ANALYTIC DEPENDENCE ON PARAMETERS FOR EVANS’ APPROXIMATED WEAK KAM SOLUTIONS

. We consider a variational principle for approximated Weak KAM solutions proposed by Evans. For Hamiltonians in quasi-integrable form h ( p )+ εf ( ϕ,p ), we prove that the map which takes the parameters ( ε,P,(cid:37) ) to Evans’ approximated solution u ε,P,(cid:37) is real analytic. In the mechanical case, we compute a recursive system of periodic partial diﬀerential equations identifying univocally the coeﬃcients for the power series of the perturbative parameter ε .


1.
Introduction. In the classical integrability theory of Hamiltonian systems, a central role is played by the Hamilton-Jacobi method. The basic idea is to integrate the Hamilton's ODE by a change of variables (x, p) → (X, P ) implicitly defined by a generating function v(x, P ). That is In particular, one looks for a function v(x, P ) and for an integrable Hamiltonian H(P ) which solve the so-called Hamilton-Jacobi equation 4626 OLGA BERNARDI AND MATTEO DALLA RIVA does, the new variables (X, P ) are not globally defined. However, most mechanical systems are quasi-integrable. That is where (ϕ, p) ∈ T d × R d are the angle-action variables, ε is a small real parameter and d ∈ N, d ≥ 1, is the fixed dimension of the ambient space. For quasi-integrable Hamiltonians, the classical perturbation approach consists in finding a canonical transformation which pushes the perturbation to the order ε 2 and then iterating the procedure. Since for ε = 0 the Hamiltonian (3) is integrable, we look for a generating function in the form v(ϕ, P ) = P · ϕ + εu(ϕ, P ) + O(ε 2 ) and possibly expand v(ϕ, P ) in a power series of ε. We note here that the εdependence of the generating function v(x, P ) is crucial also for numerical investigations, e.g. in Celestial and Quantum Mechanics. We also observe that in this context one has to deal with the resonances related to the so-called small divisors. The main strategies to handle such a problem are based on KAM and Nekhoroshev theorems (cf. [16,2,23,26]) and on Newton-Nash-Moser implicit function theorem (cf. [24,13]).
The application of such deep results leads to new intriguing questions concerning, for example, the generalization of the KAM Theory to a wider class of Hamiltonians which are not necessarily almost-integrable. The most important outcomes in this direction have been obtained by the Weak KAM Theory introduced by Mather, Mané and Fathi (see, e.g., [22,21,10]) which exploits variational and PDE's methods to treat Tonelli Hamiltonians. In particular, by the Weak KAM Theorem one can prove that, for any P in R d (and then with no non-resonance conditions) the Hamilton-Jacobi equation (2) admits global Lipschitz continuous solutions. The corresponding HamiltonianH(P ) is given bȳ and is called "effective Hamiltonian". However, since Weak KAM solutions are in general not differentiable, they cannot be used as generating functions in order to conjugate the original flow to an integrable one. In order to bypass this lack of regularity, in [7,8] Evans introduced a sort of approximated integrability for Tonelli Hamiltonians. The main result of his approach is a sequence of smooth functions uniformly converging to a Weak KAM solution and defining, for any P ∈ R d , a dynamics on T d . The properties of this torus dynamics and its relations with the original Hamiltonian flow have been discussed in [8] and in [3]. More recently, Evans returned to this subject in [9].
In the present paper, we propose a functional analytic approach to investigate the variational approximated version of Weak KAM Theory introduced by Evans. For Hamiltonians in the quasi-integrable form (3), we analyze the dependence on parameters of the sequence of Evans' approximated smooth solutions. In particular, we prove that the map which takes the perturbative parameter ε to the approximated solution is real analytic in a neighborhood of 0 (see Theorem 1 here below). As a consequence, it can be written in terms of a converging power series of ε for ε close to 0. Moreover, for mechanical Hamiltonians, we compute a recursive system of periodic partial differential equations which identifies univocally the coefficients of the power series of the parameter ε (see Section 4). We underline two possible applications of this regularity result. First, the converging power series of ε can be used in order to investigate the asymptotic behavior of the parameters involved in Evans' construction. Moreover, this series can be useful for a numerical treatment of the above sequence of smooth functions uniformly converging to a Weak KAM solution.
2. Analytical setting and main result. We start by recalling the main lines of the approach to Weak KAM Theory proposed by Evans in [7,8]. Instead of looking for minimizers u for the sup norm as suggested by formula (4), Evans considers a positive real number and looks for minimizers u of the functional Then, for all (P, ) ∈ R d × R + the corresponding Euler-Lagrange equation is div ϕ e H(ϕ,P +∂ϕu) ∂H ∂p (ϕ, P + ∂ ϕ u) = 0.
In detail: where u ij = ∂ 2 u ∂ϕi∂ϕj . Under suitable convexity hypotheses on H -see (c1), (c2) and (c3) below-and by using standard variational techniques, Evans proves the existence of minimizers u for (5) for all (P, ) ∈ R d × R + . He also shows that such minimizers are smooth and unique up to an additive constant. (So that there exists a unique minimizer with zero integral mean, i.e. such that T d udϕ = 0.) It is worth noting that the variational problem given by (5) arises in certain mean-field games. For an exhaustive discussion of these structures, we refer to [20], [11] and also to [12] for recent extensions for elliptic problems.
In the present paper we focus our attention on smooth real valued Hamiltonians H defined on the covering space where the functions h and f satisfy the following conditions: for each p, ξ ∈ R d ; (c3) (growth bounds) There exists a constant C > 0 such that We suppose that f (ϕ, p) is a jointly real analytic function of (ϕ, p) ∈ T d × R and that h is real analytic.
As proved by Evans [7, Thm. 5.2], conditions (c1) -(c3) imply the existence of a unique solution of equation (6) with zero integral mean. We shall denote such a solution by u ε,P,ρ . Then we ask the following question: what can be said on the function which takes (ε, P, ρ) to u ε,P,ρ ?
In particular, what about the ε-dependence?
In our main Theorem 1 we prove that under conditions (c1) -(c4) the map which takes (ε, P, ρ) to u ε,P,ρ is real analytic. However, one may wish to relax the regularity condition in (c4) and -for example-ask a differentiability condition on f and h instead of the real analyticity prescribed in (c4) (cf. Proposition 3 below). As one can expect, a weaker regularity assumption on f and h leads to a lower regularity of the function which takes (ε, P, ρ) to the solution u ε,P,ρ (cf. Thm. 6 below).
The proof of Theorem 1 utilizes a functional analytic approach. We identify u ε,P, as the implicit solution of a functional equationM (ε, P, , u) = 0, whereM is a (non-linear) operator acting between suitable Banach spaces (see (13) and (14) below). Then we study the dependence of u ε,P, upon (ε, P, ) by means of the Implicit Function Theorem for real analytic maps (cf., e.g., Deimling, Ch. 4 in [6]). We observe that methods based on the Implicit Function Theorem have been largely exploited for the study of nonlinear perturbation problems. We refer for example to the works of Stoppelli and Valent in nonlinear elasticity (see, e.g., [27,28,29]) and to the approach of Henry for the analysis of (regular) perturbations of the domain in boundary value problems (cf. [14]). We also mention the papers written by the second named author together with Lanza de Cristoforis and Musolino where a method based on the Implicit Function Theorem is applied to the study of singular perturbations of the domain in linear and nonlinear boundary value problems (see, for example, [5,17]).
In the present paper we will need to set the problem in the frame of Banach spaces of periodic functions with the following two properties: they have to be appropriate for the application of the standard elliptic regularity theory and, in addition, they have to be closed under the product of functions. A suitable choice is that of periodic Schauder spaces. Here below, we first introduce such spaces and then we state the main result of the paper.
For any m ∈ N and β ∈ [0, 1[, we denote by C m,β (T d ) the space of periodic functions from R d to R which have continuous partial derivatives up to the order m and β-Hölder continuous derivatives of order m. As is well known, C m,β (T d ) is a Banach space. In addition, we denote by C m,β z (T d ) the closed subspace of C m,β (T d ) consisting of the functions with zero mean, T d u dϕ = 0. For the sake of brevity we write C m (T d ) instead of C m,0 (T d ). Then, we fix once for all α ∈]0, 1[ and we have the following Theorem 1 which is an immediate consequence of Theorem 6 below. εf (ϕ, p) , where the functions h and f satisfy conditions (c1) -(c4). For any (P, ) ∈ R d ×R + , there exists ε 0 > 0 such that the map from ] − ε 0 , ε 0 [→ C 2,β z (T d ) which takes ε to the unique solution u ε,P, of equation (6) is real analytic.
We observe that by Theorem 6 one may also deduce that the map from ] − ε 0 , ε 0 [×R d × R + to C 2,β z (T d ) which takes a triple (ε, P, ) to u ε,P, is real analytic. As an immediate consequence of Theorem 1, there exists 0 < ε 1 ≤ ε 0 and a sequence {v k,P,ρ } k∈N in C 2,α z (T d ) such that where the series converges absolutely and uniformly in C 2,α z (T d ). In Section 4 we consider the mechanical case H(ϕ, p) = |p| 2 /2 + εf (ϕ) and we compute a recursive system of periodic partial differential equations which identify univocally the coefficients {v k,P,ρ } k∈N . Finally, we observe that for a numerical use of such a system, one may be interested in asymptotic approximations of u ε,P,k rather than having the complete series expansion. Under the hypothesis of Theorem 1 one can prove that for all N ∈ N. However, asymptotic approximations of such a form do not require the real analyticity of the functions f and h and can be deduced under weaker regularity assumptions (cf. Theorem 6 below).
Proof. (i) It is easily verified that L P, u ∈ C 0,α (T d ), so it remains to show that Then L P, u = div(A P, ∇u) for all u ∈ C 2,α (T d ). Thus T d L P, u dϕ = 0 by the periodicity of A P, ∇u and equality (9). (ii) Since L P, is continuous from C 2,α z (T d ) to C 0,α z (T d ) it suffices to show that it is one-to-one and onto in order to derive that it is an isomorphism by the open mapping theorem. If L P, u = 0 then a standard energy argument shows that T d ∇u · A∇u dϕ = 0. Accordingly ∇u · A∇u = 0 on T d and thus ∇u = 0 by the ellipticity of L P, . Thus u is constant and then u = 0 because T d u dϕ = 0 by the membership of u in C 2,α z (T d ). Now we have to prove that L P, is onto. Let v ∈ C 0,α z (T d ). Then we denote by N P, (v) the periodic newtonian potential defined by where S L P, ,T d denotes the periodic analog of a fundamental solution of L P, introduced in Appendix A. Then by a classical argument based on Fubini Theorem and the periodicity of S L P, ,T d one verifies that Thus, by Proposition 8 in Appendix A we have N P, (v) ∈ C 2,α z (T d ) and L P, N P, We proceed by studying the (nonlinear) operator M from R×R d ×R + ×C 2,α z (T d ) to C 0,α (T d ) which takes (ε, P, , u) to the function defined by the left hand side of (7). So that (7) is equivalent to M (ε, P, , u) = 0. In order to investigate the mapping properties of M and establish the correct regularity assumptions on the functions f and h, we exploit the following notation for the composition operators.
If F is a continuous function from T d × R d to R, then we denote by T F the (nonlinear nonautonomous) composition operator from (C(T d )) d to C(T d ) which takes a vector valued function v ≡ (v 1 , . . . , v d ) to the function T F (v) defined by Similarly, for a continuous function G from R d to R, we denote by T G the (nonlinear autonomous) composition operator from (C(T d )) d to C(T d ) which takes a vector valued function v ≡ (v 1 , . . . , v d ) to the function T G (v) defined by In the sequel we shall assume the following condition: The composition operators T f , T h , and T ∂ϕ j f , with j ∈ {1, . . . , d}, map functions of (C 1,α (T d )) d to functions of C 0,α (T d ).
In addition we shall assume either one of the following conditions (11) and (12).
Here q is fixed natural number in N \ {0}.
The maps T f and T h are of class C q+2 from (C 1,α (T d )) d to C 0,α (T d ) and the maps T ∂ϕ j f , with j ∈ {1, . . . , d}, are of class C q+1 from (C 1,α (T d )) d to The maps T f and T h are real analytic from (C 1,α (T d )) d to C 0,α (T d ) and the maps T ∂ϕ j f , with j ∈ {1, . . . , d}, are real analytic from (C 1,α (T d )) d to We observe that condition (11) implies that T ∂ 2 [17,Prop. 6.3]. Clearly condition (12) implies condition (11).
Finally, the next proposition gives some sufficient conditions for the validity of (10), (11), and (12). In the sequel we say that a function f belongs to C m (T d × R d ) if f belongs to C m (R d × R d ) and for every ξ ∈ R d fixed the map which takes x ∈ R d to f (x, ξ) is periodic. Similarly, we say that f is jointly real analytic from T d × R d to R if it is jointly real analytic from R d × R d to R and for every ξ ∈ R d fixed the map which takes x ∈ R d to f (x, ξ) is periodic. Proposition 3. The following statements hold.
We now take the limit as ε → 0 in equality (16) and apply a standard induction argument on h, verifing that equation lim ε→0 ∂ h ε (M (ε, P, k, u ε,P, )) = 0 is equivalent to lim ε→0 ∂ h ε (M (ε, P, k, u ε,P, )) = 0 for all h ∈ N, h ≥ 1. Then, by a straightforward calculation, we obtain that the equations for v 0,P,k , v 1,P,k , and v 2,P,k are as follows: v 0,P,k = 0 , L P, v 1,P,k = −P · ∂ ϕ g , while the (recursive) equations for the v h,P,k 's with h ≥ 3 are delivered by Appendix A. Appendix. For fixed (P, ) ∈ R d × R + , we consider the partial differential operator on R d defined by and the polynomial function (so that L p, = Ξ p, (∂ x1 , . . . , ∂ x d )). As is well known, there exists a periodic tempered distribution S P, ,T d on R d such that where δ z denotes the Dirac measure with mass in z (cf. e.g. [1, page 53] and [18]). The distribution S P, ,T d is determined up to an additive constant, and we can take in the sense of distributions in R d (cf. e.g., [18,Thm. 3.1]). In addition, we have the following result (for a proof we refer to [18,Thm. 3.5]).