A <span class="mathjax-formula">$\Gamma $</span>-convergence result for variational integrators of lagrangians with quadratic growth

October 2004, Vol. 10, 656–665 DOI: 10.1051/cocv:2004025

A Γ -CONVERGENCE RESULT FOR VARIATIONAL INTEGRATORS OF LAGRANGIANS WITH QUADRATIC GROWTH

Francesco Maggi

and Massimiliano Morini

Abstract. Following the Γ-convergence approach introduced by M¨uller and Ortiz, the convergence of discrete dynamics for Lagrangians with quadratic behavior is established.

Mathematics Subject Classification. 37M15, 49J45.

Received October 7, 2003.

1. Introduction

In a recent paper M¨uller and Ortiz [4] have introduced the study of convergence properties of discrete dynamics and variational integrators using the tool of Γ-convergence. The theory of discrete dynamics is a formulation of Lagrangian mechanics in which time is considered as a discrete variable. This point of view leads to studying discrete trajectories, thevariational integrators, generated by a discrete version of Hamilton’s principle in which the classical integral action is replaced by a sum, calleddiscrete action. The interest of this construction lies in the fact that, compared to the trajectories generated by more usual approximation schemes, variational integrators converge to classical solutions in a more stable way. We refer to Marsden and West [3]

for a complete account and for more references on the topic.

In order to state and comment on the result by M¨uller and Ortiz [4], we introduce some definitions.

We say that a functionf :R^N×R^N→R,f =f(s, ξ), is aLagrangian with quadratic behavior if

f ∈C²(R^N×R^N) with sup{|D²f(s, ξ)|: (s, ξ)∈R^N×R^N}<∞, (1) whereD²f is the matrix of all the second order partial derivatives off. Thus, for someC >0,

|f(s, ξ)| ≤C(1 +|s|²+|ξ|²), ∀(s, ξ)∈R^N×R^N. Theaction is the functionalF :H_loc¹ (R;R^N)× E →R, defined by

F(u, A) :=

Af(u,u)dt,˙ ∀(u, A)∈H_loc¹ (R;R^N)× E,

Keywords and phrases. Discrete dynamics, variational integrators, Gamma-convergence.

1 Dipartimento di Matematica “U. Dini”, Viale Morgagni 67/A 50134, Firenze, Italy.

2 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA, 15213 USA;

e-mail:morini@asdf4.math.cmu.edu

c EDP Sciences, SMAI 2004

whereE is the family of all open bounded intervals ofR. A functionuis said to bestationary for the action if, δF(u, ϕ, A) = 0, ∀A∈ E,∀ϕ∈C_c^∞(A;R^N),

where thefirst variation ofF(·, A) onuwith respect toϕis given by δF(u, ϕ, A) := lim

t→0

F(u+tϕ, A)−F(u, A)

t ·

For every h > 0 we fix the gridTh ={hi}i∈Z ofR of step hand define the setX_h of allTh-piecewise affine continuous functions with values inR^N. A functionu_h∈X_h is said to beh-stationary forF if

δI(u_h, ϕ, A) = 0, ∀A∈ E,∀ϕ∈X_h, with sptϕ⊂A.

The main theorem in M¨uller-Ortiz [4] is the following:

Theorem 1. Assume thatf(s, ξ) =|ξ|²/2−V(s), where V is of classC² and|∇²V|< C on R^N. Let{u_h}_h be a given sequence of h-stationary functions such that the Fourier transforms of theu_h’s are inL¹ and form a compact sequence of Radon measures in the duality with C_b(R;R^N), the space of continuous and bounded functions onR.

Then there existsu∈W^2,∞(R;R^N)such that a) u_h u^∗ inW^1,∞(R;R^N);

b) uis stationary;

c) the Fourier transforms of theu_h’s converge as measures in the flat norm to the Fourier transform of u.

The assumption made on the sequence of Fourier transforms implies that the sequence {uh}h is uniformly bounded on L^∞(R;R^N). Here we will deduce a similar property starting from the growth assumptions rather than supposing it.

The following theorem addresses these issues in a framework suitable to treat Lagrangiansf with quadratic behavior.

Theorem 2. Let f be a Lagrangian with quadratic behavior, see (1), and let f(s,·)be uniformly convex, that is there existsν >0such that

∂²f

∂ξ²(s, ξ)η, η

≥ν|η|², ∀s, ξ, η∈R^N. (2)

Then there existsh₀=h₀(C, ν) such that

a) for every (u₀, ξ₀) ∈ R^N×R^N and h < h₀, there exists an h-stationary u_h that is h-stationary, with u_h(0) =u₀,(u_h(h)−u_h(0))/h=ξ₀. Furthermore,

h<hsup0

u_{h W}1,∞(A;R^N)≤K <∞, ∀A∈ E, (3)

whereK is a constant depending only on|u0|,|ξ0|, C, ν,L¹(A);

b) every sequence{uh}h, with u_h h-stationary and such that,

suph max{|u_h(0)|,|(u_h(h)−u_h(0))/h|}<∞, (4) has a subsequence that converges weakly-∗star in W_loc^1,∞(R;R^N)to a stationary u.

In Theorem 2 we are assuming thatf has the same growth in thesandξvariables. A similar result cannot hold iff has different growths with respect tosand ξ. For example, let us considerf(s, ξ) =ξ²/2 +|s|^2+ε/(2 +ε), ε >0. The Euler-Lagrange equation that characterizes the stationary functions is ¨u=u|u|^ε, which has positive

solutions for positive initial data. By conservation of the energy, for every solutionuwith suitable initial data there exists a constantC >0 such that

t= _u(t)

0

du

2C+_2+ε² u^2+ε

, ∀t∈[0, γ],

where [0, γ] is the maximal interval of definition of u. Sinceu(γ⁻) =∞we have γ=

_∞

0

du

2C+_2+ε² u^2+ε

<∞.

The rest of the paper is devoted to the proof of Theorem 2. The first assertion in statement a) is proved in Section 2, while the second one, namely the bound (3), is proved in Section 3. In Section 4 we prove statement b) of Theorem 2 (here we follow [4]). In Section 5 we present a shorter proof of (3) that works for the prototype case wheref(s, ξ) =|ξ|²/2−V(s).

2. Existence and uniqueness of h -stationary functions

As a consequence of the assumption thatf is a Lagrangian with quadratic behavior, uniformly convex in the ξvariable, there existC, ν >0 such that: for everys, ξ∈R^N,

ν|ξ|²−C(1 +|s|²)≤f(s, ξ) ≤ C(1 +|s|²+|ξ|²), (5) ∂f

∂s(s, ξ)+∂f

∂ξ(s, ξ) ≤ C(1 +|s|+|ξ|), (6)

∂²f

∂s²(s, ξ)+∂²f

∂ξ²(s, ξ)+∂²f

∂s∂ξ(s, ξ) ≤ C; (7)

for everys₁, s₂, ξ∈R^N,

∂f

∂ξ(s₁, ξ)−∂f

∂ξ(s₂, ξ)≤C|s₁−s₂|, (8) for everys, ξ₁, ξ₂∈R^N,

∂f

∂ξ(s, ξ₁)−∂f

∂ξ(s, ξ₂), ξ₁−ξ₂

≥ν|ξ₁−ξ₂|². (9)

We introduce the following notation for functionsu_h∈X_h. We define for everyi∈Z u^h_i :=u_h(ih), ξ_i^h:=u_h(ih+h)−u_h(ih)

h =u^h_i+1−u^h_i

h ·

Step 1. We claim that u_h ∈ X_h is h-stationary if and only if, for every i ∈ Z, it satisfies the following discrete Euler-Lagrange equation:

0 = ₁

0 t∂f

∂s

[u^h_i, u^h_i+1](t),u^h_i+1−u^h_i h

+1

h

∂f

∂ξ

[u^h_i, u^h_i+1](t),u^h_i+1−u^h_i h

dt (10)

+ ₁

0 (1−t)∂f

∂s

[u^h_i+1, u^h_i+2](t),u^h_i+2−u^h_i+1 h

−1 h

∂f

∂ξ

[u^h_i+1, u^h_i+2](t),u^h_i+2−u^h_i+1 h

dt,

where [a, b](t) :=a+t(b−a) for everya, b∈R^N,t∈[0,1]. Indeed, if uis a piecewise affine function that takes the valuesa, b, c∈R^N at the pointsih,(i+ 1)h,(i+ 2)hrespectively, then

I(u,(ih,(i+ 2)h)) = ₁

0 f

[a, b](t),b−a h dt+

₁

0 f

[b, c](t),c−b

h dt=:L_h(a, b, c).

Taking a variationφ∈X_h which vanishes at ihand (i+ 2)his equivalent to considering the function Φ(d) :=

L_h(a, b+d, c) in thatuis stationary for this variation if and only if∇Φ(0) = 0, which gives exactly (10).

Step 2. Fora, b∈R^N we define

Ψ^h_a(b) :=

₁

0

∂f

∂ξ

[a, b](t),b−a h dt, M_h(a, b) :=

₁

0 th∂f

∂s

[a, b](t),b−a

h +∂f

∂ξ

[a, b](t),b−a h dt, N_h(a, b) :=

₁

0 h(1−t)∂f

∂s

[a, b](t),b−a h dt.

Then (10)_i can be written as Ψ^h_uh

i+1(u^h_i+2) =M_h(u^h_i, u^h_i+1) +N_h(u^h_i+1, u^h_i+2). (11)

We claim that for everya∈R^N, the function Ψ^h_a :R^N→R^N is a bijection with Lip((Ψ^h_a)⁻¹)→0 as h→0.

Indeed we have

Ψ^h_a(y)−Ψ^h_a(x) = ₁

0

∂f

∂ξ

[a, y](t),y−a

h −∂f

∂ξ

[a, x](t),x−a

h dt

= ₁

0 dt ₁

0

d dr

∂f

∂ξ

[a, x+r(y−x)](t),x+r(y−x)−a

h dr

= ₁

0 dt ₁

0

∂²f

∂s∂ξ

[a, x+r(y−x)](t),x+r(y−x)−a

h ·(y−x)

+ ∂²f

∂ξ²

[a, x+r(y−x)](t),x+r(y−x)−a

h ·(y−x)

h

dr.

By the mean value theorem we can finds=s(a, x, y), ξ=ξ(a, x, y, h), such that Ψ^h_a(y)−Ψ^h_a(x) = ∂²f

∂s∂ξ(s, ξ)·(y−x) +∂²f

∂ξ²(s, ξ)·(y−x) h · By (7) and (9) we have

|Ψ^h_a(y)−Ψ^h_a(x)||y−x| ≥ Ψ^h_a(y)−Ψ^h_a(x), y−x ≥ν h−C

|y−x|². (12)

Ifh < ν/C the surjectivity and injectivity of Ψ^h_a follow from (12) at once, as well as Lip((Ψ^h_a)⁻¹)≤ν

h−C ₋₁

→0 as h→0.

As a consequence for h < ν/C small enough we have that (11)_i is equivalent to the fixed point problem u^h_i+2=G^h[u^h_i, u^h_i+1](u^h_i+2), whereG^h[a, b] :R^N→R^N is defined by

G^h[a, b](x) := (Ψ^h_b)⁻¹(M_h(a, b) +N_h(b, x)).

Ifh < h₀=h₀(C, ν) this map is a contraction. Indeed by (7)

∂N_h

∂x (b, x) ≤h

₁

0

∂²f

∂s²

[b, x](t),x−b

h +1

h

∂²f

∂s∂ξ

[b, x](t),x−b h

dt≤2C,

and so

LipG^h[a, b]≤Lip(Ψ^h_b)⁻¹LipN_h(b,·)≤2C ν

h−C ₋₁

.

By the Banach-Caccioppoli theorem for everya, b∈R^N there exists a uniquex∈R^N such thatx=G^h[a, b](x).

Step 3. Leth < h₀ and fixu₀, ξ₀ ∈R^N. Then there exists a uniqueh-stationary point u_h ∈X_h such that u^h₀ =u₀, ξ₀^h=ξ₀. Indeed, under these initial conditions we haveu^h₁ =u^h₀+hξ₀^h, and then, in view of step 2, we can uniquely determineu^h₂ fromu^h₀, u^h₁ by solvingx=G^h[u^h₀, u^h₁](x). By induction we can define u_h on [0,∞) in such a way that (10)_i∈N are satisfied. Similarly, we may defineu_h on (−∞,0] so that the equations (10)_i∈Z hold.

3. Proof of local boundedness of h -stationary sequences

Step 1. For eachh < h₀we consider the h-stationary pointu_h withu^h₀=u₀ andξ₀^h=ξ₀. Let us write the ith discrete Euler-Lagrange equation asA^h_i =B_i^hwhere

A^h_i : = ₁

0

∂f

∂ξ

[u^h_i+1, u^h_i+2](t), ξ^h_i+1

−∂f

∂ξ

[u^h_i, u^h_i+1](t), ξ_i^h dt, B_i^h: = h

₁

0 t∂f

∂s

[u^h_i, u^h_i+1](t), ξ^h_i

+ (1−t)∂f

∂s

[u^h_i+1, u^h_i+2](t), ξ_i+1^h dt.

Since|[a, b](t)−[b, c](t)| ≤ |a−b|+|b−c|, we have that

|A^h_i| ≥

₁

0

∂f

∂ξ

[u^h_i+1, u^h_i+2](t), ξ_i+1^h

−∂f

∂ξ

[u^h_i+1, u^h_i+2](t), ξ_i^h dt

−

₁

0

∂f

∂ξ

[u^h_i+1, u^h_i+2](t), ξ_i^h

−∂f

∂ξ

[u^h_i, u^h_i+1](t), ξ^h_i dt

≥ν|ξ_i+1^h −ξ_i^h| −Ch(|ξ_i^h|+|ξ^h_i+1|), (13) and by (6),

|B_i^h| ≤Ch(1 +|u^h_i|+h|ξ^h_i|+|ξ_i^h|) +Ch(1 +|u^h_i+1|+h|ξ_i+1^h |+|ξ_i+1^h |). (14) Since|u^h_i| ≤ |u0|+h_i−1

j=0|ξ_j^h| andA^h_i =B_i^h, a straightforward computation leads from (13) and (14) to

|ξ^h_i+1−ξ^h_i| ≤ 2Ch ν

1 +|u₀|+h i k=0

|ξ^h_k|+|ξ_i^h|+|ξ_i+1^h |+h|ξ_i+1^h |

.

Let us defineσ^h_i := max{|ξ_j^h|: 0≤j≤i}. We have j

i=0

|ξ_i+1^h −ξ_i^h| ≤ 2Ch ν

(j+ 1)(1 +|u₀|) +h j i=0

i k=0

|ξ_k^h|+ j i=0

(|ξ_i^h|+|ξ_i+1^h |) +h j i=0

|ξ_i+1^h |

≤ 2C(j+ 1)h ν

(1 +|u0|) +h(j+ 1)σ_j^h+ 2σ^h_j +hσ^h_j

+2C

ν (h+h²)|ξ_j+1^h |.

Since|ξ_j+1^h | ≤ |ξ0|+_j

i=0|ξ_i+1^h −ξ_i^h|, we obtain

1−2C

ν (h+h²)

|ξ_j+1^h | ≤ |ξ₀|+2C(j+ 1)h ν

(1 +|u₀|) + (h(j+ 1) +h+ 2)σ_j^h

·

Choose h₀ =h₀(C, ν) such that for h < h₀ we have (1−(2C/ν)(h+h²)) ≥1/2 and for R₀ := 2h₀ we have (4(R₀+h₀)C/ν)(R₀+ 2h₀+ 2)≤1/2. For everyj∈ {0, ...,[R₀/h]}we have (j+ 1)h≤R₀+h, and thus

|ξ_j+1^h | ≤2|ξ0|+4(R₀+h)C

ν (1 +|u0|) +4(R₀+h)C

ν (R₀+ 2h+ 2)σ_j^h≤L+σ_j^h/2, (15) if we define

L=L(u₀, ξ₀, C, ν) := 2|ξ₀|+ (4(R₀+h₀)C/ν)(1 +|u₀|).

By (15) we conclude that for everyh < h₀and for everyj∈ {0, ..., Nh}, whereN_h:= [R₀/h], we have

σ^h_j+1≤max{σ_j^h, L+σ^h_j/2}· (16)

From (16) it follows that

σ_j+1^h ≤ max{σ_j^h, L+σ^h_j/2} ≤max{σ_j−1^h , L+σ_j−1^h /2, L+σ^h_j/2}

≤ max{σ_j−1^h , L+σ_j^h/2} ≤...≤max{σ₀, L+σ_j^h/2}, (17) whereσ₀^h=|ξ₀^h|=|ξ0|=:σ₀. Below we writeN instead ofN_h for simplicity of notations and we iterate (17) to get,

σ_N^h₊₁ ≤ max{σ₀, L+ 2⁻¹σ^h_N}

≤ max{σ0, L+ 2⁻¹max{σ0, L+ 2⁻¹σ_N^h₋₁}}

= max{σ0, L+ 2⁻¹σ₀, L(1 + 2⁻¹) + 2⁻²σ_N^h₋₁}

≤ max{σ₀+ 2L, L(1 + 2⁻¹) + 2⁻²σ_N^h₋₁}

≤ max{σ₀+ 2L, L(1 + 2⁻¹) + 2⁻²max{σ₀, L+ 2⁻¹σ_N^h₋₂}}

≤ max{σ₀+ 2L, L(1 + 2⁻¹) + 2⁻²(L+ 2⁻¹σ_N^h₋₂)}

= max{σ₀+ 2L, L(1 + 2⁻¹+ 2⁻²) + 2⁻³σ_N^h₋₂}

≤ ...

≤ max



σ₀+ 2L, L N j=0

2^−j+ 2^−N+1σ₀



≤σ₀+ 2L.

Thus, forh < h₀, we have

sup

[0,R0]|u˙_h| ≤ |σ_N^h_h+1| ≤σ₀+ 2L,

which implies

sup

h<h0

u_{h W}1,∞([0,R0])≤K(max{|u₀|,|ξ₀|}, C, ν), (18) withK increasing in its first variable. To complete the proof of (3) it remains to show that for everyM >0,

sup

h<h0

u_{h W}1,∞([0,M])≤K^∗(max{|u₀|,|ξ₀|}, C, ν, M). (19) Indeed we can cover [0, M] with a finite union of intervals of the formI_i:=t_i+ [0, R₀], t_i < t_i+1,t₁= 0 such that I_i∩I_i+1 has at least lengthh₀. By (18),

h<hsup0

uh W^1,∞(I1)≤K(max{|u0|,|ξ0|}, C, ν).

This is the case j = 1 of the following family of inequalities indexed by j, that we are going to prove by induction; precisely, we claim that,

h<hsup0

u_{h W}^1,∞_(Ij)≤K_j(max{|u₀|,|ξ₀|}, C, ν)<∞.

Let us assume this holds forj−1. Since the problem is autonomous we can apply (18) to get sup

h<h0

u_{h W}1,∞(Ij)≤K(max{|u_h(infI_j)|,|u˙_h((infI_j))|}, C, ν).

On the other hand, since infI_j ∈I_j−1, by the inductive hypothesis we have

K(max{|u_h(infI_j)|,|u˙_h((infI_j))|}, C, ν) ≤ K(K_j−1(max{|u₀|,|ξ₀|}, C, ν), C, ν)

=: K_j(max{|u₀|,|ξ₀|}, C, ν),

and the assertion is proved. Since with a finite number of steps we can cover [0, M], (19) is established.

4. Proof of Theorem 2, part b

The proofs of this section follow closely the ideas of M¨uller and Ortiz [4].

Step 1. We first prove the following: there exists a constantl >0 depending only onC andν such that, if Y is a linear subspace ofH_loc¹ (R;R^N) andu∈Y is stationary with respect to variations inY, i.e.,

δF(u, ϕ, A) = 0, ∀A∈ E,∀ϕ∈Y, with sptϕ⊂A, (20) thenuminimizesF(·, A) among all functionsv∈Y withu_|∂A=v_|∂A, for everyA∈ E such thatL¹(A)< l. In fact we will need to apply this property only forY =X_h.

Let ube such that (20) holds, A = (a, b)∈ E and let v ∈ Y such that ϕ :=v−uis zero on ∂A. Let us consider the function

g(r) :=

_b

a f(u+rϕ,u˙+rϕ) dt.˙ By (20) we have thatg(0) = 0 and by Taylor’s Formula we obtain

_b

a f(u+ϕ,u˙+ ˙ϕ) dt− _b

a f(u,u) dt˙ = ₁

0 g(r)(1−r) dr

= ₁

0 (1−r) _b

a Q[r, t](ϕ(t),ϕ(t)) dt˙ dr, (21)

whereQ[r, t] :Rⁿ×Rⁿ→Ris the quadratic form defined by Q[r, t](ρ, τ) :=

∂²f

∂s²

u(t) +rϕ(t),u(t) +˙ rϕ(t)˙ ρ, ρ

+ 2

∂²f

∂s∂ξ

u(t) +rϕ(t),u(t) +˙ rϕ(t)˙ ρ, τ

+ ∂²f

∂ξ²

u(t) +rϕ(t),u(t) +˙ rϕ(t)˙ τ, τ

·

By (7) and (2) it is easy to see that

Q[s, t](ρ, τ)≥ν|τ|²−C|ρ|²−2C|ρ||τ| ≥ ν

2|τ|²−(C+ 2CC)|ρ|²,

whereC is a constant depending only onν andC. Therefore by (21) and by Poincar´e’s Inequality we get _b

a

f(u+ϕ,u˙+ ˙ϕ) dt− _b

a

f(u,u) dt˙ ≥ ₁

0

(1−r) _b

a

c

2|ϕ(t)˙ |²−(C+ 2CC)|ϕ(t)|² dtdr

≥ ₁

0 (1−r) c

2 π²

(b−a)² −C−2CC _b

a |ϕ(t)|²dtdr

> 0,

where the last inequality holds provided that (b−a)²<(cπ²)/(2C+ 4CC) =:l.

Step 2. Let us consider a sequence {uh}h with eachu_h h-stationary and such that (4) hold. By (3) and by Ascoli-Arzel´a compactness criterion we infer that, up to extracting a subsequence,u_h u^∗ inW_loc^1,∞(R;R^N), where u∈ W_loc^1,∞(R;R^N). We want to prove that u ish-stationary for the action. Clearly it suffices to show that

F(u, A)≤F(v, A), ∀v∈H_loc¹ (R;R^N), with spt(v−u)⊂A,

for everyA∈ EwithL¹(A)< l, wherelis defined as in step 1. It is in this part of the proof that a Γ-convergence argument is used.

Let us fix such an interval A and a function v. By (5) and by (2) we can apply De Giorgi-Ioffe lower semicontinuity theorem ([1], [2]) to see that

F(u, A)≤lim inf

k→ ∞ F(w_k, A), ∀{w_k}_k ⊂H¹(A;R^N), w_k uw-H¹(A;R^N).

In particular

F(u, A)≤lim inf

h→0 F(u_h, A). (22)

On the other hand, by Lemma 4.2.b in [4] there exists{v_h}_h⊂X_hwithv_h=von∂Asuch thatv_h→vstrongly in H¹(A;R^N) (in particular the convergence is strong in L^∞). By (5) and the strong convergence of the v_h’s we have that

F(v, A) = lim

h→0F(v_h, A). (23)

In order to proveF(u, A)≤F(v, A) and to conclude the proof we need to compare the behavior of{F(u_h, A)}h

and {F(v_h, A)}h. Clearly, if v_h = u_h on ∂A, we conclude by a direct application of the result in step 1 for Y =X_hcombined with (22) and (23). If this is not the case we can argue as follows. Let us consider{Ah}h⊂ E such thatA_h = (a_h, b_h)⊂A= (a, b),A_h is compatible withTh and a_h→a, b_h→b. Since u(a_h)−v(a_h)→0 andu(b_h)−v(b_h)→0, there exists a sequence of affine functions{ph}h such that

v_h+p_h=u_h on ∂A_h, p_h→0 uniformly.

Letw_h:=v_h+p_h onA_h andw_h=u_honA\A_h. By the minimality ofu_h onA_hproved in step 1 we now get F(u_h, A_h)≤F(w_h, A_h).

We claim that

F(u, A)≤lim inf

h→0 F(u_h, A_h), lim

h→0F(w_h, A_h) =F(v, A),

clearly this will conclude the proof. The first inequality comes from (22) and the fact that F(u_h, A\A_h)→0 by (5) and (3). The second inequality holds since|F(w_h, A_h)−F(v_h, A_h)|=|F(v_h+p_h, A_h)−F(v_h, A_h)| →0 as h→0. This is an easy consequence of the properties off and of the uniform convergence of{p_h}_h to zero.

5. A short proof of (3) in the model case

Here we give a short proof of (3) in the model casef(s, ξ) =|ξ|²/2−V(s), whereV is as in Theorem 1. Let us consider ah-stationaryu_hsuch thatu^h₀ =u₀andξ₀^h=ξ₀. It is easy to see that the discrete Euler-Lagrange equations can now be written as

ξ_i+1^h =h ₁

0 t∇V([u^h_i, u^h_i+1](t)) + (1−t)∇V([u^h_i+1, u^h_i+2](t))dt+ξ_i^h.

Since the second derivatives ofV are uniformly bounded onR^N we have|∇V|(s)≤C(1 +|s|), and so we get

|ξ^h_i+1| ≤Ch(1 +|[u^h_i, u^h_i+1]|+|[u^h_i+1, u^h_i+2]|) +|ξ_i^h|.

As|u^h_i+1| ≤ |u^h_i|+h|ξ_i^h|it follows that

|ξ^h_i+1| ≤ Ch(1 +|u^h_i|+h|ξ_i^h|+|u^h_i|+h|ξ_i^h|+h|ξ^h_i+1|) +|ξ^h_i|

≤ 2Ch(1 +|u^h_i|) + (1 + 2Ch²)|ξ_i^h|+Ch²|ξ^h_i+1|, so that

|ξ_i+1^h | ≤A_h|ξ_i^h|+B_h(1 +|u^h_i|), A_h:=1 + 2Ch²

1−Ch² , B_h:= 2Ch

1−Ch²· (24)

Since|u^h_i| ≤ |u₀|+h_i−1

j=0|ξ_j^h|, if we defineσ_i^h:= max{|ξ^h_j|: 1≤j≤i}, we get,

|ξ_i+1^h | ≤(A_h+ihB_h)σ^h_i +B_h(1 +|u₀|).

LetR >0. For 1≤i≤[R/h] by the previous equation we have

|ξ^h_i+1| ≤(A_h+RB_h)σ^h_i +B_h(1 +|u₀|).

SinceA_h>1, settingD_h:= (A_h+RB_h)>1 we conclude

σ^h_i+1 ≤ D_hσ^h_i +B_h(1 +|u₀|)≤D²_hσ_i−1^h +B_h(1 +|u₀|)(1 +D_h)

≤ ...≤Dⁱ⁺¹_h σ₀+B_h(1 +|u₀|)ⁱ

j=0

D^j_h≤D_hⁱ⁺¹(σ₀+iB_h(1 +|u₀|)).

Then

˙

u_{h L}∞(0,R)≤σ^h_[R/h]+1≤D^[R/h]+2_h (σ₀+ ([R/h] + 1)B_h(1 +|u0|)). (25) Since

D^[R/h]+2_h = exp (([R/h] + 2) log(A_h+RB_h)) = exp (([R/h] + 2)(2CRh+o(h))),

and{([R/h] + 1)B_h}his bounded, we have that the right hand side of (25) is bounded forhsmall by a constant depending onC, R,|u₀|and|ξ₀|. This proves (3) in this special case.

Acknowledgements. We thank Irene Fonseca for proposing us the problem. This work was supported by the Center for Nonlinear analysis under NSF Grant DMS-9803791. The first author was supported by the Ph.D. program of the University of Florence and by MIUR Cofin 2003-2004 through the research project “Equazioni e sistemi differenziali nonlineari degeneri: esistenza e regolarit´a”.

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