Variational solutions of Hamilton-Jacobi equations - 1 Prologue - PowerPoint PPT Presentation

Maslov-H¨ ormander Theorem • Maslov-H¨ ormander Theorem : In any situation with respect transversality to the fibers of π M , locally, Lagrangian submanifolds are necessarily descripted by Generating Families 1 W ( q, u ) : M × R k ∋ ( q i , u A ) �− → W ( q i , u A ) ∈ R • in the following way: p i = ∂W 0 = ∂W ∂q i ( q i , u A ) , ∂u B ( q i , u A ) } Λ = { ( q, p ) : ( ∗ ) • Furthermore, zero in R k is a regular value for the map Q × R k ∋ ( q, u ) �→ ∂W ∂u ∈ R k , that is � ∂ 2 W � � ∂ 2 W � rk ( ∗ ) = k (= maximal) . ( ∗∗ ) � ∂u A ∂q i ∂u A ∂u B 1 Sometimes said Morse Families Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Maslov-H¨ ormander Theorem • u = ( u A ) A =1 ,...,k ∈ R k : auxiliary parameters • In the case of transversality, we can choose k = 0 , so W = W ( q ) • In general, we have to choose: k ≥ dim M − rk[ D ( π M ◦ j )( λ 0 )] ⇒ We cannot involve a number of aux. parameters smaller than the loss of the rank of D ( π M ◦ j )( λ 0 ) Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Maslov-H¨ ormander Theorem - Uniqueness • The above description of the Generating Families is unique up to the following three operations 2 3 : • 1. Addition of constant: W ( q ; u ) = W ( q ; u ) + const. ≈ W ( q ; u ) (trivial) • 2. Stabilization (i.e., addition of n.deg. quadratic forms): W ( q ; u, v ) = W ( q ; u ) + v T Av ¯ k , ∀ det A � = 0 ≈ W ( q ; u ) v ∈ R • 3. Fibered diffeomorphism: For any fibered diffeomorphism M × R k − → M × R k ( q, v ) �− → ( q, ¯ u ( q, v )) W ( q ; v ) := W ( q ; ¯ u ( q, v )) ≈ W ( q ; u ) 2 A. Weinstein, Lectures on symplectic manifolds , 1976 3 P. Libermann, C.-M. Marle, Symplectic geometry and analytical mechanics , 1987 Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Maslov-H¨ ormander Theorem - Uniqueness Checking the invariance under fibered diffeomorphisms (Operation 3.): M × R k − → M × R k ( q, v ) �− → ( q, ¯ u ( q, v )) W ( q ; v ) := W ( q ; ¯ u ( q, v )) � � p = ∂W 0 = ∂W Λ = ( q, p ) : ∂q ( q ; v ) , ∂v ( q ; v ) � ∂ ¯ u � p = ∂W ∂q + ∂W ∂ ¯ u 0 = ∂W = ( q, p ) : ∂q , ∂u ∂u ∂v ���� det � =0 � � p = ∂W 0 = ∂W = ( q, p ) : ∂q , ∂u = Λ ⇒ W ( q ; v ) ≈ W ( q ; u ) Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Maslov-H¨ ormander Theorem - Full reduction of parameters • By Operation 2., i.e., Stabilization by adding quadratic forms, the number of aux. parameters can increase • The number of aux. parameters can also decrease: Whenever the max rank of � ∂ 2 W � � ∂ 2 W � rk ∂uA =0 = k (= maximal) � ∂W ∂u A ∂q i ∂u A ∂u B � ∂ 2 W � can be detected from k × k -matrix : det ∂uA =0 � = 0 � ∂W ∂u A ∂u B it is possible, locally, fully to remove all the aux. par.; by implicit function th., ∂W u A = ˆ u A ( q ) ∂u A = 0 ⇒ • so ˆ u A ( q )) W ( q ) := W ( q, ˆ is a Generating Function for the same Lagrangian submanifold: p = ∂ ˆ ∂q = ∂W W u ( q )) + ∂W ∂ ˆ u ∂q ( q, ˆ ∂u ( q, ˆ u ( q )) ∂q � �� � =0 Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Maslov-H¨ ormander Theorem - Partial reduction of parameters • By Operation 2., i.e., Stabilization by adding quadratic forms, the number of aux. parameters can increase • The number of aux. parameters can also decrease: ∂ 2 W Whenever from the k × k -matrix ∂u A ∂u B � ∂ 2 W � � ∂ 2 W � rk ∂uA =0 = k (= maximal) � ∂W ∂u A ∂q i ∂u A ∂u B it is possible to detect some (smaller) non-degenerate h × h -sub-matrix, h ≤ k , ∂ 2 W det ∂u α ∂u β � = 0 , α, β = 1 , . . . , h ≤ k then, by implicit function th., ∂W u α = ˆ ∂u α ( q, u α , u Γ ) = 0 u α ( q, u Γ ) , ⇒ Γ = h + 1 , . . . , k • W ( q, u Γ ) := W ( q, ˆ ˆ u α ( q, u Γ )) is a Generating Function for the same Lagrangian submanifold. Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Canonical Transformations • Consider two manifolds, or two copies of a same manifold Q : • Q 1 and Q 2 • T ∗ Q 1 T ∗ Q 2 and • ( T ∗ Q 1 , ω 1 ) ( T ∗ Q 2 , ω 2 ) and • Diffeomorphisms f : T ∗ Q 1 − → T ∗ Q 2 (¯ q, ¯ p ) �− → f (¯ q, ¯ p ) = ( q, p ) preserving the respective symplectic structures , that is, • ω 1 = f ∗ ω 2 • are said Canonical Transformations or Symplectomorphisms • Main example: At any fixed time t ∈ R , flows of Hamilton ode systems φ t X H are Canonical Transformations: d dtφ t X H = X H ( φ t X H ) Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Other Symplectic Manifolds: a frame for Canonical Transformations • Consider the following graph-structure: P := T ∗ Q 1 × T ∗ Q 2 with projections: PR 1 PR 2 T ∗ Q 1 T ∗ Q 1 × T ∗ Q 2 T ∗ Q 2 ← − − → • Equip T ∗ Q 1 × T ∗ Q 2 with the closed 2-form Ω := PR ∗ 2 ω 2 − PR ∗ 1 ω 1 • It turns out that ( T ∗ Q 1 × T ∗ Q 2 , Ω ) is a symplectic manifold • Theorem A diffeomorphism f : T ∗ Q 1 → T ∗ Q 2 is Canonical iff Λ := graph ( f ) ⊂ T ∗ Q 1 × T ∗ Q 2 is a Lagrangian submanifold of the symplectic manifold ( T ∗ Q 1 × T ∗ Q 2 , Ω ). • In fact: 0 = Ω | graph ( f ) = f ∗ ( ω 2 ) − ω 1 Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

� � � � � � � T ∗ Q 1 × T ∗ Q 2 is isomorphic to T ∗ ( Q 1 × Q 2 ) • Observe that T ∗ Q 1 × T ∗ Q 2 is isomorphic in a natural way to T ∗ ( Q 1 × Q 2 ) , T ∗ Q 1 × T ∗ Q 2 � � ����������� � � P R 1 � P R 2 � ϕ � � � � � � T ∗ Q 1 T ∗ ( Q 1 × Q 2 ) T ∗ Q 2 T pr 1 T pr 2 � TQ 2 TQ 1 T ( Q 1 × Q 2 ) τ Q 1 τ Q 1 × Q 2 τ Q 2 pr 1 pr 2 � Q 2 Q 1 Q 1 × Q 2 in local charts: ϕ ( q (1) , q (2) ; p (1) , p (2) ) = ( q (1) , p (1) ; q (2) , p (2) ) . Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Maslov-H¨ ormander in ( T ∗ Q 1 × T ∗ Q 2 , Ω ) ormander’s th. in T ∗ Q was laid down • We recall that the setting of Maslov-H¨ on j π Q T ∗ Q Λ ֒ → − → Q λ �→ ( q ( λ ) , p ( λ )) �→ q ( λ ) , W : Q × R k → R ( q, u ) �→ W ( q, u ) � � p = ∂W 0 = ∂W Λ = ∂q ( q, u ) , ∂u ( q, u ) • Now, in the new environment T ∗ Q 1 × T ∗ Q 2 , a version of Maslov-H¨ ormander’s th. goes in this line: Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Maslov-H¨ ormander in ( T ∗ Q 1 × T ∗ Q 2 , Ω ) j π Q 1 × Q 2 Λ = graph( f ) ∼ T ∗ Q 1 × T ∗ Q 2 ∼ = T ∗ Q 1 = T ∗ ( Q 1 × Q 2 ) ֒ → − → Q 1 × Q 2 λ = ( q (1) , p (1) ) �→ ( q (1) , p (1) ; f q ( λ ) , f p ( λ )) ∼ = ( q (1) , f q ( λ ); p (1) , f p ( λ )) �→ ( q (1) , f q ( λ )) W : Q 1 × Q 2 × R k → R ( q 1 , q 2 , u ) �→ W ( q 1 , q 2 , u ) � � p 1 = − ∂W p 2 = ∂W 0 = ∂W graph( f ) = ∂q 1 ( q 1 , q 2 , u ) , ∂q 2 ( q 1 , q 2 , u ) , ∂u ( q 1 , q 2 , u ) Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

A little algebra for Canonical Transformations -1 • (i) Canonical Transformations send Lagrangian submanifolds into Lagrangian submanifolds: Theorem Let f : ( M, ω ) − → ( N, ¯ ω ) be a symplectomorphism, f ∗ ¯ ω = ω , and j : Λ ֒ → ( M, ω ) an embedded Lagrangian submanifold. Then f ◦ j (Λ) is Lagrangian in ( N, ¯ ω ) . Proof. � ω = j ∗ ◦ f ∗ ¯ � f ◦ j (Λ) = ( f ◦ j ) ∗ ¯ = j ∗ ω = ω ¯ ω ω Λ = 0 � � ���� = ω Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

A little algebra for Canonical Transformations - 2 (ii) The Composition Rule CT 1 CT 2 T ∗ M T ∗ M , T ∗ M T ∗ M − → − → ( x 0 , p 0 ) �→ (¯ x 1 , ¯ p 1 ) ( x 1 , p 1 ) �→ ( x 2 , p 2 ) , Given two Generating Functions: CT 1 : T ∗ M → T ∗ M W 1 ( x 0 , ¯ x 1 ; u ) for CT 2 : T ∗ M → T ∗ M W 2 ( x 1 , x 2 ; v ) for then the canonical transformation CT 21 = CT 2 ◦ CT 1 is generated by W 21 ( x 0 , x 2 ; u, v, w ) := W 1 ( x 0 , w ; u ) + W 2 ( w, x 2 ; v ) Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

A little algebra for Canonical Transformations - 3 The Composition Rule CT 1 CT 2 T ∗ M T ∗ M , T ∗ M T ∗ M − → − → ( x 0 , p 0 ) �→ (¯ x 1 , ¯ p 1 ) ( x 1 , p 1 ) �→ ( x 2 , p 2 ) , Proof. W 21 ( x 0 , x 2 ; u, v, w ) := W 1 ( x 0 , w ; u ) + W 2 ( w, x 2 ; v ) p 0 = − ∂W 21 p 2 = ∂W 21 p 0 = − ∂W 1 p 2 = ∂W 2 : ∂x 0 ( x 0 , w ; u ) ∂x 2 ( w, x 2 ; v ) ∂x 0 ∂x 2 0 = ∂W 21 0 = ∂W 21 0 = ∂W 1 0 = ∂W 2 : ∂u ∂v ∂u ∂v 0 = ∂W 21 0 = ∂W 1 x 1 + ∂W 2 : ∂w ∂ ¯ ∂x 1 that is: 0 = p 1 ¯ − p 1 ���� ���� the ‘final’ impulse of T C 1 the ‘starting’ impulse of T C 2 Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

A little algebra for Canonical Transformations - 4 • The Identity The generating function for the trivial canonical transformation ‘identity’ is given by W Id ( x, X ; u ) := ( X − x ) · u • The Inverse Given a Generating Function W ( x, X ; u ) for CT , then ( CT ) − 1 is generated by W ( − 1) ( X, x ; u ) := − W ( x, X ; u ) Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Geometrical synopsis of Hamilton-Jacobi equation • The Characteristics Methods Let H : T ∗ Q → R � and a real number E s.t. H − 1 ( E ) � = ∅ , better: rk dH H − 1 ( E ) = 1 , � • a classical ( C 1 ) solution S ( q ) of the related H-J equation H ( q, ∂S ∂q ( q )) = E • (if there exists...maybe just only local... and so on) • can be thought as an exact Lagrangian submanifold Λ = im ( dS ) globally transverse to the fibers of π Q : T ∗ → Q Λ = im ( dS ) ⊂ H − 1 ( E ) Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Geometrical synopsis of Hamilton-Jacobi equation • How to (geometrically) generalize ? • Def. - Geometrical solutions of H-J: We say that a Λ , Lagrangian in T ∗ Q , is a geometrical solution of H-J H = E if Λ ⊂ H − 1 ( E ) • Recalling dimensions: ⊂ H − 1 ( E ) ⊂ T ∗ Q dim Q = n, Λ ���� � �� � ���� n 2 n − 1 2 n • By relaxing transversality, we accept the possible ‘multivalued’ character of the Lagrangian submanifolds as solutions of H-J • We are saying nothing now about the topology of j : immersion/embedding,..., j (Λ) could be dense into... Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Geometrical synopsis of Hamilton-Jacobi equation • What’s the recipe to build Lagrangian submanifolds Λ into H − 1 ( E ) ? • The 2-form ω is represented by the skw 2 n × 2 n matrix � O � I E T = E − 1 = − E E 2 = − I , E = , and − I O • consider the Hamiltonian vector field X H related to H : • it is defined as an equality between 1-forms : i X H ω = − dH �� O �� � � � � X q � � ∂H I H ∂q , · = − , · X p ∂H − I O H ∂p � � � X q � ∂H H ∂p ⇒ X H = = X p − ∂H H ∂q • Theorem (Origin of Characteristics Method) If the Lagrangian → T ∗ Q solves geometrically H-J: H = E , that is Λ ⊂ H − 1 ( E ) , j : Λ ֒ then the Hamiltonian vector field is tangent to Λ : X H ( j ( λ )) ∈ T λ Λ ∀ λ ∈ Λ Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Characteristics Method for Hamilton-Jacobi equation • Theorem (Origin of Characteristics Method) If the Lagrangian → T ∗ Q solves geometrically H-J: H = E , that is Λ ⊂ H − 1 ( E ) , j : Λ ֒ then the Hamiltonian vector field is tangent to Λ : X H ( j ( λ )) ∈ T λ Λ ∀ λ ∈ Λ ———– • Proof. Since any tangent vector to Λ is also (we adopt trivial identifications: Dj ( λ ) v ≈ v )) a tangent vector to H − 1 ( E ) , it is on the kernel of dH , v ∈ T λ Λ ⇒ dH ( j ( λ )) v = 0 i X H ω = − dH i X H ω v = − dH v = 0 ω ( X H , v ) = 0 ∀ v ∈ T λ Λ ⇒ X H is ω -orthogonal to T λ Λ ; since (i) the space of the vectors ω -orthogonal to T λ Λ is of dimension 2 n − n = n ( ω is not degenerate), and since (ii) all the vectors of T λ Λ are ω -orthogonal to T λ Λ itself ( j ∗ ω = 0 ), necessarily X H is in T λ Λ . Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Characteristics Method for Hamilton-Jacobi equation • As a consequence: • take into H − 1 ( E ) a n - 1 -submanifold ℓ 0 , ⊂ H − 1 ( E ) ℓ 0 ���� � �� � n − 1 2 n − 1 • such that: X H / ∈ Tℓ 0 (Transversality Condition) • the candidate solution ‘starting’ from ℓ 0 is � φ λ Λ = X H ( ℓ 0 ) λ ∈ R • ⇒ dimension is correct (i.e. n ), • ⇒ and surely Λ ⊂ H − 1 ( E ) , from the conservation of H along φ λ X H • ⇒ at the end, we can also check that it is effectively Lagrangian: ω | Λ = 0 Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Characteristics Method for evolutive Hamilton-Jacobi equation • The evolutive case Let Q be a smooth, connected and closed (i.e: compact & ∂Q = ∅ ) manifold. • Take a Hamiltonian H : R × T ∗ Q → R ( H ∈ C 2 , σ ∈ C 1 ): • The Classical Cauchy Problem  � � ∂S t, q, ∂S ∂t ( t, q ) + H ∂q ( t, q ) = 0 ,   ( Cauchy Pr. )   S (0 , q ) = σ ( q ) , • We proceed by a space-time ‘homogeneization’: • H : T ∗ ( R × Q ) − → R ( t, q ; τ, p ) �− → H := τ + H ( t, q, p ) • with the symplectic form on T ∗ ( R × Q ) : ω = dτ ∧ dt + dp ∧ dq Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Characteristics Method for evolutive Hamilton-Jacobi equation ∂S ∂t ( t, q ) + H ( t, q, ∂S • the evolutive H-J ∂q ( t, q )) = 0 reads: • H = 0 • take into H − 1 (0) the following n -submanifold ℓ 0 , ⊂ H − 1 (0) ⊂ T ∗ ( R × Q ) ℓ 0 ���� � �� � � �� � n 2 n +1 2 n +2 • ℓ 0 encodes the initial data : �� � � � 0 , q, ∂σ , ∂σ � ⊂ H − 1 (0) ⊂ T ∗ ( R × Q ) ℓ 0 := 0 , q, − H ∂q ( q ) ∂q ( q ) : q ∈ Q •  ˙ t = 1       ∂H  q = ˙   ∂p the flow φ t X H of X H is ‘substantially’ the flow of X H :  − ∂H τ = ˙   ∂t       − ∂H p = ˙ ∂q Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Characteristics Method for evolutive Hamilton-Jacobi equation • the ( n +1)-dimensional Lagrangian submanifold, geometrical solution of the Cauchy Problem for t ∈ [0 , T ] , is • � φ t X H ( ℓ 0 ) ⊂ T ∗ ( R × Q ) Λ = t ∈ [0 ,T ] • Some remarks: • Λ is the collection of the wave front sets at any t ∈ [0 , T ] : φ t X H ( ℓ 0 ) the φ t X H ( ℓ 0 ) are Lagrangian submanifolds in T ∗ Q • Furthermore, Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Generating Functions for evolutive Hamilton-Jacobi equation • Generating Function for Λ ? • Under suitable conditions, from the above Lagrangian solution Λ , we have to provide a global Generating Function • if W t ( q 0 , q 1 ; u ) is a global Generating Function for the symplectomorphism φ t X H : T ∗ Q → T ∗ Q , • then: S t ( q ; u, ξ ) := σ ( ξ ) + W t ( ξ, q ; u ) ���� ���� � �� � aux. p. Geometric Propagator , Green kernel g . f . of im(d σ ) φ t • is generating the wave front set for t ∈ [0 , T ] : X H ( ℓ 0 ) Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Generating Functions for evolutive Hamilton-Jacobi equation • (Overcaming) Drawback: note that for W t ( ξ, q ; u ) , and so for S t ( q ; u, ξ ) , the dimension k of the space of the aux. par., R k , is depending of t , growing with t . • A new strategy: v := ���� ξ, u ) for Λ with a space v ∈ R k , k (i) To provide a Generating Function S t ( q ; uniform (independent) of t ∈ [0 , T ] (ii) under suitable hypotheses on H and σ , for any fixed ( t, q ) ∈ [0 , T ] × Q , to pick out a well precise (among many) critical value for S , • ∂S t ∂v ( q ; v ) = 0 • call it: S ( t, q ) • this will be the candidate weak function we are looking for! Variational solutions of Hamilton-Jacobi equations -2 geometrical setting: the

Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology, Palais-Smale, Min-max solutions INDAM - Cortona, Il Palazzone September 12-17, 2011 Franco Cardin Dipartimento di Matematica Pura e Applicata Universit` a degli Studi di Padova Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Aims and motivation To sum up: • Maslov-H¨ ormander theorem claims that (locally) every Lagrangian submanifold admits Generating Functions W ( q, ξ ) : p = W ,q , 0 = W ,ξ • There exist three operations linking (again locally) all the Generating Functions for a same Lagrangian submanifold • Now, our task is to derive, from a Generating Function of a Lagrangian submanifold geom. sol. of H-J equ., a suitable weak (true) function • we have to pick out, to select, Hamiltonians providing H-J equ. and relative geom. solutions with a • (i) unique global Generating Function, • and (ii) such that it admits, for any q , a well precise (universal, in a sense) critical value W ∗ : 0 = W ,ξ Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Aims and motivation • (i) unique global Generating Function? • → by Amman-Conley-Zehnder method (a sort of Lyapunov-Schmidt with a Fourier cut-off) or • → by Chaperon method (said of broken geodesics) surely Hamiltonians with quadratic p -dependence and q ∈ M compact with ∂M = ∅ , admit unique global Generating Function • more, these last Generating Functions are Quadratic at Infinity (GFQI) : W ( q, ξ ) = ξ t Aξ, for | ξ | > C (large) : det A � = 0 • they are Palais-Smale, • so min-max and Lusternik-Schnirelman theory does work: • and finally a well precise –the above point (ii)– min-max critical value can be achieved. Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Generating Functions Quadratic at Infinity • DEF 1 W is GFQI iff: W ( q, ξ ) = ξ t Aξ, for | ξ | > C (large) : det A � = 0 • A generalization of the above def., introduced by Viterbo and studied in detail by Theret, is the following: • DEF 2 A generating function W : M × R k → R , ( q, ξ ) �→ S ( q, ξ ) , is asymptotically quadratic if for every fixed q ∈ M || W ( q, · ) − P (2) ( q, · ) || C 1 < + ∞ , (1) where P (2) ( q, ξ ) = ξ t A ( q ) ξ + b ( q ) · ξ + a ( q ) and A ( q ) is a nondegenerate quadratic form. • The two defs are equivalent , up to the above three operations! Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Synopsis on min-max and Lusternik-Schnirelman theory by Relative Cohomology • Let f be a C 2 function on a manifold X . We shall assume either that X is compact or that f satisfies the Palais-Smale condition: • P-S Any sequence ( x n ) such that f ′ ( x n ) → 0 and f ( x n ) is bounded has converging subsequence. • Note that if x is the limit of such a subsequence, it is a critical point of f . • The aim of Lusternik-Schnirelman theory (L-S theory. , for short) will be to give a lower bound to the set of critical points of f on X in terms of the topological complexity of X . • We denote the sub-level sets by X a = { x ∈ X | f ( x ) ≤ a } . • We now define this topological complexity in terms of cohomology • The idea of utilizing forms in order to construct critical values of f comes back to Birkhoff and Morse. Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Synopsis on Relative Cohomology • Let Y ⊂ X be two manifolds, ι : Y ֒ → X . Define the complex of relative forms Ω q ( X, Y ) = Ω q ( X ) ⊕ Ω q − 1 ( Y ) and the following relative exterior differential (we will keep using the symbol d to indicate it) d q : Ω q ( X ) ⊕ Ω q − 1 ( Y ) − → Ω q +1 ( X ) ⊕ Ω q ( Y ) d ( ω, θ ): = ( dω, ι ∗ ω − dθ ) ∈ Ω q +1 ( X ) ⊕ Ω q ( Y ) . Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Synopsis on Relative Cohomology • The relative form ( ω, θ ) is relatively closed if d ( ω, θ ) = ( dω, ι ∗ ω − dθ ) = (0 , 0) that is, if ω is closed in X , its restriction to Y is exact, and θ is a primitive. • The relative form ( ω, θ ) is relatively exact if there exists ω, ¯ ω, ¯ θ ) ∈ Ω q − 1 ( X ) ⊕ Ω q − 2 ( Y ) such that d (¯ (¯ θ ) = ( ω, θ ) , more precisely, ω − d ¯ ω and θ = ι ∗ ¯ ω = d ¯ θ . • Observe that d 2 = 0 : d 2 ( ω, θ ) = d ( dω, ι ∗ ω − dθ ) = ( d 2 ω, ι ∗ dω − d ( ι ∗ ω − dθ )) = (0 , 0) . Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Synopsis on Relative Cohomology • The relative cohomology is by definition the space of quotients H q ( X, Y ) = Ker d q Im d q − 1 = Z q ( X, Y ) B q ( X, Y ) . Using the notation � � B ∗ ( X, Y ) = B q ( X, Y ) , H ∗ ( X, Y ) = H q ( X, Y ) , etc . q ≥ 0 q ≥ 0 The elements of H ∗ ( X, Y ) are equivalence classes of elements ( ω, θ ) + B ∗ ( X, Y ) , with ( ω, θ ) ∈ Z ∗ ( X, Y ) . We have seen that ω must be closed in X and exact in Y with θ a primitive. Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

� � � Synopsis on Relative Cohomology Let X, X ′ , Y, Y ′ be manifolds, f : Y → X an application • Theorem 1 (e.g. an embedding f : Y ֒ → X ) and ϕ : X → X ′ , ψ : Y → Y ′ two diffeomorphisms. Define f ′ := ϕ ◦ f ◦ ψ − 1 , f Y X ϕ ψ � X ′ Y ′ f ′ Then H ∗ ( X, Y ) ≡ H ∗ ( X ′ , Y ′ ) ( invariance by diffeomorphisms ) Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

� � � Synopsis on Relative Cohomology • Theorem 2 For every diffeomorphism f : Y → X , one has H ∗ ( X, Y ) = 0 . Proof. One can apply theorem 1, f Y X f id X � X X id X and observe that the closed forms on X , that also are exact on X , vanish in H ∗ ( X, X ) = 0 . (trivial cohomology between diffeomorphic manifolds) Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Synopsis on Relative Cohomology • Theorem 3 Let Z ⊆ Y ⊆ X , i and j be the inclusions: i j Z ֒ → Y ֒ → X The sequence i j H ∗ ( X, Y ) H ∗ ( X, Z ) H ∗ ( Y, Z ) − → − → is exact, which means: Im i = Ker j . • Proof. The map i takes a ( ω, θ ) in H ∗ ( X, Y ) and maps it to an element of H ∗ ( X, Z ) by restricting the domain of θ , from Y to Z . The map j takes an ( ω, θ ) in H ∗ ( X, Z ) and maps it in H ∗ ( Y, Z ) by restricting the domain of ω , from X to Y . The kernel of j are all the closed forms ω in X , that vanish (so that are exact, think of equivalence classes) in Y , and hence in Z . The image of i are all the closed forms ω in X , that are exact in Y , hence remaining exact after restriction to Z . Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Synopsis on Relative Cohomology • In summary: Relative Cohomology is invariant • (i) under diffeomorphisms , • and also • (ii) under retractions : Given ι : S ֒ → X , S is a retract of X if ∃ a continuous map (called retraction) r : X → S such that r ( y ) = y, ∀ y ∈ S . In other terms: r ◦ ι = id S , that is the inclusion ι admits a continuous left inverse, ι r → X − → S S ֒ r ◦ ι = id S • (iii) under excisions : ∃ isomorphism j ∗ : H ∗ ( X, Y ) − → H ∗ ( X \ U, Y \ U ) if the open U is disjoint from the boundary of Y , then U can be eliminated without changing cohomology Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Synopsis on min-max and Lusternik-Schnirelman theory by Relative Cohomology • Take the pair ( f, X ) P-S, f : X → R • Take a < b , X a , and f − 1 [ a, b ] = X b \ X a • Consider X b , • Suppose no critical value of f in [ a, b ] • ⇒ Theorem: X b and X a are diffeomorphic. • ⇒ by ( invariance by diffeomorphisms ) H ∗ ( X b , X a ) = 0 • A sketch of proof of the above Theor: ∇ f � = 0 in f − 1 [ a, b ] , by the flow of a vector field, which in f − 1 [ a, b ] is ∇ f dt f ◦ φ t d X = − �∇ f � 2 , X = ∇ f · X = − 1 , so : X b → X a φ b − a X is the diffeomorphism we are looking for. • (Rem: by P-S, �∇ f � is ‘bounded away from zero’, so X is Lip) Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Synopsis on min-max and Lusternik-Schnirelman theory by Relative Cohomology • Thus, something more interesting may occur in X b \ X a if there exists some non vanishing class α � = 0 in H ∗ ( X b , X a ) • More precisely, the following Theorem holds: • Theorem (min-max) Let α � = 0 in H ∗ ( X b , X a ) . For any a ≤ λ ≤ b we write: ι λ : X λ ֒ → X b and denote the induced map between relative cohomologies by ι ∗ λ : H ∗ ( X b , X a ) → H ∗ ( X λ , X a ) • (Note that: ι ∗ ι ∗ b α = α, and a α = 0 ) • Then � � λ ∈ [ a, b ] : ι ∗ c ( α, f ) := inf λ α � = 0 • is a critical value for f . Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

� A proof of the min-max Theorem • Proof. By contradiction : for α ∈ H ∗ ( X b , X a ) , α � = 0 , the value c ( α, f ) is a regular value for f . • ( PS ) ⇒ the set of critical points of f in f − 1 ([ a, b ]) is a compact set, then closed. There exists a ε (small) such that [ c − ε, c + ε ] does not contain critical values 4 of f . Hence, in view of an above theorem, H ∗ ( X c + ε , X c − ε ) = 0 X a ⊆ X c − ε ⊆ X c + ε • Consider now the exact sequence based on: (rem. Th. 3 above): � H ∗ ( X c + ε , X a ) ⋆ � H ∗ ( X c − ε , X a ) 0 = H ∗ ( X c + ε , X c − ε ) i ∗ c + ε α ∈ H ∗ ( X b , X a ) • Since the horizontal sequence is exact, one has that the kernel of ⋆ is the null space, hence ⋆ is injective. By definition of c , α � = 0 in H ∗ ( X c + ε , X a ) , hence its image under the map ⋆ should be non-zero: α � = 0 in H ∗ ( X c − ε , X a ) , this fact contradicts the definition of c . 4 in other words, c cannot be an accumulation point of critical values Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

It is time to come back to GFQI • Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Quadraticity at infinity • We will see that, at least for Hamiltonians which are quadratic (a generic hyperbolic q.f.) on p ’s and with possible compactly supported ‘perturbation’ on [0 , T ] × T ∗ T n , like: H = 1 2 p t Ap + V ( t, q, p ) • the Lagrangian submanifold Λ , geometrical solution of the Cauchy Problem for The evolutive case  � � ∂S t, q, ∂S ∂t ( t, q ) + H ∂q ( t, q ) = 0 ,   ( Cauchy Pr. )   S (0 , q ) = σ ( q ) , • is generated by a Generating Function Quadratic at Infinity, S t ( q ; ξ, U ) , with respect to the aux. parameters ( ξ, U ) Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Quadraticity at infinity: uniqueness & Palais-Smale • A Theorem by Viterbo globalizes to the GFQI the already known (local) theorem characterizing, by three operations, all the (local) GF of a same Lagrangian submanifold Λ . • ⇒ In essence: the GFQI are unique, up to the three operations • Together with uniqueness, we gain also the following crucial property: • GFQI are Palais-Smale • This is a crucial step in order to define the minmax or variational solution of H-J Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Quadraticity at infinity: Palais-Smale • Theorem Let f : M × R k → R , f : ( q, ξ ) �→ f ( q, ξ ) be a GFQI. Then, for any fixed q , f ( q, · ) is Palais-Smale. • Proof. For every fixed q , let { ξ j } j ∈ N be a sequence such that ∂f | f ( q, ξ j ) | ≤ ¯ C < + ∞ , lim ∂ξ ( q, ξ j ) = 0 j → + ∞ If the sequence { ξ j } j ∈ N is, from a certain index on, in a compact set Ω , then there must be a converging subsequence, let say that ¯ ξ is its limit. This limit must obviously be a critical point. Let us verify that nothing different can happen. Since f is a GFQI, then for | ξ | > C , f ( q, ξ ) = ξ T Aξ , where ξ T Aξ is a non-degenerate quadratic form. If there were only finite terms of the sequence in some Ω compact set, it would follow that lim j → + ∞ | ξ j | = + ∞ . Then the terms ξ j would end up outside from the ball B ( C ) , and this would contradicts the hypothesis, since in such case ∂f ∂ξ ( q, ξ j ) = 2 Aξ j Recalling that A is non-degenerate, ∂f ∂ξ ( q, ξ j ) would then tend to ∞ and not to zero. Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

GfQI: sub-level sets for great | c | • Let f ( q, ξ ) be a GFQI: if | ξ | > K then f ( q, ξ ) = ξ t Aξ with A t = A non-degenerate. Let R be the spectral radius of A , i.e.the supremum of the absolute value | λ | of the eigenvalues λ of A , Aξ λ = λξ λ , − R | u | 2 ≤ ξ t Aξ ≤ R | u | 2 . If for chosen (large enough) c > 0 such that R K 2 < c, − c < ξ ∈ B ( K ) f ( ξ ) ≤ min ξ ∈ B ( K ) f ( ξ ) < c, max and then f c = A c , f − c = A − c hence: H ∗ ( f c , f − c ) = H ∗ ( A c , A − c ) Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Variational min-max solutions for H-J equations • We utilize a result from algebraic geometry: it is well know that the relative cohomology for A is (see next paragraph for further clarifications)  R , if h = i, Morse index (: # of neg . eigenvalues) of A,  H h ( A c , A − c ) =  0 , if h � = i. Let α be precisely the generator of the 1-dimensional H i ( A c , A − c ) . We define the Variational min-max solutions for H-J equations: S ( t, q ) := c ( α ; S ( t, q ; · )) • Proceeding in this way for every ( t, q ) , the solution defined with this technique is known as the min-max , or variational solution, by Chaperon Sikorav Viterbo. It comes out that it is a Lipschitz-continuous function (see the unpublished work of Ottolenghi-Viterbo) and the beautiful book of Siburg. This last fact is rather surprising, it is the same regularity of the viscosity solutions. Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Interlude: Relative Cohomology of quadratic forms • Rem: Q := ξ T Qξ, Q c := { ξ ∈ R N : Q ≤ c } , A := Q − ( c + ε ) • A ‘graphical’ explanation of H ∗ ( Q c , Q − c ) ∼ = H ∗ ( D k − , ∂D k − ) : Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Interlude: Relative Cohomology of quadratic forms • A := Q − ( c + ε ) , We have seen: ◦ ◦ H ∗ ( Q c , Q − c ) ∼ = by excision H ∗ ( Q c \ A, Q − c \ A ) ∼ = by retraction H ∗ ( D k − , ∂D k − ) ◦ c ( D k − \ ∂D k − , ∅ ) ∼ c ( D k − , ∂D k − ) ∼ • H ∗ ( D k − , ∂D k − ) = H ∗ = H ∗ = H ∗ D k − ) , c ( ◦ D k − ) ∼ • H ∗ = H ∗ c ( R k − ) , c (  R , if p = k −  H p c ( R k − ) = A classical theorem says:  if p � = k − 0 , • finally: H ∗ ( Q c , Q − c ) ∼ = H ∗ ( D k − , ∂D k − ) ∼ = R Variational solutions of Hamilton-Jacobi equations - 3 Relative Cohomology,

Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder reduction - Minmax & viscosity INDAM - Cortona, Il Palazzone September 12 -17, 2011 Franco Cardin Dipartimento di Matematica Pura e Applicata Universit` a degli Studi di Padova Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

A ∞ -parameter Generating Function • Here the construction of a global generating function for the geometric 2 | p | 2 + V ( q ) on T ∗ R n solution for H ( q, p ) = 1 2 p T Bp + V , B hyperbolic) : (then, the case 1 • ∂q ( t, q ) | 2 + V ( q ) = 0  ∂S ∂t ( t, q ) + 1 2 | ∂S  ( CP ) H  S (0 , q ) = σ ( q ) , • Let us consider the set of curves: � γ ( · ) = ( q ( · ) , p ( · )) ∈ H 1 � [0 , T ] , R 2 n � � Γ := : p (0) = dσ ( q (0)) • Sobolev imbedding theorem, H 1 � (0 , T ) , R 2 n � → C 0 � [0 , T ] , R 2 n � ֒ • The candidate gen. funct. is the Hamilton-Helmholtz functional Action : A : [0 , T ] × Γ − → R � t ( t, γ ( · )) �→ A [ t, γ ( · )] := σ ( q (0)) + [ p ( r ) · ˙ q ( r ) − H ( r, q ( r ) , p ( r ))] dr 0 Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

A ∞ -parameter Generating Function • A : [0 , T ] × Γ − → R γ = Φ ( velocities ) ∈ L 2 , • Since ˙ • we introduce the following bijection representation g for [0 , T ] × Γ : g : [0 , T ] × R n × L 2 � (0 , T ) , R 2 n � − → [0 , T ] × Γ ( t, q, Φ) �− → g ( t, q, Φ) = ( t, γ ( · )) , γ ( · ) = γ t,q ( · ) Φ = (Φ q , Φ p ) •   � t � t � s � �   ∂σ   γ ( s ) :=  q − Φ q ( r ) dr, q − Φ q ( r ) dr + Φ p ( r ) dr   ∂q s 0 0  � �� � q (0) • To be more clear, we remark that the second value of the map g ( t, q, Φ) is the curve γ ( · ) = ( q ( · ) , p ( · )) which is 1) starting from ( q (0) , dσ ( q (0))) , such that 2) ˙ γ ( · ) = Φ ( · ) , and 3) q ( t ) = q . Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

A ∞ -parameter Generating Function • The geometrical solution is realized by the web of the characteristics coming out from the n -dim initial manifold : � � � � 0 , q, ∂σ , ∂σ ⊂ T ∗ R n +1 (Γ H ) σ = 0 , q ; − H ∂q ( q ) ∂q ( q ) • Theorem The infinite-parameter generating function: W = A ◦ g : [0 , T ] × R n × L 2 − → R , (2) ( t, q, Φ) �− → W ( t, q, Φ) := A ◦ g ( t, q, Φ) , generates L H = � � � 0 ≤ t ≤ T ϕ t (Γ H ) σ , the geometric solution for the H 2 | p | 2 + V ( q ) : Hamiltonian H ( q, p ) = 1 � � � � ∂W t, q, ∂W � � H-J : ∂t ( t, q, Φ) D Φ ( t,q, Φ)=0 + H ∂q ( t, q, Φ) = 0 � � Φ: DW Φ: DW D Φ ( t,q, Φ)=0 � ∂W D Φ ( t,q, Φ)=0 = ∂σ � Initial data : ∂q (0 , q, Φ) ∂q ( q ) � Φ: DW • Note: L 2 is the ∞ -dimensional space of auxiliary parameters Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

The finite reduction (A-C-Z method) • Since DW D Φ ( t, q, Φ) = 0 selects in Γ characteristic curves, we will reduce the L 2 -set of { (Φ q , Φ p ) } to the smaller L 2 -set of the alone { Φ q } : it is substantially the Legendre transformation at work. •  � � � t � s Φ q ( s ) = ∂σ q − 0 Φ q ( r ) dr + 0 Φ p ( r ) dr   ∂q �   � t � � q = p ˙  DW Φ p ( s ) = − ∂V D Φ ( t, q, Φ) = 0 ≈ ⇒ q − Φ q ( r ) dr p = − ∂V ∂q ˙ ∂q ( q )  s   � �� �   Φ p is determined by Φ q Hence � t � s � t � � � � Φ q ( s ) = ∂σ ∂V q − Φ q ( r ) dr − q − Φ q ( τ ) dτ dr ( • ) ∂q ∂q 0 0 r • ⇒ Here ( • ) is a fixed point problem for Φ q ( · ) Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

The finite reduction (A-C-Z method) • By simplicity, in the following, we set the initial data: σ ≡ 0 • Actually, this is not restrictive: Consider the canonical transformation: � q = q ˜ ( easily we see : dp ∧ dq = d ˜ p ∧ d ˜ q ) p + ∂σ p = ˜ ∂q (˜ q ) p + ∂σ � K (˜ q, ˜ p ) = H ( q, p ) q ) = H (˜ q, ˜ ∂q (˜ q )) � p + ∂σ q =˜ q, p =˜ ∂q (˜ If (˜ q ( t ) , ˜ p ( t ) is a characteristic for K , starting from ˜ p (0) = 0 , then p ( t ) + ∂σ ( q ( t ) , p ( t )) = (˜ q ( t ) , ˜ ∂q (˜ q ( t )) is a characteristic for H , starting from p (0) = ∂σ ∂q ( q (0)) . Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

The finite reduction (A-C-Z method) • For every Φ q ∈ L 2 ((0 , T ) , R n ) , Fourier expansion: � (Φ q ) k e i (2 πk/T ) s Φ q ( s ) = k ∈ Z • For an arbitrarily fixed cut-off N ∈ N , the projection maps P N and Q N on the � e i (2 πk/T ) s � k ∈ Z of L 2 ((0 , T ) , R n ) , basis � � (Φ q ) k e i (2 πk/T ) s , (Φ q ) k e i (2 πk/T ) s P N Φ q ( s ) := Q N Φ q ( s ) := | k |≤ N | k | >N • P N L 2 ⊕ Q N L 2 = L 2 ((0 , T ) , R n ) We will write u := P N Φ q and v := Q N Φ q ⇒ Φ q = u + v Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

The finite reduction (A-C-Z method) • Theorem (Lip-contractive map) ′′ ( q ) | = C ( < + ∞ ) . Fix the cut-off N . Let sup q ∈ R n | V For fixed ( t, q ) ∈ [0 , T ] × R n and fixed u ∈ P N L 2 ((0 , T ) , R n ) , the map (try to recall ( • ) ...) F : Q N L 2 ((0 , T ) , R n ) − → Q N L 2 ((0 , T ) , R n ) � s � t � � � � ∂V v �− → Q N − q − ( u + v ) ( τ ) dτ dr ∂q 0 r is Lipschitz with constant √ Lip ( F ) ≤ T 2 C � � 1 + 2 N 2 πN √ � � T 2 C • We will choose N such that 1 + 2 N < 1 2 πN • Denote by F ( t, q, u ) ( s ) , shortly F ( u ) , the fixed point map : � s � t � � � � ∂V F ( u ) = Q N − q − ( u + F ( u )) ( τ ) dτ dr . ∂q 0 r Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

The finite reduction (A-C-Z method) • Recall the above fixed point equation for σ ≡ 0 : � s � t � � ∂V Φ q ( s ) = − q − Φ q ( τ ) dτ dr ( • ) ∂q 0 r • � s � t � � � � ∂V → for any u : F ( u ) = Q N − q − ( u + F ( u )) ( τ ) dτ dr ∂q 0 r ( ∗ ) • � s � t � � � � ∂V → search for some u : − q − ( u + F ( u )) ( τ ) dτ u = P N dr ∂q 0 r ( ∗∗ ) • summing m. by m., we restore –and solve– ( • ) : � t q t,q ( s ) = q − s Φ q ( r ) dr | Φ q = u + F ( u ) • equation ( ∗∗ ) is a finite dimensional equation , sometimes said ‘bifurcation equation’ in some analogous Lyapunov-Schmidt procedure. Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

The finite reduction (A-C-Z method) • P N L 2 ((0 , T ) , R n ) ≈ R n (2 N +1) : is the (new, finitely reduced) finite-dim. space of aux. parameters u • Theorem The finite-parameter function: W := [0 , T ] × R n × R n (2 N +1) − → R , ( t, q, u ) �− → W ( t, q, u ) = �� t � � � [ p ( s ) · ˙ q ( s ) − H ( s, q ( s ) , p ( s ))] ds = ( q ( s ) ,p ( s )) , � 0 where ( q ( s ) , p ( s )) is obtained by the finite reduction, depending on t, q, u : ( q ( s ) , p ( s )) =   � t  � �  , − ∂V   = pr Γ ◦ g t, q, [ u + F ( u )] ( s ) q − ( u + F ( u )) ( τ ) dτ ,   ∂q   � �� � r   � �� � Φ q ( s ) Φ p ( s ) 2 | p | 2 + V ( q ) . generates the geometric solution for H ( q, p ) = 1 Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

The finite reduction (A-C-Z method) • The last task in order to prove the theorem, is to see that the ‘biforcation equation’: � s � t � � � � ∂V − q − ( u + F ( u )) ( τ ) dτ ( ∗∗ ) u = P N dr ∂q 0 r is precisely given by ∂W ∂u ( t, q, u ) = 0 • The other relations hold: ∂W ∂t ( t, q, u ) + H ( t, q, ∂W p (0) = ∂W ∂q ( t, q, u )) = 0 ∂q (0 , q, u ) = 0 , Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

The finite reduction (A-C-Z method) • Since the fixed point map F can also obtained by the implict function Th. , more smoothness is gained for the generating function: • Smoothness: For fixed ( t, q ) ∈ [0 , T ] × R n , u �→ ∂ F u �→ F ( u ) and ∂u ( u ) are uniformly bounded . • Theorem (G.F. Quadratic at ∞ ): The finite-parameters function W := A ◦ g : [0 , T ] × R n × R n (2 N +1) − → R , → ¯ ( t, q, u ) �− W ( t, q, u ) = �� t � 1 � � � q ( s ) | 2 − V ( q ( s )) � = 2 | ˙ ds � � t q ( s )= q − s [ u ( r )+( F ( u ))( r )] dr 0 is asymptotically quadratic: there exists an u -polynomial P (2) ( t, q, u ) such that for any fixed ( t, q ) ∈ [0 , T ] × R n || W ( t, q, · ) − P (2) ( t, q, · ) || C 1 < + ∞ and, in this specific mechanical case, its leading term is positive defined (Morse index is 0 ) . Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

The finite reduction (A-C-Z method), the NON-CONVEX case • Whenever the Lagrangian L is non convex, e.g. q ) = 1 q T B ˙ L ( q, ˙ 2 ˙ q − V ( q ) , where B is a generically hyperbolic matrix , • a global Legendre transformation still does work (even thought Young-Fenchel is gone) • Theorem (G.F. Quadratic at ∞ ): The finite-parameters function W := A ◦ g : [0 , T ] × R n × R n (2 N +1) − → R , → ¯ ( t, q, u ) �− W ( t, q, u ) = �� t � 1 � � � q T B ˙ � = 2 ˙ q − V ( q ( s )) ds � � t q ( s )= q − s [ u ( r )+( F ( u ))( r )] dr 0 is asymptotically quadratic: there exists an u -polynomial P (2) ( t, q, u ) such that for any fixed ( t, q ) ∈ [0 , T ] × R n || W ( t, q, · ) − P (2) ( t, q, · ) || C 1 < + ∞ and its leading term is has the Morse index � = 0 (it will be related to the Morse index of B ) . Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

The finite reduction (A-C-Z method) • Proof (trace, for the convex case ) Through the Legendre transformation, �� t � 1 � � � q ( s ) | 2 − V ( q ( s )) � 2 | ˙ W ( t, q, u ) = ds � � t q ( s )= q − s [ u ( r )+( F ( u ))( r )] dr 0 � t � t � 1 � �� 2 | u ( s ) + ( F ( u )) ( s ) | 2 − V = q − [ u ( r ) + ( F ( u )) ( r )] dr ds. 0 s As a consequence of the compactness of V , of the uniformely boundness of F and its derivatives, for fixed ( t, q ) ∈ [0 , T ] × R n we obtain that || W ( t, q, · ) − P (2) ( t, q, · ) || C 1 < + ∞ , where P (2) ( t, q, u ) is polynomial with positive defined leading term � t 1 | u ( s ) | 2 ds = u T Qu 2 0 (hence with Morse index 0 ) and linear term with uniformly bounded coefficient, so that, W ( t, q, u ) is an asymptotically quadratic generating function. � Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

Minmax-variational and viscosity solutions for convex Hamiltonians • Now, given W ( t, q, u ) , we construct the variational solution: • For any fixed ( t, q ) , u �→ W ( t, q, u ) is Palais-Smale ⇒ L.-S. does work, • Relative Cohomology of the sub-level sets of W ( t, q, u ) and u T Qu are equivalent for large c > 0 : H ∗ ( W ( t, q, · ) c , W ( t, q, · ) − c ) ≈ H ∗ ( Q c , Q − c ) • We recall that the relative cohomology of quadrics is 1-dim:  R , if h = i : Morse index (# of neg . eigenvalues) of Q,  H h ( Q c , Q − c ) =  if h � = i. 0 , H 0 ( Q c , Q − c ) = R • In the convex case we are concerning, we have i = 0 : Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

Minmax-variational and viscosity solutions for convex Hamiltonians • Let α = 1 be the generator of the 1-dimensional H 0 ( Q c , Q − c ) ≈ R • ( Note that, concerning with the absolute deRham cohomology, for any manifold M with k connected components H 0 dR ( M ) = R k This follows from the fact that any smooth function on M with zero derivative (i.e. locally constant) is constant on each of the connected components of M .) • For large c , H 0 ( W ( t, q, · ) c , W ( t, q, · ) − c ) = H 0 ( Q c , Q − c ) = R but, for suitable small λ < c , some other connected components can arise for W ( t, q, · ) λ so that H 0 ( W ( t, q, · ) λ , W ( t, q, · ) − λ ) � = H ∗ ( Q λ , Q − λ ) Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

Minmax-variational and viscosity solutions for convex Hamiltonians see pictures, H 0 ( W ( t, q, · ) λ , W ( t, q, · ) − λ ) = R 2 = span ( α 1 , α 2 ) and, in such a case, ι ∗ λ 1 = R ( α 1 + α 2 ) that is, a same constant is assigned to both connected components. Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

Minmax-variational and viscosity solutions for convex Hamiltonians • We define the variational min-max solution for H-J: ι ∗ S ( t, q ) = minmax( W ( t, q, ; · )) := inf { λ ∈ [ − c, c ] : λ 1 � = 0 } • Finally : ⇒ S ( t, q ) = min u W ( t, q, ; u ) Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

Minmax-variational and viscosity solutions for convex Hamiltonians • We have to recall some facts: • ( i ) u �→ W ( t, q, · ) is a finite reduction of � t � � 1 � � � q ( s ) | 2 − V ( q ( s )) � Φ q ( · ) �→ σ ( q (0)) + 2 | ˙ ds � � t q ( s )= q − s Φ q ( r ) dr 0 • ( ii ) Critical points of u �→ W ( finite ) are one-to-one related to the critical points of � Φ q �→ σ + L ds ( infinite ) • ( iii ) some more is true: Morse indices related to ( infinite ) are precisely Morse indices related to ( finite ) • ⇒ This is sufficient to say that the variational min-max solution: � t � � q ( s ) , ˙ S ( t, q ) = u ∈ P N L 2 W ( t, q ; u ) = min inf σ (˜ q (0)) + L (˜ q ( s )) ds ˜ q ( · ):˜ ˜ q ( t )= q 0 = Lax-Oleinik semi-group : viscosity solution! Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

Variational solutions for general Hamiltonians Minmax are Lipschitz • Lipschitz property in the following slides: x = ( q, t ) , u : aux. parameters Theorem (minmax are Lipschitz) Let W ( x, u ) be the GFQI for a geometrical solution (a Lagrangian submanifold) for a H-J problem, W ( x, u ) = u T Qu, | u | > K (: large ) Let ι ∗ S ( x ) = minmax W ( x, · ) = inf { λ ∈ [ − c, c ] : λ α � = 0 } where α is the class generator of H i ( Q c , Q − c ) , i : Morse index of A be the related variational minmax solution of the H-J equation H ( x, ∂S ∂x ( x )) = e Then S ( x ) is Lipschitz. Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

Variational solutions for general Hamiltonians are Lipschitz • Proof. Denote by C > 0 the Lipschitz constant of the GFQI in U = T n × [0 , T ] , uniformely for ξ ∈ R k : � � ∂W � � C = sup ∂x ( x, u ) � � � � x ∈ U u ∈ R k so that | W ( x, u ) − W ( y, u ) | ≤ C | x − y | x, y ∈ U ( ∗ ) Def.: For fixed x , let now to define, for ε > 0 arbitrary small, a x ( y ) := S ( x ) + ε + C | x − y | , ∀ y ∈ U Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

Variational solutions for general Hamiltonians are Lipschitz • a x ( y ) := S ( x ) + ε + C | x − y | , ∀ y ∈ U � � x := { u ∈ R k : W ( x, u ) ≤ c } recall the notation for the sublevel sets: W c • We notice that W a x ( x ) ⊆ W a x ( y ) ( ∗∗ ) x y In fact, if u ∈ W a x ( x ) , x W ( x, u ) ≤ a x ( x ) = S ( x ) + ε ���� by definition of a x ( y ) for y = x from ( ∗ ) , W ( y, u ) ≤ W ( x, u ) + C | x − y | ≤ S ( x ) + ε + C | x − y | = a x ( y ) . ���� by definition of a x ( y ) Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

Variational solutions for general Hamiltonians are Lipschitz • By the very definition of S ( x ) , and S ( x ) < S ( x ) + ε = a x ( x ) , the relative , W − c ) contains 5 a non vanishing class α , so, by cohomology H ∗ ( W a x ( x ) x W a x ( x ) ⊆ W a x ( y ) , the same is true for H ∗ ( W a x ( y ) , W − c ) . This means that x y y S ( y ) ≤ a x ( y ) then, S ( y ) ≤ a x ( y ) = S ( x ) + ε + C | x − y | for the arbitrarity of ε > 0 , S ( y ) ≤ S ( x ) + C | x − y | : S ( y ) − S ( x ) ≤ C | x − y | By interchanging the role of x and y , we finally obtain | S ( y ) − S ( x ) | ≤ C | x − y | , ∀ x, y ∈ U • In other words: S ( x ) inherits the same Lip constant C > 0 from W ( x, u ). 5 W − c = W − c , ∀ x ∈ U and c large x Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

Variational solutions are not ‘Markovian’ • There is a ‘drawback’ of the variational solution C 0 , 1 : it is inherited from a generating function of a Lagrangian submanifold, starting from a smooth , C 1 , initial function σ : N → R , Consider the application J : J : C 1 , 1 ([0 , T ] × T ∗ N ) × C 1 ( N ) → C 0 , 1 ([0 , T ] × N ) ( H, σ ) �→ u =: J ( H, σ )( t ) = S ( t, q ) Theorem The application J is uniformly continuous if all the spaces are equipped with the C 0 topology. Thus it extends to an uniformly continuous map, still denoted by J , J : C 0 , 1 ([0 , T ] × T ∗ N ) × C 0 ( N ) → C 0 ([0 , T ] × N ) in particular, fixed H , � J ( H, σ 1 ) − J ( H, σ 2 ) � C 0 ≤ � σ 1 − σ 2 � C 0 . • Note: it is the same non-expansive property of the Lax-Oleinik semi-group ! Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

Variational solutions are not ‘Markovian’ Every continuous σ ∈ C 0 ( N ) can be approximated in the uniform convergence by a sequence of differentiable σ n ∈ C 1 ( N ) , The related variational solution is J ( H, σ n ) = u σ n . By continuity of J , i ) u σ n is a Cauchy sequence and ii ) its limit is independent of the approximating sequence σ n . ⇒ : Definition: C 0 -variational solution Given a continuous initial datum σ ∈ C 0 ( N ) , the C 0 -variational solution for the Cauchy problem is the unique function u σ ∈ C 0 ([0 , T ] × N ) such that, for any arbitrary C 1 approximating sequence σ n : C 0 C 1 ( N ) ∋ σ n → σ ∈ C 0 ( N ) , − with related C 0 , 1 -variational solutions J ( H, σ n ) = u σ n , we have that n → + ∞ � u σ n − u σ � C 0 = 0 on [0 , T ] × N. lim (3) Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

A topological algebraic framework • Main Theorem: Let N be compact, and S : N × R n ∋ ( x, ξ ) �− → S ( x, ξ ) ∈ R be a GFQI. Then, up to a shift of the degree by k − : H ∗ ( S ∞ , S −∞ ) ∼ = H ∗ ( N ) Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

Meaning of the Main Theorem: H ∗ ( S ∞ , S −∞ ) ∼ = H ∗ ( N ) ⇒ For compact N , the absolute cohomology H ∗ ( N ) , is precisely the relative cohomology of the sublevel sets of generic functions on f : N → R : for c > 0 : − c < min f ≤ max f < c, H ∗ ( f ∞ , f −∞ ) = H ∗ ( f c , f − c ) = H ∗ ( N, ∅ ) ∼ = H ∗ ( N ) In other words: To look for critical values and critical points of GFQI S : N × R k → R is like looking for critical values and critical points of f : N → R ! Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

A topological algebraic framework • Proof. Since S is a GFQI, for c > 0 big enough, S ± c ∼ = N × Q ± c =: S ±∞ � It follows remembering Kunneth formula: � H n ( M × Q c , M × Q − c ) ≃ � p + q = n H p ( M ) ⊗ H q ( Q c , Q − c ) : H ∗ ( S ∞ , S −∞ ) ∼ = H ∗ ( N × Q ∞ , N × Q −∞ ) ∼ = H ∗ ( N ) ⊗ H ∗ ( Q ∞ , Q −∞ ) ∼ = ∼ = H ∗ ( N ) ⊗ H ∗ ( D k − , ∂D k − ) ∼ c ( R k − ) ∼ c ( N × R k − ) ∼ = H ∗ ( N ) ⊗ H ∗ = H ∗ = H ∗ ( N ) where the last one is realized by the Thom isomorphism : giving the negative bundle π : N × R k − − → N and denoting by t k − the Poincar´ e dual cohomological class of the null section (= N ) of π , we get the k − -shifted isomorphism: → T ( α ) := π ∗ α ∧ t k − ∈ H h + k − H h ( N ) ∋ α �− ( N × R k − ) c Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

A metric on the Lagrangian submanifolds set L • We denoted by c ( α, L ) the min-max critical value of a GFQI relative to a Lagrangian submanifold L ∈ L • L : the set of Lagrangian submanifolds of T ∗ N which are Hamiltonian isotopic to O T ∗ N • L 1 , L 2 ∈ L be generated by the GFQI S 1 ( x ; ξ ) and S 2 ( x ; η ) respectively. • We denote by ( S 1 ♯ S 2 )( x ; ξ, η ) the GFQI ( S 1 ♯ S 2 )( x ; ξ, η ) := S 1 ( x ; ξ ) + S 2 ( x ; η ) • Considering ( S 1 ♯ ( − S 2 ))( x ; ξ, η ) = S 1 ( x ; ξ ) − S 2 ( x ; η ) we note that its critical points of ( S 1 ♯ ( − S 2 )) are precisely marking the � L 2 : intersections L 1 0 = ∂S 1 ∂x − ∂S 2 0 = ∂S 1 0 = ∂S 2 ( x, p 1 ) ∈ L 1 , ( x, p 2 ) ∈ L 2 : ∂x = p 1 − p 2 , ∂ξ , ∂η Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder

A metric on the group of Hamiltonian diffeomorphisms of T ∗ N • The main theorem, here in the form H ∗ � ( S 1 ♯ ( − S 2 )) ∞ , ( S 1 ♯ ( − S 2 )) −∞ � ∼ = H ∗ ( N ) is telling us that we have simply to look at (the cohomology of) the base manifold N in order to find global critical points of S 1 ♯ ( − S 2 ) . This leads us to the • Definition: γ ( L 1 , L 2 ) := c ( µ, S 1 ♯ ( − S 2 )) − c (1 , S 1 ♯ ( − S 2 )) , where 1 ∈ H 0 ( N ) and µ ∈ H n ( N ) are generators. • γ ( L 1 , L 2 ) is a metric on L . Variational solutions of Hamilton-Jacobi equations - 4 Amann-Conley-Zehnder