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 iff p ∈ ∂φ(x) iff x ∈ ∂φ∗(p).

◮ 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−2u

= f

• n Ω ,

u =

• n ∂Ω.

It suffices to minimize, on H1

0(Ω), the convex functional

φ(u) = 1 2

• Ω

|∇u|2dx + 1 p

• Ω

|u|pdx +

• Ω

fudx But what about the following non-selfadjoint equation ??

• − ∆u + |u|p−2u + Σn

i=1ai(x) ∂u ∂xi

= f

• n Ω ,

u =

• n ∂Ω.

Is there a variational resolution?

Variational approach to gradient flows (Brezis-Ekeland,1976)

• − ˙

v(t) ∈ ∂φ(v(t)) a.e.

• n

[0, T], v(0) = v0. φ convex l.s.c on Hilbert space H. (e.g., φ(u) = 1

2

• Ω |∇u(x)|2dx.)

Minimize the “selfdual functional" I(u) = T

• φ(u(t)) + φ∗(−˙

u(t))

• dt + 1

2|u(T)|2

• ver all path u ∈ L2

H, with ˙

u ∈ L2

H and u(0) = v0.

Using that T

0 u(t), ˙

u(t)dt = 1

2|u(T)|2 − 1 2|u(0)|2,

I(u) = T

• φ(u(t)) + φ∗(−˙

u(t)) + u(t), ˙ u(t)

• dt + 1

2|v0|2 ≥ 0, Now if (BIG IF) we can prove inf I(u) = I(¯ u) = 1

2|v0|2, then we are

done by Legendre duality.

Bogomolnyi’s trick of completing the square

(Yang-Mills, Chern-Simon, Seiberg-Witten, Ginsburg-Landau) Associate to any connection A ∈ Ω1(Ad E) where E is a vector bundle over an oriented closed 4-manifold M, the curvature FA = dA + 1

2[A ∧ A] ∈ Ω2(Ad E), and the exterior differential on k-forms

dAw = dw + [A ∧ w]. I(A) :=

• M

FA2 =

• M

1 2(FA2 + ∗ FA2) = 1 2

• M

FA + ∗FA2 −

• M

FA ∧ ∗FA = 1 2

• M

FA + ∗FA2 + 8π2c2(P) ≥ 8π2c2(P). c2(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 = 8π2c2(P) and if it is attained for some

connection A, then FA = − ∗ FA, the anti-selfdual Yang-Mills.

◮ They are first order equations obtained variationally but not via

Euler-Lagrange.

◮ In particular, YM equations d∗

AFA = 0 via Bianchi identities.

Self-dualize the problem: Ghoussoub-Tzou (2004)

Minimize the “selfdual functional" I(u) = T

• φ(u(t)) + φ∗(−˙

u(t))

• dt+1

2|u(0)|2 − 2u(0), v0 + |v0|2 + 1 2|u(T)|2 I(u) = T

• φ(u(t)) + φ∗(−˙

u(t)) + u(t), ˙ u(t)

• dt + |u(0) − v0|2 ≥ 0,

The Lagrangian on L2

X × L2 X∗ defined by

L(u, p) = T

0 φ(u(t)) + φ∗(−˙

u(t) + p(t))dt + ℓ (u(0), u(T)) if ˙ u ∈ L 2

X

+∞

• therwise

is selfdual! that is for every pair (u(t), p(t)) ∈ L2

X × L2 X∗,

L∗(p, u) = L(u, p). Why is inf I(u) = inf L(u, 0) = 0? Because by basic convex duality: α = sup

v∈X

−L∗(0, v) ≤ inf

u∈X L(u, 0) = β

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)

• r

(p, x) = ∂L(x, p).

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 Ip(x) = L(x, p) − x, p and by showing that infx∈X Ip(x) = 0.

◮ infx∈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

LΓ(x, p) = L(x, −Γx + 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: Γx + ∂φ(x) = ∂L(x) where L(x, p) = ϕ(x) + ϕ∗(−Γx − p). On can then solve Γx + ∂φ(x) = p by minimizing Ip(x) = L(x, p) − x, p = ϕ(x) + ϕ∗(−Γx + p) − x, p Simple example −∆u + |u|p−2u + Σn

i=1ai(x) ∂u ∂xi

= f on Ω u =

• n ∂Ω.
• 3. T : D(T) ⊂ X → 2X∗ 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−2u + Σn

i=1ai(x) ∂u ∂xi

= f

• n Ω ,

u =

• n ∂Ω.

Assuming div(a) ≥ 0 on Ω, then it suffices to minimize, on the same H1

0(Ω), the new convex functional

I(u) = Ψ(u) + Ψ∗(a · ∇u + 1 2div(a) u), where ψ is the convex functional Ψ(u) = 1 2

• Ω

|∇u|2dx+1 p

• Ω

|u|pdx+

• Ω

fudx+1 4

• Ω

div(a) |u|2dx, and ψ∗ is its Fenchel-Legendre transform.

◮ Equation is not an Euler-Lagrange equation. ◮ It is derived from the fact that I(¯

u) = infu∈H1

0(Ω) I(u) = 0.

Porous media

For m > 0, u0 ∈ Lm+1 and f ∈ H−1, the infimum of

I(u) =

1 m+1

T e2ωt

• Ω

|u(t, x)|m+1dxdt +

m m+1

T e2ωt

• Ω

|(−∆)−1(f(x) − ωu(t, x) − ∂u ∂t (t, x))|

m+1 m dxdt

− T e2ωt

• Ω

u(x, t)((−∆)−1f)(x)dxdt +

• Ω

|∇(−∆)−1u0(x)|2 dx −2

• Ω

u0(x)(−∆)−1u(0, x) dx + 1 2

• u(0)2

H−1 + e2wTu(T)2 H−1

• n A 2

H−1 is equal to zero and is attained uniquely at an L m+1(Ω)-valued u:

• ∂u

∂t

= ∆um + ωu + f on Ω × [0, T], u(0, x) = u0(x)

• n

Ω. Note that Euler-Lagrange equations (for heat flow m = 1, w = 0)

• ( ∂

∂t − ∆)( ∂ ∂t + ∆)u

= a.e.

• n

[0, T], u(0) = u0.

Semi-groups of contractions associated to selfdual Lagrangians

To any selfdual Lagrangian L such that Dom(∂L) is non-empty,

• ne can associate a semi-group of 1-Lipschitz maps (Tt)t∈R+
• n Dom(∂L) such that T0 = Id and for any x0 ∈ Dom(∂L), the

path x(t) = Ttx0 satisfies: ˙ x(t) ∈ −¯ ∂L(t, x(t)) for all t ∈ [0, T] and, x(t)2

H = x02 − 2

t

0 L(x(s), ˙

x(s))ds for every t ∈ [0, T]. The path (x(t))t = (Ttx0)t is obtained as a minimizer on A2

H of

the functional I(u) = T L(u(t), ˙ u(t))dt+1 2u(0)2−2x0, u(0)+x02+1 2u(T)2, whose infimum is equal to zero.

Stationary incompressible Navier-Stokes

On the space V = {u ∈ H1

0(Ω; R3); divv = 0}, consider the

nonlinear operator Λu, v :=

• Ω

Σ3

j,k=1

∂vj ∂xk ujuk dx = (u · ∇)v, u. For f ∈ Lp(Ω, R3) with p > 6

5 and ν > 0 consider the convex

function φ(u) = ν 2

• Ω

Σ3

j,k=1( ∂uj

∂xk )2 dx +

• Ω

Σ3

j=1fjuj.

The functional I(u) = φ(u) + φ∗(−(u · ∇)u) has an infimum equal to zero on V and is attained at a solution

• f the NS equation

−(u · ∇)u = −ν∆u + ∇p + f

• n Ω

u =

• n ∂Ω

div(u) = 0.

The nonlinear case (1)

In order to solve −Λu = ∂φ(u) where Λ : X → X∗ is linear or not, and φ : X → ¯ R is convex lower semi-continuous. Try minimizing the non-negative functional functional I(u) = φ(u) + φ∗(−Λu) + Λu, u ≥ 0. If you can prove that I(¯ u) = inf

u∈X I(u) = 0

then you are done! If Λu, u = 0 (Λ conservative) then it suffices to minimize I(u) = φ(u) + φ∗(−Λu).

Navier-Stokes evolution

The minimum of

I(u) = T

• Φ(u(t)) + Φ∗(−(u · ∇)u(t) + f − ˙

u(t)) −

• Ω

f, udx

• dt

+

• Ω

1 2(|u(0, x)|2 + |u(x, T)|2) − 2u(0, x), u0(x) + |u0(x)|2

• dx
• n A2

V is zero if n = 2 (≤ 0 when n = 3) and is attained at a

solution of the NS equation             

∂u ∂t + (u · ∇)u − f ∈ ∂Φ(u)

= ν∆u − ∇p

• n [0, T] × Ω

divu =

• n [0, T] × Ω

u(t, x) =

• n [0, T] × ∂Ω

u(0, x) = u0(x)

• n Ω

Hamiltonian systems with general boundary conditions

Let φ : [0, T] × R2N → ¯ R be convex lsc in u for a.e. t ∈ [0, T], and ψ : X → ¯ R ∪ {∞} be convex lsc on R2N. Assume:

• 1. There is β ∈ (0, π

2T ) and γ, α ∈ L2(0, T; R+) such that

−α(t) ≤ φ(t, u) ≤ β

2|u|2 + γ(t) for every u ∈ H.

2. T

0 φ(t, u) dt → +∞

as |u| → +∞.

• 3. ψ is bounded from below and 0 ∈ Dom(ψ).

The infimum of the functional J(u) = T

• φ(t, u(t)) + φ∗(t, −J ˙

u(t)) + J ˙ u(t), u(t)

• dt

+ u(T) − u(0), J u(0)+u(T)

2

+ ψ

• u(T) − u(0)
• + ψ∗

− J u(0)+u(T)

2

• n A2

X is then equal to zero and is attained at a solution of

       −J ˙ u(t) = ∂φ

• t, u(t)
• ,

−J u(T)+u(0)

2

∈ ∂ψ

• u(T) − u(0)
• .

Connecting Lagrangian manifolds

Suppose H : R2N → R is a convex lsc Hamiltonian with −α ≤ H(p, q) ≤ β

2(|p|2 + |q|2) + γ, where β < 1 2max(2T2,1).

(1) Let ψ1 and ψ2 be two convex lsc and coercive functions on RN such that one of them satisfies: lim inf

|p|→+∞ ψi(p) |p|2 > 2T

for i = 1 or 2. (2) Let X = W1,2(0, T; RN), then the minimum of the functional I(p, q) : = T

• H
• p(t), q(t)
• + H∗

− ˙ q(t), ˙ p(t)

• + 2 ˙

q(t) · p(t)

• dt

+ψ2

• q(T)
• + ψ∗

2

• − p(T)
• + ψ1
• p(0)
• + ψ∗

1

• q(0))
• n X × X is zero and is attained at a solution of

           ˙ p(t) ∈ ∂2H

• p(t), q(t)
• t ∈ (0, T),

− ˙ q(t) ∈ ∂1H

• p(t), q(t)
• t ∈ (0, T),

q(0) ∈ ∂ψ1

• p(0)
• &

−p(T) ∈ ∂ψ2

• q(T)
• .

Unexpected surprise: All maximal monotone operators are selfdual vector fields and vice-versa

(i) Let L be a proper selfdual Lagrangian L on a reflexive Banach space X × X∗, then the vector field x → ¯ ∂L(x) is maximal monotone. (ii) Conversely, if β : D(β) ⊂ X → 2X∗ is a maximal monotone

• perator with a non-empty domain, then there exists a selfdual

Lagrangian Lβ on X × X∗ such that β = ¯ ∂Lβ. (1) One can then solve p ∈ β(u) by minimizing on X the functional I(u) = Lβ(u, p) − u, p. (2) One can also solve −div(β(∇u(x))) = p(x) on Ω by minimizing on H1

0(Ω) the functional

I(u) := inf

f∈L2(Ω;RN) −div(f)=p

• Ω
• Lβ
• ∇u(x), f(x)
• − u(x), p(x)RN
• dx

Nonlinear inverse problems

Consider the following problem: Given f0 ∈ H1

0(Ω), find a maximal monotone vector field T in a

given class C such that f0 is a solution of −div(T(∇f(x)) = g(x). (3) Least square approach: Minimize

• Ω

|f(x) − f0(x)|2dx

• ver all f ∈ H1

0(Ω), T ∈ C, such that −div(T(∇f)) = g on Ω.

The constraint set is not easily tractable.

Penalized least square

Let L = {L selfdual on ¯ Rn × ¯ Rn; ¯ ∂L = T for some T ∈ C}. For each ǫ > 0, consider the minimization problem: inf

• Pǫ(L, f, v); L ∈ L, f ∈ H1

0(Ω), v ∈ L2(Ω; ¯

Rn), div v = 0

• , where

Pǫ(L, f, v) =

• Ω

|f(x) − f0(x)|2dx +1 ǫ

• Ω
• LT
• ∇f, v + p0
• − f(x)g(x)
• dx

Pǫ is convex and lsc in all variables. So if L is a convex compact class of selfdual Lagrangians, then there exists a minimizer (Lǫ, fǫ, vǫ) ∈ L × H1

0(Ω).

Now when ǫ is small enough, the non-negative penalization has to be small at (Lǫ, fǫ, vǫ), and a weak cluster point (L0, f0, v0) is a solution with T0 := ¯ ∂L0 being the optimal maximal monotone

• perator, since the penalty term has to be zero.

A basic homogenization problem

We consider the conductivity equation with a given heat source u∗

n in a heterogenous medium defined by the

non-homogeneous conductivity vector field β.          τn(x) ∈ β( x

ǫn , ∇un(x))

x ∈ Ω, −div(τn(x)) = u∗

n(x)

x ∈ Ω, un(x) = x ∈ ∂Ω, (4) where Ω is a bounded domain of RN, and β : Ω × RN → RN is a measurable map on Ω × RN such that:

◮ β(x, ·) is maximal monotone on RN for almost all x ∈ Ω ◮ β(., ξ) is Q-periodic for an open non-degenerate

parallelogram Q in Rn. This problem has been investigated in recent years by many authors: Francfort, Murat, Tartar, Damlamian, Meunier, Van Shaftingen, Braides, Chiado Piat, Dal Maso, Defranscheshi.

Representation of a family of maximal monotone fields

• τ(x)

∈ β(x, ∇u(x)) a.e. x ∈ Ω, −div(τ(x)) = p(x) a.e. x ∈ Ω. The class MΩ,p(RN) introduced by Chiado Piat, Dal Maso, Defranscheshi consists of all possibly multi-valued functions β : Ω × RN → RN with closed values, which satisfy: (i) β is measurable with respect to L(Ω) × B(RN) and B(RN) where L(Ω) is is the σ-field of all measurable subsets of Ω and B(RN) is the σ-field of all Borel subsets of RN. (ii) For a.e. x ∈ Ω, the map β(x, .) : RN → RN is maximal monotone. (iii) There exist non-negative constants m1, m2, c1 and c2 such that for every ξ ∈ RN and η ∈ β(ξ), ξ, ηRN ≥ max

• c1

p |ξ|p − m1, c2 q |η|q − m2

• ,

(5)

Selfdual Lagrangians associated to maximal monotone operators

(1) If β ∈ MΩ,p(RN) for p > 1, then there exists a state-dependent selfdual Lagrangian L : Ω × RN × RN → R such that (*) β(x, .) = ¯ ∂L(x, .) for a.e. x ∈ Ω and for all a, b ∈ RN, (∗∗) C0(|a|p + |b|q − n0(x)) ≤ L(x, a, b) ≤ C1(|a|p + |b|q + n1(x)) where C0 and C1 are two positive constants and n0, n1 ∈ L1(Ω). (2) Conversely, if L : Ω × RN × RN → R is a state-dependent selfdual Lagrangian satisfying (**), then ¯ ∂L(x, .) ∈ MΩ,p(RN).

Lifting Self-dual Lagrangians from ¯ Rn × ¯ Rn to W1,p

0 (Ω) × W−1,q(Ω)

Theorem

Suppose L is a state-dependent selfdual Lagrangian on Ω × RN × RN such that for all a, b ∈ RN, (∗∗) C0(|a|p +|b|q −n0(x)) ≤ L(x, a, b) ≤ C1(|a|p +|b|q +n1(x)) where C0, C1 > 0 and n0, n1 ∈ L1(Ω). Then the Lagrangian defined on W1,p

0 (Ω) × W−1,q(Ω) by

F(u, u∗) := inf{

• Ω

L

• x, ∇u(x), f(x)
• dx; f ∈ Lq(Ω; RN), −div(f) = u∗},

is selfdual.

Variational resolution of the main equation

Let β ∈ MΩ,p(RN) for some p > 1, then for every u∗ ∈ W−1,q(Ω) with 1

p + 1 q = 1, there exist ¯

u ∈ W1,p

0 (Ω) and ¯

f(x) ∈ Lq(Ω; RN) such that ¯ f ∈ β(x, ∇¯ u(x)) a.e. x ∈ Ω −div(¯ f) = u∗. (6) It is obtained by minimizing the functional I(u) := inf

f∈Lq(Ω;RN) −div(f)=u∗

• Ω
• L
• x, ∇u(x), f(x)
• − u(x), u∗(x)RN
• dx
• n W1,p(Ω), where L is a state-dependent selfdual Lagrangian
• n Ω × RN × RN associated to β in such a way that

¯ ∂L(x, ·) = β(x, ·) for a.e x ∈ Ω.

Variational formula for the homogenized maximal monotone vector field

Given now a maximal monotone family β in MΩ,p(RN) that is Q-periodic for an open non-degenerate parallelogram Q in Rn, its homogenization βhom can now be given by a variational formula in terms of a homogenized selfdual Lagrangian Lhom derived from the state-dependent selfdual Lagrangian Lβ associated to β.

Theorem

If β ∈ MΩ,p(RN) is Q-periodic and L is a state-dependent selfdual Lagrangian on Ω × RN × RN such that β(x, .) = ¯ ∂L(x, .). The homogenized maximal monotone vector field βhom is given by βhom = ¯ ∂Lhom where Lhom is the selfdual Lagrangian Lhom(ξ, η) = min

φ∈W1,p

# (Q)

g∈Lq

#(Q;RN)

1 |Q|

• Q

L

• x, ξ + ∇φ(x), η + g(x)
• dx.

Happy Birthday Schachermayer and Many Happy Returns

But do take some time off to improve your backgammon!!!!