Lawrence Craig EVANS

(For Haim Brezis,in continuing admiration)

1 Introduction

1.1 Weak KAM for a model Hamiltonian

Thisisa follow-up to two of my earlier papers[2–3]that proposea PDE/variational approach to weak KAM theory,originating with Mather and Fathi(see[5–6,10–11],etc.).In this paper,we specialize to the classical Hamiltonian

where the potential W is smooth and Tn-periodic,where Tn=[0,1]ndenotes the unit cube with opposite faces identified.Given a vector P=(P1,···,Pn) ∈ Rn,the corresponding cell PDE reads

whereis the effective Hamiltonian corresponding to H,as introduced in the important,but unpublished,paper of Lions-Papanicolaou-Varadhan[9].Here v=v(P,x)denotes a Tn-periodic viscosity solution.As shown for instance in[2],there exists also a Radon probability measureσon Tnsolving the transport PDE

in an appropriate weak sense.

A central goal of weak KAM theory is developing a nonperturbative methods to identify“integrable structures” within the otherwise possibly chaotic dynamics generated by a given Hamiltonian H=H(p,x),and in particular to understand if and how the effective Hamiltonianencodes such information.The PDE approach to weak KAM aims at extracting such information from the two coupled PDE(1.2)–(1.3).

This paper extends previous work by discovering for the particular case of the Hamiltonian(1.1)several new integral identities,especially for the variational approximations introduced below.We also record how some previously derived general formulas simplify in this case,and provide in Section 4 some applications.

1.2 Variational approximation

We consider for fixedε>0 the problem of minimizing the functional

Standard regularity theory shows that vεis a smooth function of x and also of the parameters ε and P=(P1,···,Pn).

It is convenient to change notation,writing

Theorem 1.1(i)We haveσε≥0,

and

(ii)Furthermore,

for eachε>0,and

As above,D=Dxmeans the first derivatives in x,andthe second derivatives in x.Likewise,∆=∆xmeans the Laplacian in the x-variables.

ProofThe termis introduced to achieve the normalization(1.6).The PDEs(1.7)and(1.8)are the Euler-Lagrange equation(1.4)rewritten in respective divergence and nondivergence forms respectively.

The assertion(1.9)follows upon our using a solution of(1.2)in the variational principle,and the limit(1.10)is demonstrated in[2].

Remark 1.1As shown in[2],we have the uniform estimates

for a constant C independent ofε.Hence we can extract a subsequence,such that

A main assertion of[2]is that v,σsolve the transport equation(1.3)and the cell PDE(1.2)on the support ofσ.In particular,Dv makes senseσ-almost everywhere,even ifσ has a singular part with respect to Lebesgue measure.

See also Bernardi-Cardin-Guzzo[1],Gomes-Sanchez Morgado[8],Gomes-Iturriaga-Sanchez Morgado-Yu[7],etc.for more on this variational method.

2 Identities and Estimates

The ideas are to extract useful information from the two forms(1.7)–(1.8)of the Euler-Lagrange PDE.This section records various relevant integral identities,mostly derived by differentiating with respect to different variables.Some of the resulting formulas are special cases of those in[2–3]and some are new.

2.1 Differentiations in x

We start by differentiating with respect to xkfor k=1,···,n.

Theorem 2.1We have the identities

and

ProofIn view of(1.1)and(1.5),we have

Differentiating in xkonce,and then twice,we learn that

and

Multiply(2.4)by σε,integrate by parts and recall(1.7)to derive the first identity in(2.1).The second follows upon our multiplying(1.8)byσεand integrating.To get(2.2),multiply(2.5)byσεand integrate.

Remark 2.1Aswe obtain from(2.2)the estimate

for a constant C independent ofε.Recall that we write DHε=DxHε.

We next generalize Theorem 2.1.

Theorem 2.2For each smooth functionϕ:R→R,we have the identity∫

Proof(1)We multiply the Euler-Lagrange equation(1.7)byand integrate by parts over Tnas follows:

Now,according to(1.7),and furthermore.We can therefore simplify,obtaining the identity

Sinceit follows that

2.2 Differentiations in P

We next differentiate with respect to the parameters Pk,for k=1,···,n.In the following expressions,we write

Hereafter DPmeans the gradient in P,andmeans the mixed second derivatives in x and P.To minimize notational clutter,we can safely writeandsincedoes not depend upon x.

Theorem 2.3These following further identities hold:

Remark 2.2Formula(2.10)means that for k,l=1,···,n,

Therefore,for allξ=(ξ1,···,ξn),and hence

Proof(1)Differentiating(2.3)in Pk,and then in Pl,we find

We now multiply(2.12)by σεand integrate,using(1.6)–(1.7)and(2.14)to derive(2.9).

In addition,(2.14)implies

So the identity(2.10)follows,if we multiply(2.13)byσεand integrate.

Remark 2.3It follows from(2.10)that

where “tr” means trace.

2.3 Differentiations inε

In the following,subscriptsεdenote derivatives with respect toε.

Theorem 2.4We have

and so

In addition,

Remark 2.4The identity(2.18)implies that

Differentiating in x and then in P,we can likewise show that

Proof(1)We differentiate(2.3)twice inε,to learn that

and then

Multiply(2.21)byσεand recall(1.7),to derive(2.16).

(2)Next multiply(2.22)by σε.We observe that

and thus

This gives the first equality in(2.18),and the second follows when we explicitly calculate

2.4 Estimates for D uε−D u

A key question is how well vεandσεapproximate asε→ 0 particular solutions v,σ of the weak KAM PDE(1.2)–(1.3).

Now let v be a viscosity solution of(1.2)andσa corresponding weak solution of(1.3).To allow for changes in P we also assume,for this subsection only,that vεsolves the variational problem for the vector Pε.Consequently,we have

Theorem 2.5These following identities hold:

and

Observe that right-hand sides involve Taylor expansions ofandThus ifapproximatessufficiently well for small ε and if Pεis close to P,thenis small on the support ofσε.

Proof(1)We have

and so

Since|a|2−|b|2=−|a−b|2+2a·(a−b),we calculate that

the second equality resulting from(1.7).Consequently,(2.25)implies

(2)To prove(2.24),we next integrate(2.25)with respect to the measureσ(recall(1.3)),

Hence,(2.25)gives

3 Linearizations and Adjoints

3.1 Linearizing the PDE

The linearization about vεof the Euler-Lagrange equation(1.8)is the operator

defined for smooth,periodic functions w:Tn→R.

Lemma 3.1We have the alternative formulas

and

Proof(1)Formula(3.2)follows immediately from(3.1).

(2)Recall from(1.7)that div((P+Dvε)σε)=0.Consequently,the expression on the right-hand side of(3.3)equals

The linearization Lεis useful,as it appears when we differentiate the nonlinear PDE(1.8).

Theorem 3.1These following identities hold:

Proof(1)According to(3.1)and(1.8),

This is(3.4).

(2)We obtain(3.5)upon differentiating(1.8)with respect to xk:The left-hand side appears when the differentiation falls upon vεand the right-hand side appears when the differentiation falls upon the term involving the potential W.

Similarly,(3.6)results from our differentiating(1.8)with respect to Pk,and(3.8)from our differentiating inε.We directly compute from the definition(3.1)that(3.9)is also valid.

Remark 3.1We observe from(3.6)–(3.7)that

But note also that x+DPvεis not Tn-periodic.We will return to this point in Subsection 4.2.

3.2 The ad joint op erator

We introducenext the adjointof Lεwith respect to the standard inner product in L2(Tn),so that

for all smooth,Tn-periodic functions f and g.

Theorem 3.2(i)We have

(ii)Therefore

and

ProofThe identity(3.12)follows from(3.3)and an integration by parts.

Remark 3.2It follows from(3.13)(or(3.3))that the operator Lε,acting on smooth functions,is symmetric with respect to the L2inner product weighted by σε:

for smooth,Tn-periodic functions f,g.Perhaps the spectrum of Lεcontains useful dynamical information in the limit

3.3 M ore id entities

We can employ the foregoing formulas to rewrite some of the expressions from Section 2.

Theorem 3.3We have the identity

and consequently the estimate

Proof(1)Owing to(3.6),we have

We multiply by σεand integrate,recalling from(3.14)that

The last equality follows from(1.7).

(2)In view of(2.15),

We use(3.18)to see that the second term equals

Therefore

The formula(3.16)follows,as does the inequality(3.17),since

Theorem 3.4We have

and therefore

Proof(1)According to(3.8),we have

We multiply byσεand integrate as follows:

The formula(2.18)implies

Recalling yet again(1.7),we observe that the second integral term equals

the last equality following from(1.8).We substitute(3.21)and rewrite,obtaining(3.19).

4 Some Applications

We collect in the concluding section some applications of the foregoing formulas,of which those in Subsection 4.2 concerning nonresonance are the most interesting.

4.1

An overall goal is understanding howand its approximationsfor smallε>0 provide analytic control of vε,σε,and thus in the limit of v,σ.

As an illustration,we show next that ifis nice enough as a function ofεnear zero,then we can construct a limit measureσthat is absolutely continuous with respect to Lebesgue measure.

Theorem 4.1If

then

andσ∈L1(Tn)solves(1.3).

ProofIf 0< ε1< ε2,we have

according to(2.18).Consequently,(4.1)implies that{σε}ε>0is a Cauchy sequence in L1(Tn)asε→0.

4.2 Nonresonance phenomena

We assume hereafter that we can select Pεso that

doesnot depend upon ε.Write V=(V1,···,Vn).We suppose also the nonresonance condition that for some constant c>0,

Next,take g:Tn→R to be smooth and have zero mean

Then using a standard Fourier series representation and the nonresonance condition(4.4),we have the following lemma.

Lemma 4.1There exists a smooth Tn-periodic solution f=f(X)of the linear elliptic PDE

Furthermore,we have for each s≥0 the estimate

for a constant Cs.

Theorem 4.2Assume that

Then for each smooth function g:,we have

Remark 4.1This is a variant of a theorem in[2].The formal interpretation is that under the symplectic change of variable

defined implicitly by the formulas p=P+Dxv,X=x+DPv the dynamics become linear:X(t)=X0+tV for t≥ 0.Since V ·k0 for all k ∈ Zn−{0},the flow is therefore asymptotically equidistributed with respect to Lebesgue measure.The rigorous assertion(4.8)is consistent with this picture.

Proof(1)Subtracting a constant if necessary,we may assume that the average of g is zero.Now let f solve the linear PDE(4.5),and define

The functionis Tn-periodic,although x+DPvεis not.

Recalling from(3.10)that,we compute

Here and afterwards f is evaluated at x+DPvε.It follows that

for

(2)Selecting s large enough,we deduce from(4.6)thatis bounded.Consequently,(2.15)implies the estimate

Likewise,

(3)It follows now from(4.9)and(3.14)that

asε→0.

AcknowledgementThe author would like to thank the referees for very careful reading.

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