New Identities for Weak KAM Theory

Transcription

1 New Identities for Weak KAM Theory Lawrence C. Evans Department of Mathematics University of California, Berkeley Abstract This paper records for the Hamiltonian H = p + W (x) some old and new identities relevant for the PDE/variational approach to weak KAM theory. For Haim Brezis, in continuing admiration. Introduction. Weak KAM for a model Hamiltonian. This is a follow-up to two of my earlier papers [E], [E] that propose a PDE/variational approach to weak KAM theory, originating with Mather and Fathi ([M], [M] [F], [F], etc.) In this paper we specialize to the classical Hamiltonian (.) H(p, x) = p + W (x), where the potential W is smooth and -periodic, where = [0, ] n denotes the unit cube with opposite faces identified. Given a vector P = (P,..., P n ) R n, the corresponding cell PDE reads (.) P + Dv + W = H(P ) in, where H : R n R is the effective Hamiltonian corresponding to H, as introduced in the important, but unpublished, paper of Lions Papanicolaou Varadhan [L-P-V]. Here v = v(p, x) denotes a -periodic viscosity solution. As shown for instance in [E] there exists also a Radon probability measure σ on solving the transport PDE (.3) div(σ(p + Dv)) = 0 in Class of 96 Collegium Chair. Supported in part by NSF Grant DMS-3066 and the Miller Institute for Basic Research in Science

2 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 Hamiltonian H encodes such information. The PDE approach to weak KAM aims at extracting such information from the two coupled PDE (.), (.3). This paper extends previous work by discovering for the particular case of the Hamiltonian (.) 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. I thank the referees for careful reading and in particular for finding an error in one of my proofs, now removed.. Variational approximation. We consider for fixed > 0 the problem of minimizing the functional I [v] := dx, e H(P +Dv,x) among Lipschitz continuous functions v : R with mean zero: v dx = 0. We write D = D x to denote the gradient in the variables x. PDE and calculus of variations theory (see [E]) implies that this problem has a unique, smooth minimizer v = v (P, x), which satisfies therefore the Euler-Lagrange PDE (.4) div(e H(P +Dv,x) (P + Dv )) = 0 in. Standard regularity theory shows that v is a smooth function of x and also of the parameters and P = (P,..., P n ). It is convenient to change notation, writing H := H(P + Dv (, x) = P + Dv + W ) (.5) H (P ) := log e H T n dx σ := e H H (P ). THEOREM. (i) We have σ 0, (.6) σ dx =, (.7) div(σ (P + Dv )) = 0,

3 and (.8) (P + Dv ) DH + v = (P i + v x i )(P j + v x j )v x i x j + (P i + v x i )W xi + v = 0. (ii) Furthermore, (.9) H (P ) H(P ) (P R n ) for each > 0, and (.0) lim 0 H (P ) = H(P ) (P R n ). As above, D = D x means the first derivatives in x, and D = D x the second derivatives in x. Likewise, = x is the Laplacian in the x-variables. Proof. The term H (P ) is introduced to achieve the normalization (.6). The PDE (.7) and (.8) are the Euler-Lagrange equation (.4) rewritten in respective divergence and non-divergence forms. The assertion (.9) follows upon our using a solution of (.) in the variational principle, and the limit (.0) is demonstrated in [E]. Remark. As shown in [E], we have the uniform estimates max D xv, v C for a constant C independent of. Hence we can extract a subsequence such that v k v uniformly on, σ k σ weakly as measures. A main assertion of [E] is that v, σ solve the transport equation (.3) and the cell PDE (.) on the support of σ. In particular Dv makes sense σ-almost everywhere, even if σ has a singular part with respect to Lebesgue measure. See also Bernardi Cardin Guzzo [B-C-G], Gomes Sanchez Morgado [G-SM], Gomes Iturriaga Sanchez Morgado Yu [G-I-SM-Y], etc for more on this variational method. Identities and estimates The ideas is to extract useful information from the two forms (.7), (.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 [E], [E] and some are new.. Differentiations in x. We start by differentiating with respect to x k for k =,..., n: 3

4 THEOREM. We have the identities (.) DW σ dx = 0, and v σ dx = 0 (.) Tn ( D v + W )σ + Dσ σ dx = 0. Proof. In view of (.) and (.5), we have (.3) P + Dv + W = H (P ) + log σ. Differentiating in x k once, and then twice, we learn that (.4) (P i + v x i )v x i x k + W xk = σ x k σ and (.5) (P + Dv ) D( v ) + D v + W = div x ( Dσ Multiply (.4) by σ, integrate by parts and recall (.7) to derive the first identity in (.). The second follows upon our multiplying (.8) by σ and integrating. To get (.), multiply (.5) by σ and integrate. (.6) Remark. As Dσ = DH σ, we obtain from (.) the estimate ( D v + DH ) σ dx C, σ ). for a constant C independent of. Recall that we write DH = D x H. We next generalize Theorem.: THEOREM. For each smooth function φ : R R, we have the identity D v φ(σ )σ + ( v ) φ (σ )(σ ) dx T (.7) n + DH [φ(σ )σ + φ (σ )(σ ) ] dx = W φ(σ )σ dx. 4

5 Proof.. We multiply the Euler Lagrange equation (.7) by div(φ(σ )(P + Dv )) and integrate by parts over : 0 = (σ (P + vx i )) xi (φ(σ )(P + vx j )) xj dx = (σ (P + vx i )) xj (φ(σ )(P + vx j )) xi dx = (σx j (P + vx i ) + σ vx i x j )(φ (σ )σx i (P + vx j ) + φ(σ )vx i x j ) dx = D v φ(σ )σ + φ (σ ) Dσ (P + Dv ) + [φ(σ ) + σ φ (σ )]v x i x j (P + v x j )σ x i dx. Now Dσ (P + Dv ) = σ v, according to (.7), and furthermore vx j x i (P + vx j ) = Hx i W xi. We can therefore simplify, obtaining the identity 0 = D v φ(σ )σ + φ (σ )(σ ) ( v ) + [φ(σ ) + σ φ (σ )](DH DW ) Dσ dx. Since Dσ = DH σ, it follows that D v φ(σ )σ + ( v ) φ (σ )(σ ) dx + DH [φ(σ )σ + φ (σ )(σ ) ] dx T n = [φ(σ ) + σ φ (σ )]DW Dσ dx = W σ φ(σ ) dx.. Differentiations in P. We next differentiate with respect to the parameters P k, for k =,..., n. In the following expressions we write (.8) V := D H (P ). Hereafter D P means the gradient in P, and Dx,P means the mixed second derivatives in x and P. To minimize notational clutter, we can safely write D H = D H P and D H = DP H, since H = H (P ) does not depend upon x. THEOREM.3 These further identities hold: (.9) D H (P ) = V = (P + Dv )σ dx, Tn (.0) D H (P ) = (I + D xp v ) (I + D xp v )σ + D P σ D P σ dx. 5 σ

6 Remark. Formula (.0) means that for k, l =,..., n, H P k P l (P ) = Tn (δ ik + v xipk )(δ il + v xipl )σ + σ P k σp l dx. σ Therefore ξ D H(P )ξ = H Pk P l (P )ξ k ξ l 0 for all ξ = (ξ,..., ξ n ) and hence (.) P H (P ) is convex. Proof.. Differentiating (.3) in P k, and then in P l, we find (.) (P i + v x i )(δ ik + v x i P k ) = H P k + σ P k σ, (.3) (P i + v x i )v x i P k P l + (δ il + v x i P l )(δ ik + v x i P k ) = H P k P l + ( σ Pk σ. Since σ dx =, we have (.4) 0 = σp k dx = Tn σ P k σ σ dx. We now multiply (.) by σ and integrate, using (.6), (.7) and (.4) to derive (.9). In addition, (.4) implies ( σ Pk σ ) P l σ dx = Tn σp k σp l dx. σ So the identity (.0) follows, if we multiply (.3) by σ and integrate. Remark. It follows from (.0) that Tn (.5) tr(d H (P )) = I + D x,p v σ + D P σ dx σ = ( I + D x,p v + ) (P + Dv )(I + D x,p v ) V σ dx where tr means trace..3 Differentiations in ɛ. In the following, subscripts denote derivatives with respect to : 6 ) P l.

7 THEOREM.4 We have (.6) H (P ) = σ log σ H dx = H (P ) σ dx, T n and so (.7) In addition, H σ dx = H (P ) H (P ). (.8) Tn H (P ) = Dv σ + σ dx σ ( = Dv + (P + Dv ) Dv H H (P ) + H (P ) ) σ dx. Remark. The identity (.8) implies (.9) H (P ) is convex. Differentiating in x and then in P, we can likewise show that (.0) (P, ) H (P ) is jointly convex. Proof.. We differentiate (.3) twice in, to learn that (.) (P + Dv ) Dv = H (P ) + log σ + σ σ and then ( ) (.) (P + Dv ) Dv + Dv = H (P ) + σ σ σ +. σ Multiply (.) by σ and recall (.7), to derive (.6).. Next multiply (.) by σ. We observe that 0 = σ σ dx = T σ n σ dx; and thus ( ) σ Tn σ σ dx = dx. σ σ 7

8 This gives the first equality in (.8), and the second follows when we explicitly calculate σ..5 Estimates for Du Du. A key question is how well v and σ approximate as 0 particular solutions v, σ of the weak KAM PDE (.), (.3). Now let v be a viscosity solution of (.) and σ a corresponding weak solution of (.3). To allow for changes in P we also assume, for this subsection only, that v solves the variational problem for the vector P. Consequently we have P + Dv + W = H (P ) + log σ. THEOREM.5 These identities hold: (.3) (P +Dv ) (P +Dv) σ dx = H(P ) H (P )+ H (P )+D H (P ) (P P ) and (.4) (P + Dv ) (P + Dv) dσ = H dσ H(P ) + D H(P ) (P P ). Observe that right hand sides involve Taylor expansions of H and H. Thus if H approximates H sufficiently well for small and if P is close to P, then Dv Dv is small on the support of σ. and so (.5) Proof.. We have P + Dv + W = H (P ) + log σ, P + Dv + W = H(P ); ( P + Dv P + Dv ) = H (P ) H(P ) + log σ. Since a b = a b + a (a b), we calculate that ( P + Dv P + Dv ) σ dx = (P + Dv ) (P + Dv) σ dx + (P + Dv ) ((P + Dv ) (P + Dv))σ dx = (P + Dv ) (P + Dv) σ dx + (P P ) (P + Dv )σ dx = (P + Dv ) (P + Dv) σ dx + (P P ) D H (P ), 8

9 the second equality resulting from (.7). Consequently (.5) implies (P + Dv ) (P + Dv) σ dx = H(P ) H (P ) + (P P ) D H (P ) log σ σ dx. The identity (.3) follows, since H (P ) = log σ σ dx according to (.6).. To prove (.4), we next integrate (.5) with respect to the measure σ (recall (.3)): ( P + Dv P + Dv ) dσ = (P + Dv ) (P + Dv) dσ + (P + Dv) ((P + Dv) (P + Dv )) dσ = (P + Dv ) (P + Dv) dσ + (P P ) P + Dv dσ = (P + Dv ) (P + Dv) dσ + (P P ) D H(P ). Hence (.5) gives (P + Dv ) (P + Dv) dσ = H (P ) H(P ) + (P P ) D H(P ) + log σ dσ = H dσ H(P ) + (P P ) D H(P ). 3 Linearizations and adjoints 3. Linearizing the PDE. The linearization about v of the Euler-Lagrange equation (.8) is the operator (3.) L [w] := (P i + v x i )(P j + v x j )w xi x j ((P j + v x j )v x i x j + W xi )w xi w, defined for smooth, periodic functions w : R. 9

10 LEMMA 3. We have the alternative formulas (3.) L [w] = (P i + v x i )((P j + v x j )w xj ) xi H x i w xi w. and (3.3) L [w] = σ ( [(P i + v x i )(P j + v x j ) + δ ij ]σ w xj )x i. Proof.. Formula (3.) follows immediately from (3.).. Recall from (.7) that div((p + Dv )σ ) = 0. Consequently the expression on the right hand side of (3.3) equals (P i + v x i )((P j + v x j )w xj ) xi σ x i σ w x i w. But this the formula (3.) for L [w], since σ = e H H. The linearization L is useful, as it appears when we differentiate the nonlinear PDE (.8): THEOREM 3. These identities hold: (3.4) L [v ] = (P i + v x i )v x j v x i x j + P i W xi. (3.5) L [v x k ] = (P i + v x i )W xi x k (k =,..., n). (3.6) L [v P k ] = (P i + v x i )v x i x k + W xk (k =,..., n). (3.7) L [x k ] = (P i + v x i )v x i x k W xk (k =,..., n). (3.8) L [v ] = v. (3.9) L [H ] = DH D v W. 0

11 Proof.. According to (3.) and (.8) This is (3.4). L [v ] = (P i + v x i )(P j + v x j )v x i x j v ((P j + v x j )v x i x j + W xi )v x i = (P i + v x i )W xi ((P j + v x j )v x i x j + W xi )v x i = P i W xi (P j + v x j )v x i x j v x i.. We obtain (3.5) upon differentiating (.8) with respect to x k : the left hand side appears when the differentiation falls upon v and the right hand side when the differentiation falls upon the term involving the potential W. Similarly, (3.6) results from our differentiating (.8) with respect to P k, and (3.8) from our differentiating in. We directly compute from the definition (3.) that (3.9) is also valid. Remark. We observe from (3.6) and (3.7) that (3.0) L [x k + v P k ] = 0 (k =,..., n). But note also that x + D P v is not -periodic. We will return to this point in Section The adjoint operator. We introduce next the adjoint L of L with respect to the standard inner product in L ( ); so that (3.) L [f]g dx = fl [g] dx for all smooth, -periodic functions f and g. THEOREM 3.3 (i) We have ( wσ ) (3.) L [w] = ([(P i + v xi )(P j + v xj ) + δ ij ]σ (ii) Therefore (3.3) L [σ w] = σ L [w] and (3.4) L [σ ] = 0. x i ) x j.

12 Proof. The identity (3.) follows from (3.3) and an integration by parts. Remark. It follows from (3.3) (or (3.3)) that the operator L, acting on smooth functions, is symmetric with respect to the L inner product weighted by σ : (3.5) L [f]g σ dx = fl [g] σ dx for smooth, -periodic functions f, g. Perhaps the spectrum of L contains useful dynamical information in the limit More identities. We can employ the foregoing formulas to rewrite some of the expressions from Section : THEOREM 3.4 We have the identity (3.6) ( tr(d H (P )) = n + P + Dv V (P + Dv ) DxP v DxP v ) σ dx and consequently the estimate ( (3.7) (P + Dv ) DxP v + DxP v ) σ dx n + P + Dv V σ dx. Proof.. Owing to (3.6), we have L [ D P v ] = v P k ((H ) xk W xk ) (P + Dv ) D xp v D xp v. We multiply by σ and integrate, recalling from (3.4) that L [σ ] = 0: ( (P + Dv ) DxP v + DxP v )σ dx = vp k ((P i + vx i )vx i x k + W xk )σ dx = vp k ((P i + vx i )vx i x k + Hx k )σ dx T (3.8) n = vp k (P i + vx i )vx i x k σ dx + vp k σx k dx T n T n = vp k x i (P i + vx i )vx k σ dx vp k x k σ dx = vp k x i (P i + vx i )(P k + vx k Vk )σ dx vp k x k σ dx,

13 the last equality following from (.7).. In view of (.5), ( tr(d H (P )) = I + D x,p v + (P + Dv )Dx,P v + P + Dv V ) σ dx + (P + Dv )Dx,P v (P + Dv V )σ dx. We use (3.8) to see that the second term equals ( (P + Dv ) DxP v + DxP v )σ dx vp k x k σ dx. Therefore ( tr(d H (P )) = I + D x,p v + (P + Dv )Dx,P v + P + Dv V ) σ dx ( (P + Dv ) DxP v + DxP v )σ dx vp k x k σ dx ( = P + Dv V (P + Dv )Dx,P v + I Dx,P v ) ) σ dx. The formula (3.6) follows; as does the inequality (3.7), since tr(d H (P )) 0. THEOREM 3.5 We have ( (3.9) H H (P ) = H (P ) + H ) (P ) (P + Dv ) Dv Dv σ dx, and therefore (3.0) ( (P + Dv ) Dv + Dv )σ H dx (P ) + H (P ) σ dx. T n Proof.. According to (3.8), we have L [ (v ) ] = v v (P + Dv ) Dv Dv. We multiply by σ and integrate: (3.) ( (P + Dv ) Dv + Dv )σ dx = v v σ dx. 3

14 The formula (.8) implies H (P ) = ( Dv + (P + Dv ) Dv + H H (P ) + H ) (P ) σ dx (P + Dv ) Dv H H (P ) + H (P ) σ dx. Recalling yet again (.7), we observe that the second integral term equals Tn v (P + Dv ) DH σ dx = v v σ dx, the last equality following from (.8). We substitute (3.) and rewrite, obtaining (3.9). 4 Some applications We collect in the concluding section some applications of the foregoing formulas, of which those in Section 4. concerning nonresonance are the most interesting. 4. H as 0. An overall goal is understanding how H and its approximations H for small > 0 provide analytic control of v, σ, and thus in the limit of v, σ. As an illustration, we show next that if H (P ) is nice enough as a function of near zero, then we can construct a limit measure σ that is absolutely continuous with respect to Lebesgue measure: THEOREM 4. If (4.) then ( ) / H (P ) is integrable near = 0, (4.) σ σ in L ( ) and σ L ( ) solves (.3). Proof. If 0 < <, we have σ σ dx σ dxd (Tn σ σ 4 ) ( ) dx σ dx d Tn ( ) H d,

15 according to (.8). Consequently, (4.) implies that {σ } >0 is a Cauchy sequence in L ( ) as Nonresonance phenomena. We assume hereafter that we can select P so that (4.3) V := D H (P ) does not depend upon. Write V = (V,..., V n ). We suppose also the nonresonance condition that for some constant c > 0 (4.4) V k c k γ for all vectors k Z n, k 0. Next, take g : R to be smooth and have zero mean: g(x) dx = 0. Then using a standard Fourier series representation and the nonresonance condition (4.4), we have LEMMA 4. There exists a smooth -periodic solution f = f(x) of the linear elliptic PDE (4.5) V k V l f Xk X l = g in. Furthermore, we have for each s 0 the estimate (4.6) f H s ( ) C s g H s+γ ( ), for a constant C s. THEOREM 4.3 Assume that (4.7) tr(d H (P )) = o() as 0. Then for each smooth function g : R we have (4.8) lim g(x + D P v )σ dx = g(x) dx. 0 5

16 Remark. This is a variant of a theorem in [E]. The formal interpretation is that under the symplectic change of variable (x, p) (X, P ), defined implicitly by the formulas p = P + D x v, X = x + D P v the dynamics become linear: X(t) = X 0 + tv for t 0. Since V k 0 for all k Z n {0}, the flow is therefore asymptotically equidistributed with respect Lebesgue measure. The rigorous assertion (4.8) is consistent with this picture. Proof.. Subtracting a constant if necessary, we may assume the average of g is zero. Now let f solve the linear PDE (4.5), and define f(x) := f(x + D P v ). The function f is -periodic, although x + D P v is not. Recalling from (3.0) that L [x + D P v ] = 0, we compute L [ f] = D X f L [x + D P v ] [(P i + v x i )(P j + v x j ) + δ ij ](δ ik + v x i P k )(δ jl + v x j P l )f Xk X l = [(P i + v x i )(P j + v x j ) + δ ij ](δ ik + v x i P k )(δ jl + v x j P l )f Xk X l. Here and afterwards f is evaluated at x + D P v. It follows that (4.9) L [ f] + g(x + D P v ) = E + E, for E := [(P i + v x i )(δ ik + v x i P k )(P j + v x j ))(δ jl + v x j P l ) V k V l ]f Xk X l, E := (δ ik + vx i P k )(δ il + vx i P l )f Xk X l. Selecting s large enough, we deduce from (4.6) that f C, is bounded. Consequently, (.5) implies the estimate E σ dx C ( + I + Dx,P v ) ) (P + Dv )(I + Dx,P v ) V σ dx C( tr(d H (P ))) = o(). Likewise, E σ dx C tr(d H (P )) = o(). 6

17 3. It follows now from (4.9) and (3.4) that g(x + D P v )σ dx = ( L [ f] E E )σ dx T n = fl [σ ] dx + o() = o() as 0. References [B-C-G] [E] [E] [E-G] [F] [F] O. Bernardi, F. Cardin and M. Guzzo, New estimates for Evans variational approach to weak KAM theory, Comm in Contemporary Math 5 (03). L. C. Evans, Some new PDE methods for weak KAM theory, Calculus of Variations and Partial Differential Equations, 7 (003), L. C. Evans, Further PDE methods for weak KAM theory, Calculus of Variations and Partial Differential Equations 35 (009), L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics I, Archive Rational Mech and Analysis 57 (00), 33. A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sr. I Math. 34 (997), A. Fathi, Weak KAM Theorem in Lagrangian Dynamics (Cambridge Studies in Advanced Mathematics), to be published. [G-I-SM-Y] D. Gomes, R. Iturriaga, H Sanchez-Morgado, Y. Yu, Mather measures selected by an approximation scheme, Proc. Amer. Math. Soc. 38 (00), [G-SM] [L-P-V] D. Gomes and H. Sanchez Morgado, A stochastic Evans-Aronsson problem, Trans. Amer. Math. Soc. 366 (04), P.-L. Lions, G. Papanicolaou, and S. R. S. Varadhan, Homogenization of Hamilton Jacobi equations, unpublished, circa 988. [M] J. Mather, Minimal measures, Comment. Math Helvetici 64 (989), [M] J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Zeitschrift 07 (99)

18 [Y] Y. Yu L variational problems and weak KAM theory, Comm. Pure Appl. Math. 60 (007) 47. 8