Ambrosio L., Caffarelli L., Crandall M.G., Evans L.C., Fusco N. Calculus of Variations and Nonlinear Partial Differential Equations

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130 L.C. Evans

and assume

(H1)

⎧

⎪

⎨

⎪

⎩

there exists p

∗

∈ R

such that

∗

= DH

∗

)

is type (L, γ)forsomeL, γ > 0;

(H2) D

∗

) is invertible;

(H3) H

are real analytic.

Theorem 1.4 (KAM). Under hypotheses (H1)–(H3), there exists ε

> 0

such that for each 0 <ε≤ ε

, there exists P

∗

(close to p

∗

)andasmooth

mapping

Φ(P

∗

, ·):R

→ R

× R

such that for each x

∈ R

Φ(P

∗

+ tω

∗

)=:(x(t), p(t))

solves the Hamiltonian dynamics (H).

The idea of the proof is for k =0, 1,..., iteratively to construct Φ

,ε

so that P

= p

∗

= D

) for all k;

k+1

:= H

◦ Φ

;

and

(p, x)=

(p)+K

(p, x),

where K

≤ε

and the error estimates ε

converge to 0 very rapidly. Then

→ P

∗

) →

H(P

∗

);

and

Φ := lim

k→∞

◦ Φ

k−1

◦···◦Φ

is the required change of variables.

The full details of this procedure are very complicated. See Wayne’s dis-

cussion [W] for much more explanation and for references to the vast literature

on KAM.

Remark. We have



X(t)=x

+ tω

∗

P(t) ≡ P

∗

(1.18)

for

∗

= D

H(P

∗

)

But note that, in spite of my notation, Φ is really only deﬁned for P = P

∗

Weak KAM Theory and Partial Diﬀerential Equations 131

2 Weak KAM Theory: Lagrangian Methods

Our goal in this and the subsequent sections is extending the foregoing clas-

sical picture into the large. The resulting, so-called “weak KAM theory” is

a global and nonperturbative theory (but is in truth pretty weak, at least

as compared with the assertions from the previous section). There are two

approaches to these issues: the Lagrangian, dynamical systems methods (dis-

cussed in this section) and the nonlinear PDE methods (explained in the next

section).

The following discussion follows Albert Fathi’s new book [F5], which the

interested reader should consult for full details of the proofs. Related exposi-

tions are Forni–Mather [Fo-M] and Ma˜n´e[Mn].

2.1 Minimizing Trajectories

Notation. If x :[0,T] → R

and

[x(·)] ≤ A

[y(·)]

for all y(0) = x(0), y(T )=x(t), we call x(·)aminimizer of the action A

[·]

on the time interval [0,T].

Theorem 2.1 (Velocity Estimate). For each T>0, there exists a constant

such that

max

0≤t≤T

x(t)|≤C

(2.1)

for each minimizer x(·) on [0,T].

Idea of Proof. This is a fairly standard derivative estimate for solutions of

the Euler–Lagrange equation (E-L). !"

2.2 Lax–Oleinik Semigroup

Deﬁnition. Let v ∈ C(T

) and set

−

v(x) := inf{v(x(0)) +



x, x) ds | x(t)=x}

We call the family of nonlinear operators {T

−

}

t≥0

the Lax–Oleinik semigroup.

Remark. The inﬁmum above is attained. So in fact there exists a curve x(·)

such that x(t)=x and

−

v(x)=v(x(0)) +



x, x) dt. !"

132 L.C. Evans

Theorem 2.2 (Regularization). For each time t>0 there exists a constant

such that

−

Lip

≤ C

(2.2)

for all v ∈ C(T

Idea of Proof. Exploit the strict convexity of v → L(v, x), as in similar

arguments in Chapter 3 on my PDE text [E1]. !"

We record below some properties of the Lax–Oleinik semigroup acting on

the space C(T

), with max norm .

Theorem 2.3 (Properties of Lax–Oleinik Semigroup).

(i) T

−

◦ T

−

= T

−

t+s

(semigroup property).

(ii) v ≤ ˆv implies T

−

v ≤ T

−

ˆv.

(iii) T

−

(v + c)=T

−

v + c.

(iv) T

−

v −T

−

ˆv≤v − ˆv (nonexpansiveness).

(v) For al l v ∈ C(T

), lim

t→0

−

v = v uniformly.

(vi) For al l v, t → T

−

v is uniformly continuous.

2.3 The Weak KAM Theorem

We begin with an abstract theorem about nonlinear mappings on a Banach

space X:

Theorem 2.4 (Common Fixed Points). Suppose {φ

}

t≥0

is a semigroup

of nonexpansive mappings of X into itself. Assume also for all t>0 that

(X) is precompact in X and for all x ∈ X that t → φ

(x) is continuous.

Then there exists a point x

∗

such that

∗

)=x

∗

for all times t ≥ 0.

Idea of Proof. Fix a time t

> 0. We will ﬁrst show that φ

has a ﬁxed

point.

Let 0 <λ<1. Then there exists according to the Contraction Mapping

Theorem a unique element x

satisfying

λφ

)=x

By hypothesis {φ

) | 0 <λ<1} is precompact. So there exist λ

→ 1for

which x

→ x and

(x)=x.

Weak KAM Theory and Partial Diﬀerential Equations 133

Now let x

be ﬁxed point of φ

1/2

.Thenx

is ﬁxed point also of φ

k/2

for

k =1, 2,.... Using compactness, we show that x

→ x

∗

and that φ

∗

)=x

∗

for all t ≥ 0. !"

Next is an important result of Albert Fathi [F1]:

Theorem 2.5 (Weak KAM Theorem). There exists a function v

−

∈

C(T

) and a constant c ∈ R such that

−

+ ct = v

−

for all t ≥ 0. (2.3)

Idea of proof. We apply Theorem 2.4 above. For this, let us write for func-

tions v, ˆv ∈ C(T

)

v ∼ ˆv

if v − ˆv ≡ constant; and deﬁne also the equivalence class

[v]:={ˆv | v ∼ ˆv}.

Set

X := {[v] | v ∈ C(T

)}

with the norm

[v] := min

a∈R

v + a11,

where 11 denotes is the constant function identically equal to 1.

We have T

: X → X, according to property (iii) of Theorem 2.3. Hence

there exists a common ﬁxed point

[v]

∗

=[v]

∗

(t ≥ 0).

Selecting any representative v

−

∈ [v]

∗

, we see that

−

= v

−

+ c(t)

for some c(t). The semigroup property implies c(t + s)=c(t)+c(s), and

consequently c(t)=ct for some constant c. !"

2.4 Domination

Notation. Let w ∈ C(T

). We write

w ≺ L + c (“w is dominated by L + c”)

w(x(b)) −w(x(a)) ≤



x, x)ds + c(b −a) (2.4)

for all times a<band for all Lipschitz continuous curves x :[a, b] → R

134 L.C. Evans

Remark.

w ≺ L + c if and only if w ≤ ct + T

−

w for all t ≥ 0.

Theorem 2.6 (Domination and PDE).

(i) If w ≺ L + c and the gradient Dw(x) exists at a point x ∈ T

, then

H(Dw(x),x) ≤ c. (2.5)

(ii) Conversely, if w is Lipschitz continuous and H(Dw, x) ≤ c a.e., then

w ≺ L + c.

Idea of Proof. (i) Select any curve x with x(0) = x,

x(0) = v.Then

w(x(t)) −w(x)

≤



x, x) ds + c.

Let t → 0, to discover

v ·Dw ≤ L(v, x)+c;

and therefore

H(Dw(x),x)=max

(v ·Dw − L(v, x)) ≤ c.

(ii) If w is smooth, we can compute

w(x(b)) −w(x(a)) =



w(x(t)) ds



Dw(x) ·

x dt

≤



x, x)+H(Dw(x), x) dt

≤



x, x) dt + c(b −a).

See Fathi’s book [F5] for what to do when w is only Lipschitz. !"

Weak KAM Theory and Partial Diﬀerential Equations 135

2.5 Flow invariance, characterization of the constant c

Notation. (i) We will hereafter write

T (T

):=R

× T

= {(v, x) | v ∈ R

,x∈ T

}

for the tangent bundle over T

,and

∗

):=R

× T

= {(p, x) | p ∈ R

,x∈ T

}

for the cotangent bundle. We work in the tangent bundle for the Lagrangian

viewpoint, and in the cotangent bundle for the Hamiltonian viewpoint.

(ii) Consider this initial-value problem for the Euler-Lagrange equation:



−

x, x)) + D

x, x)=0

x(0) = x,

x(0) = v.

(2.6)

We deﬁne the ﬂow map {φ

}

t∈R

on T (T

) by the formula

(v, x):=(v(t), x(t)), (2.7)

where v(t)=

x(t).

Deﬁnition. A probability measure µ on the tangent bundle T (T

)isﬂow

invariant if



T (T

)

Φ(φ

(v, x)) dµ =



T (T

)

Φ(v, x) dµ

for each bounded continuous function Φ.

Following is an elegant interpretation of the constant c from the Weak

KAM Theorem 2.5, in terms of action minimizing ﬂow invariant measures:

Theorem 2.7 (Characterization of c). The constant from Theorem 2.5 is

given by the formula

c = −inf





T (T

)

L(v, x) dµ | µ ﬂow invariant, probability measure

. (2.8)

Deﬁnitions. (i) The action of a ﬂow-invariant measure µ is

A[µ]:=



T (T

)

L(v, x) dµ.

(ii) We call µ a minimizing (or Mather) measure if

−c = A[µ]. (2.9)

136 L.C. Evans

Idea of Proof. 1. Recall that

−

= v

−

+ ct and v

−

≺ L + c.

Let us write

(v, x)=(

x(s), x(s))

for (v, x) ∈ T (T

), and put

π(v, x):=x

for the projection of T (T

)ontoT

Then

−

(πφ

(v, x)) − v

−

(π(v, x)) ≤



L(φ

(v, x)) ds + c. (2.10)

Integrate with respect to a ﬂow-invariant probability measure µ:



T (T

)

−

(πφ

(v, x)) − v

−

(π(v, x)) dµ

≤



T (T

)

L(φ

(v, x)) dµds + c



T(T

)

L(v, x) dµ + c.

Therefore

−c ≤



T (T

)

L(v, x) dµ for all ﬂow-invariant µ.

2. We must now manufacture a measure giving equality above. Fix x ∈ T

Find a curve x :(−∞, 0] → R

such that x(0) = x,

x(0) = v; and for all

times t ≤ 0:

−

(x(0)) − v

−

(x(t)) =



L(φ

(v, x)) ds −ct.

Deﬁne for t ≥ 0 the measure µ

by the rule

(Φ)=



−t

Φ(φ

(v, x)) ds

for each continuous function Φ.Then

−

(x) − v

−

(x(−t))



T (T

)

Ldµ

+ c.

Weak KAM Theory and Partial Diﬀerential Equations 137

Sending t

→−∞, we deduce that

dµ

dµ

weakly as measures, for some ﬂow invariant probability measure µ that

satisﬁes

−c =



T (T

)

Ldµ.

2.6 Time-reversal, Mather set

It is sometimes convenient to redo our theory with time reversed, by intro-

ducing the backwards Lax–Oleinik semigroup:

Deﬁnition.

v(s):=sup



v(x(t)) −



x, x)ds | x(0) = x



As above, there exists a function v

∈ C(T

) such that

− ct = v

for all t ≥ 0

for the same constant c described in Theorem 2.7.

Deﬁnition. We deﬁne the Mather set.



spt(µ),

the union over all minimizing measures µ as above. The projected Mather set

:= π(

One goal of weak KAM theory is studying the structure of the Mather set (and

the related Aubry set), in terms of the underlying Hamiltonian dynamics. See

Fathi [F5] for much more.

3 Weak KAM Theory: Hamiltonian and PDE Methods

3.1 Hamilton–Jacobi PDE

In this section, we reinterpret the foregoing ideas in terms of the theory of

viscosity solutions of nonlinear PDE:

138 L.C. Evans

Theorem 3.1 (Viscosity Solutions).

(i) We have

H(Dv

,x)=c a.e.

(ii) In fact



H(Dv

−

,x)=c

−H(Dv

,x)=−c

(3.1)

in sense of viscosity solutions.

Idea of Proof. 1. There exists a minimizing curve x :(∞, 0] → R

such that

x(0) = x,

x(0) = v,and

−

(x)=v

−

(x(−t)) +



−t

x, x) ds + ct (t ≥ 0).

If v is diﬀerentiable at x, then we deduce as before that

v ·Dv

−

(x)=L(v, x)+c,

and this implies H(Dv

−

,x) ≥ c. But always H(Dv

−

,x) ≤ 0.

2. The assertions about v

being viscosity solutions follow as in Chapter

10 of my PDE book [E1]. !"

3.2 Adding P Dependence

Motivated by the discussion in Section 1 about the classical theory of canonical

transformation to action-angle variables, we next explicitly add dependence

on a vector P.

So select P ∈ R

and deﬁne the shifted Lagrangian

L(v, x):=L(v, x) − P · v. (3.2)

The corresponding Hamiltonian is

H(p, x)=max

(p · v −

L(v, x)) = max

((p + P ) · v − L(v, x)),

and so

H(p, x)=H(P + p, x). (3.3)

As above, we ﬁnd a constant c(P ) and periodic functions v

= v

(P, x)so

that



H(P + Dv

−

,x)=

H(Dv

−

,x)=c(P )

−H(P + Dv

,x)=−

H(Dv

,x)=−c(P )

in the viscosity sense.

Weak KAM Theory and Partial Diﬀerential Equations 139

Notation. We write

H(P ):=c(P ),u

−

:= P · x + v

−

:= P · x + v

Then



H(Du

−

,x)=

H(P )

−H(Du

,x)=−

H(P )

(3.4)

in the sense of viscosity solutions.

3.3 Lions–Papanicolaou–Varadhan Theory

A PDE construction of

We next explain an alternative, purely PDE technique, due to Lions -

Papanicolaou - Varadhan [L-P-V], for ﬁnding the constant c(P ) as above and

v = v

−

Theorem 3.2 (More on viscosity solutions). For each vector P ∈ R

there exists a unique real number c(P ) for which we can ﬁnd a viscosity solu-

tion of



H(P + D

v, x)=c(P )

v is T

-periodic.

(3.5)

Remark. In addition v is semiconcave, meaning that D

v ≤ CI for some

constant C. !"

Idea of Proof. 1. Existence. Routine viscosity solution methods assert the

existence of a unique solution v

εv

+ H(P + D

,x)=0.

Since H is periodic in x, uniqueness implies v

is also periodic.

It is now not very diﬃcult to derive the uniform estimates

max |D

|, |εv

|≤C.

Hence we may extract subsequences for which

→ v uniformly,ε

→−c(P ).

It is straightforward to conﬁrm that v is a viscosity solution of H(P +

v, x)=c(P ).

2. Uniqueness of c(P ). Suppose also

H(P + D

ˆv, x)=ˆc(P ).