |∇ u | 2 dx + 1 fudx + 1 | u | p dx + div ( a ) | u | 2 dx , 2 p 4 Ω Ω Ω Ω and ψ ∗ is its Fenchel-Legendre transform. ◮ Equation is not an Euler-Lagrange equation. ◮ It is derived from the fact that I (¯ u ) = inf u ∈ H 1 0 (Ω) I ( u ) = 0.

Porous media For m > 0, u 0 ∈ L m + 1 and f ∈ H − 1 , the infimum of � T � e 2 ω t | u ( t , x ) | m + 1 dxdt 1 I ( u ) = m + 1 0 Ω � T � | ( − ∆) − 1 ( f ( x ) − ω u ( t , x ) − ∂ u m + 1 m dxdt m e 2 ω t + ∂ t ( t , x )) | m + 1 0 Ω � T � � |∇ ( − ∆) − 1 u 0 ( x ) | 2 dx e 2 ω t u ( x , t )(( − ∆) − 1 f )( x ) dxdt + − 0 Ω Ω � u 0 ( x )( − ∆) − 1 u ( 0 , x ) dx + 1 � � � u ( 0 ) � 2 H − 1 + e 2 wT � u ( T ) � 2 − 2 2 H − 1 Ω on A 2 H − 1 is equal to zero and is attained uniquely at an L m + 1 (Ω) -valued u : ∆ u m + ω u + f on Ω × [ 0 , T ] , � ∂ u = ∂ t u ( 0 , x ) = u 0 ( x ) on Ω . Note that Euler-Lagrange equations (for heat flow m = 1, w = 0) � ( ∂ ∂ t − ∆)( ∂ ∂ t + ∆) u = 0 a . e . on [ 0 , T ] , u ( 0 ) = u 0 .

Semi-groups of contractions associated to selfdual Lagrangians To any selfdual Lagrangian L such that Dom ( ∂ L ) is non-empty, one can associate a semi-group of 1-Lipschitz maps ( T t ) t ∈ R + on Dom ( ∂ L ) such that T 0 = Id and for any x 0 ∈ Dom ( ∂ L ) , the path x ( t ) = T t x 0 satisfies: x ( t ) ∈ − ¯ ˙ ∂ L ( t , x ( t )) for all t ∈ [ 0 , T ] and, � t H = � x 0 � 2 − 2 � x ( t ) � 2 0 L ( x ( s ) , ˙ x ( s )) ds for every t ∈ [ 0 , T ] . The path ( x ( t )) t = ( T t x 0 ) t is obtained as a minimizer on A 2 H of the functional � T u ( t )) dt + 1 2 � u ( 0 ) � 2 − 2 � x 0 , u ( 0 ) � + � x 0 � 2 + 1 2 � u ( T ) � 2 , L ( u ( t ) , ˙ I ( u ) = 0 whose infimum is equal to zero.