On the validity of the Euler Lagrange equation

Transcription

1 J. Math. Anal. Appl. 304 (2005) On the validity of the Euler Lagrange equation A. Ferriero, E.M. Marchini Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via R. Cozzi 53, Milano, talia Received 9 September 2003 Available online 14 October 2004 Submitted by B.S. Mordukhovich Abstract The purpose of the present paper is to establish the validity of the Euler Lagrange equation for the solution ˆx to the classical problem of the calculus of variations Elsevier nc. All rights reserved. Keywords: Euler Lagrange equation 1. ntroduction The purpose of the present paper is to establish (under Carathéodory s conditions) the validity of the Euler Lagrange equation (E L) for the solution ˆx to the classical problem of the calculus of variations consisting in minimizing the functional J(x)= L ( t,x(t),x (t) ) dt, * Corresponding author. addresses: ferriero@matapp.unimib.it (A. Ferriero), elsa@matapp.unimib.it (E.M. Marchini) X/$ see front matter 2004 Elsevier nc. All rights reserved. doi: /j.jmaa

2 A. Ferriero, E.M. Marchini / J. Math. Anal. Appl. 304 (2005) where = (a, b) R, on the set of those absolutely continuous functions x : R s satisfying the boundary conditions x(a) = A, x(b) = B. Establishing the validity of the Euler Lagrange equation amounts to proving that [ ξ L ( t, ˆx(t), ˆx (t) ),η (t) + x L ( t, ˆx(t), ˆx (t) ),η(t) ] dt = 0 for every variation η in some suitable class. A large number of papers has been devoted to this classical problem, e.g., [4 6,8 10]. The example obtained by Ball Mizel [1], modifying an earlier example of Maniá [7], provides a variational problem where the integrability of x L(, ˆx( ), ˆx ( )) does not hold, as a consequence, (E L) is not true along the solution. Hence, some condition on the term x L(, ˆx( ), ˆx ( )) has to be imposed in order to ensure the validity of (E L). A result of Clarke [5] implies that the following assumption on the term x L(, ˆx( ), ˆx ( )): there exists a function S(t) integrable on such that, for y in a neighborhood of the solution, x L ( t,y, ˆx (t) ) S(t) is sufficient to establish the validity of (E L). This condition implies that, locally along the solution, x L(t, x, x ) is Lipschitzian of Lipschitz constant S(t). However, there are simple meaningful examples of variational problems where this Lipschitzianity condition is not verified. Consider the Lagrangian defined by L(x, ξ) = (ξ x 2/3) 2, the problem (P) of minimizing 1 0 L ( x(t),x (t) ) dt over the absolutely continuous functions x with x(0) = 0, x(1) = 1. One can easily verify that ˆx(t) = t 2/3 is a minimizer for (P) (indeed, L( ˆx(t), ˆx (t)) = 0 on [0, 1], L is non-negative everywhere). n this case, although L is not differentiable everywhere, L x ( ˆx(t), ˆx (t)) exists a.e. (it is a.e. zero) it is integrable. The purpose of the present paper is to provide a result on the validity of (E L) that is satisfied by Lagrangians that are Lipschitzian in x, but that applies as well to the non-lipschitzian cases as the example before. n the proof we first show that the fact that ˆx is a solution implies the integrability of ξ L(, ˆx( ), ˆx ( )). Then, using this result, we establish the validity of (E L) under Carathéodory s condition. Note that we do not assume any convexity hypothesis on the Lagrangian. Moreover, no growth condition whatsoever is assumed so that, as far as we know, relaxation theorems cannot be applied.

3 358 A. Ferriero, E.M. Marchini / J. Math. Anal. Appl. 304 (2005) ntegrability of ξ L(, ˆx( ), ˆx ( )) Consider the problem of minimizing the functional J(x)= L ( t,x(t),x (t) ) dt on the set of those absolutely continuous functions x : R s satisfying the boundary conditions x(a) = A, x(b) = B. Let ˆx be a (weak local) minimizer yielding a finite value for the functional J, set µ = sup t [a,b] ˆx(t). Our results will depend on the following assumption. Assumption A. (i) L is differentiable in x along ˆx, for a.e. t, the map x L(, ˆx( ), ˆx ( )) is integrable on ; (ii) there exists a function S(t) integrable on such that, for any y B(0,µ+ 1), L ( t,y, ˆx (t) ) L ( t, ˆx(t), ˆx (t) ) + S(t) y ˆx(t). Consider problem (P) as presented in the ntroduction. L ˆx satisfy Assumption A: L x ( ˆx(t), ˆx (t)) exists a.e. (identically zero, hence integrable), S(t) = t 2/3 verifies the inequality L ( y, ˆx (t) ) S(t) y ˆx(t), since (2 3 t 1/3 y 2 ) 2 = 4( y t 1/3 ) 3 9t 2/3 ( ( y t 2/3 ) y +t 1/3 4 ) 9t 2/3 y t2/3. This is our first result on the term ξ L(, ˆx( ), ˆx ( )). n what follows, R denotes R {+ }. Theorem 2.1. Suppose that L : R s R s R is an extended valued function, finite on its effective domain of the form dom L = R s G, where G R s is an open set, that it satisfies Carathéodory s conditions, i.e., L(,x,ξ) is measurable for fixed (x, ξ) L(t,, ) is continuous for almost every t. Moreover assume that L is differentiable in ξ on dom L that ξ L satisfies Carathéodory s conditions on dom L. Suppose that Assumption A holds. Then, ξ L ( t, ˆx(t), ˆx (t) ) dt < +. Proof. (1) By assumption, L(, ˆx( ), ˆx ( )) L 1 (), hence setting S 0 ={t : ˆx (t) / G}, we have m(s 0 ) = 0. Given ɛ>0, we can cover S 0 by an open set O 1 of measure m(o 1 )< ɛ/2. We have also that ξ L is a Carathéodory s function that ˆx is measurable in. Hence, by the theorems of Scorza Dragoni of Lusin, for the given ɛ>0 there exists an open set O 2 such that m(o 2 )<ɛ/2 at once ˆx is continuous in \ O 2, ξ L is continuous in ( \ O 2 ) R s G. By taking K ɛ = \ (O 1 O 2 ), we have that K ɛ

4 A. Ferriero, E.M. Marchini / J. Math. Anal. Appl. 304 (2005) is a closed set such that on it ˆx is continuous with values in G, ξ L is continuous on K ɛ R s G m( \ K ɛ )<ɛ.forn 1setɛ n = (b a)/2 n+1 K n = K ɛn ; set also C n = n j=1 K j. Then C n are closed sets, C n C n+1, ˆx is continuous on C n with values in G, ξ L is continuous in C n R s G lim n + m( \ C n ) = 0. From these properties it follows that there exists k n > 0 such that, for all t C n, ξ L ( t, ˆx(t), ˆx (t) ) <kn. There is no loss of generality in assuming k n k n 1. Moreover, we have that m(c 1 ) (b a)/2 n=2 m(c n \ C n 1 ) (b a)/2. For all n>1, we set = C n \ C n 1. Hence we obtain that C m = C mn=2 1 that = E C 1 n>1, where m(e) = 0. (2) Consider the function { 0 if ξ L(t, ˆx(t), ˆx (t)) = 0, θ(t)= v n = θ(t)dt, ξ L(t, ˆx(t),ˆx (t)) ξ L(t, ˆx(t),ˆx (t)) otherwise, so that v n m( ). There exists a closed set B n C 1 such that m(b n ) = v n. Set θ n (t) = θ(t)χ (t) + v n v n χ B n (t). We have that θ n (t) dt = θ(t)dt + v n = 0. Hence, setting θ n (t) = t a θ n (τ) dτ, we see that the functions θ n(t) are admissible variations. Moreover we obtain θ n sup t t a θ n (τ) dτ θ n (τ) dτ 2m(An ). (3) For t in,wehave ξ L(t, ˆx(t), ˆx (t)) <k n ;fort in B n, ξ L(t, ˆx(t), ˆx (t)) < k 1 k n. Recalling that Ā n C n, we infer that, for all t Ā n B n, ξ L ( t, ˆx(t), ˆx (t) ) k n. We wish to obtain an uniform bound for ξ L computed in a suitable neighborhood of the solution ( ˆx( ), ˆx ( )). Consider the set (Ā n B n ) R s G as a metric space M n with distance d((t, x, ξ), (t,x,ξ )) = sup( t t, x x, ξ ξ ). On M n, ξ L is continuous. Moreover, its subset G n = {( t, ˆx(t), ˆx (t) ) } : t Ā n B n is compact, on G n, ξ L is bounded by k n. Hence there exists δ n > 0 such that, for (t,x,ξ) M n with d((t, x, ξ), (t, ˆx(t), ˆx (t))) < δ n,wehave ξ L(t, x, ξ) <k n + 1.

5 360 A. Ferriero, E.M. Marchini / J. Math. Anal. Appl. 304 (2005) (4) For < min{1/2m( ), δ n /2m( ), δ n }, consider the integrals L t, ˆx(t) + θn (t), ˆx (t) + θ n (t)) L ( t, ˆx(t), ˆx (t) )] dt = L t, ˆx(t) + θn (t), ˆx (t) + θ n (t)) L ( t, ˆx(t) + θ n (t), ˆx (t) )] dt B n + L t, ˆx(t) + θn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt. For every t B n there exists ζ (t) (0,)such that L t, ˆx(t) + θn (t), ˆx (t) + θ n (t)) L ( t, ˆx(t) + θ n (t), ˆx (t) )] = ξ L ( t, ˆx(t) + θ n (t), ˆx (t) + ζ (t)θ n (t)),θ n (t) ξ L ( t, ˆx(t) + θ n (t), ˆx (t) + ζ (t)θ n (t)) from the choice of, ξ L(t, ˆx(t)+ θ n (t), ˆx (t) + ζ (t)θ n (t)) <k n + 1. Hence, we can apply the dominated convergence theorem to obtain that lim L t, ˆx(t) + θn (t), ˆx (t) + θ n 0 (t)) L ( t, ˆx(t) + θ n (t), ˆx (t) )] dt = B n B n ξ L ( t, ˆx(t), ˆx (t) ),θ n (t) dt. (5) Set f + (s) = max{0,f(s)}, f (s) = max{0, f(s)}. Since 0 1 [ L ( t, ˆx(t) + θn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] + S(t) θ n (t), by the dominated convergence theorem, lim L t, ˆx(t) + θn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] + dt 0 = lim L t, ˆx(t) + θn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] + dt. 0 By the Fatou s lemma, lim inf L t, ˆx(t) + θn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt 0 lim inf 0 We have obtained that L t, ˆx(t) + θn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt.

6 lim sup 0 lim 0 lim inf 0 A. Ferriero, E.M. Marchini / J. Math. Anal. Appl. 304 (2005) L t, ˆx(t) + θn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt lim sup 0 L t, ˆx(t) + θn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] + 1 L t, ˆx(t) + θn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt [ L ( t, ˆx(t) + θn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt = x L ( t, ˆx(t), ˆx (t) ),θ n (t) dt. (6) Since ˆx is a minimizer, we have 0 ξ L ( t, ˆx(t), ˆx (t) ),θ n (t) dt B n + lim sup L t, ˆx(t) + θn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt 0 ξ L ( t, ˆx(t), ˆx (t) ),θ n (t) dt + x L ( t, ˆx(t), ˆx (t) ),θ n (t) dt. ( ) B n Since θ n (t) = ξ L(t, ˆx(t), ˆx (t))/ ξ L(t, ˆx(t), ˆx (t)), for any t in, it follows that ξ L(t, ˆx(t), ˆx (t)), θ n (t) χ (t) = ξ L(t, ˆx(t), ˆx (t)) χ An (t). Hence, we obtain that ( ) can be written as ξ L ( t, ˆx(t), ˆx (t) ) dt B n ξ L ( t, ˆx(t), ˆx (t) ),θ n (t) dt + x L ( t, ˆx(t), ˆx (t) ),θ n (t) dt. On B n, ξ L(t, ˆx(t), ˆx (t)) is bounded by k 1 ; from Hölder s inequality the estimate on θ n obtained in (2) we have that there exists a constant C (independent of n) such that ξ L ( t, ˆx(t), ˆx (t) ) dt Cm( ). (7) As m +, the sequence of functions ( ξ L(t, ˆx(t), ˆx (t)) χ { mn=2 }(t)) m converges monotonically to the function ξ L(t, ˆx(t), ˆx (t)) χ { n>1 }(t). From the estimate above monotone convergence, we obtain

7 362 A. Ferriero, E.M. Marchini / J. Math. Anal. Appl. 304 (2005) \C 1 ξ L ( t, ˆx(t), ˆx (t) ) dt = On C 1, ξ L(t, ˆx(t), ˆx (t)) <k 1. Hence ξ L ( t, ˆx(t), ˆx (t) ) dt < +. ξ L ( t, ˆx(t), ˆx (t) ) χ { n>1 } dt ( Cm ). n>1 3. Additional regularity the validity of the Euler Lagrange equation Corollary 3.1. Under the same assumptions as in Theorem 2.1, for every variation η, η(a) = 0, η(b) = 0 η L (), we have [ ξ L ( t, ˆx(t), ˆx (t) ),η (t) + x L ( t, ˆx(t), ˆx (t) ),η(t) ] dt = 0. Proof. We shall prove that, for every η in AC() with bounded derivative, such that η(a) = η(b) = 0, we have [ ξ L ( t, ˆx(t), ˆx (t) ),η (t) + x L ( t, ˆx(t), ˆx (t) ),η(t) ] dt 0. Fix η,let η (t) K for almost every t in. (1) Define C n k n as in point (1) of the proof of Theorem 2.1. Set v n = η (t) dt. \C n We have that lim n + v n =0. n particular, for n ν, there exists B n C 1 such that m(b n ) = v n. Set 0 for t \ C n, (η n ) (t) = η (t) for t C n \ B n, v n v n + η (t) for t B n. We obtain [ ] η n (t) dt = η vn (t) dt + v n + η (t) dt = η (t) dt + v n C n \B n B n C n = η (t) dt = 0. Hence, setting η n (t) = t a η n (τ) dτ, we have that the functions η n(t) are variations that, for almost every t in, η n (t) (1 + K), so that η n (1 + K)(b a).

8 A. Ferriero, E.M. Marchini / J. Math. Anal. Appl. 304 (2005) (2) As in point (3) of the proof of Theorem 2.1, there exists δ n > 0 such that for (t,x,ξ) C n R s G, with d((t, x, ξ), (t, ˆx(t), ˆx (t))) < δ n,wehave ξ L(t, x, ξ) < k n + 1. (3) For < min{1/(1 + K)(b a),δ n /(1 + K)(b a),δ n /(1 + K)}, consider the integrals L t, ˆx(t) + ηn (t), ˆx (t) + η n (t)) L ( t, ˆx(t), ˆx (t) )] dt = C n + L t, ˆx(t) + ηn (t), ˆx (t) + η n (t)) L ( t, ˆx(t) + η n (t), ˆx (t) )] dt L t, ˆx(t) + ηn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt. For almost every t C n, there exists ζ (t) (0,)such that L t, ˆx(t) + ηn (t), ˆx (t) + η n (t)) L ( t, ˆx(t) + η n (t), ˆx (t) )] = ξ L ( t, ˆx(t) + η n (t), ˆx (t) + ζ (t)η n (t)),η n (t) ξ L ( t, ˆx(t) + η n (t), ˆx (t) + ζ (t)η n (t)) (1 + K) < (k n + 1)(1 + K). Hence, we can apply the dominated convergence theorem to obtain that lim L t, ˆx(t) + ηn (t), ˆx (t) + η n 0 (t)) L ( t, ˆx(t) + η n (t), ˆx (t) )] dt = ξ L ( t, ˆx(t), ˆx (t) ),η n (t) dt = ξ L ( t, ˆx(t), ˆx (t) ),η n (t) dt. C n (4) Following the point (5) of Theorem 2.1, we obtain that lim sup L t, ˆx(t) + ηn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt 0 x L ( t, ˆx(t), ˆx (t) ),η n (t) dt. (5) Since ˆx is a minimizer, we have 0 ξ L ( t, ˆx(t), ˆx (t) ),η n (t) dt + lim sup L t, ˆx(t) + ηn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt 0 [ ξ L ( t, ˆx(t), ˆx (t) ),η n (t) + x L ( t, ˆx(t), ˆx (t) ),η n (t) ] dt.

9 364 A. Ferriero, E.M. Marchini / J. Math. Anal. Appl. 304 (2005) (6) Since lim n + η n (t) = η (t) ξ L ( t, ˆx(t), ˆx (t) ) η n (t) ξ L ( t, ˆx(t), ˆx (t) ) (1 + K), lim η n(t) = η(t) n + x L ( t, ˆx(t), ˆx (t) ) ηn (t) x L ( t, ˆx(t), ˆx (t) ) (1 + K)(b a), by the dominated convergence we obtain lim ξ L ( t, ˆx(t), ˆx (t) ),η n (t) dt = ξ L ( t, ˆx(t), ˆx (t) ),η (t) dt n + lim n + x L ( t, ˆx(t), ˆx (t) ),η n (t) dt = x L ( t, ˆx(t), ˆx (t) ),η(t) dt. t follows that [ ξ L ( t, ˆx(t), ˆx (t) ),η (t) + x L ( t, ˆx(t), ˆx (t) ),η(t) ] dt 0. Even if the validity of the Euler Lagrange equations already follows by the previous Corollary 3.1 the DuBois Reymond s lemma [3], we give an alternative proof in Corollary 3.3. n the following theorem we prove an additional regularity result for the Lagrangian evaluated along the minimizer. Theorem 3.2. Under the same assumptions as in Theorem 2.1,themap ξ L(, ˆx( ), ˆx ( )) is in L (). Proof. Using an iteration process, we shall prove that for every p in N, ξ L(, ˆx( ), ˆx ( )) is in L p (). Since the L p are nested, this proves that ξ L(, ˆx( ), ˆx ( )) p 1 Lp (). At the same time, we shall prove that there exists a constant K>0 such that, for every 1 p<+, ξ L p K, thus proving that ξ L(, ˆx( ), ˆx ( )) is in L (). Suppose that: (1) From Theorem 2.1, we know that ξ L(, ˆx( ), ˆx ( )) is in L 1 ().Starting the iteration process, fix p N suppose that ξ L(, ˆx( ), ˆx ( )) L p (), to prove that ξ L(, ˆx( ), ˆx ( )) L p+1 (). (2) We can assume ξ L p 0. Define C n,,k n as in point (1) of the proof of Theorem 2.1. For all n>1, set vn p (b a) ( = 2 ξ L p ξ L t, ˆx(t), ˆx (t) ) ξ L ( t, ˆx(t), ˆx (t) ) p 1 dt. p

10 A. Ferriero, E.M. Marchini / J. Math. Anal. Appl. 304 (2005) Since m(c 1 ) (b a)/2 vn p (b a) 2 ξ L p p ξ L ( t, ˆx(t), ˆx (t) ) p (b a)/2, there exists a set Bn p C 1 such that m(bn p ) = vn p, so that 2 ξ L p p v p n (b a) vn p dt = ( ξ L t, ˆx(t), ˆx (t) ) ξ L ( t, ˆx(t), ˆx (t) ) p 1. Set B p n ( θ p n ) (t) = ξ L ( t, ˆx(t), ˆx (t) ) ξ L ( t, ˆx(t), ˆx (t) ) p 1 χ An (t) + 2 ξ L p p (b a) v p n v p n χ B p n (t) θn p (t) = t a (θ n p ) (τ) dτ. We obtain that θn p 2 ξ L(t, ˆx(t), ˆx (t)) p dt.the variations θn p have bounded derivatives so we can apply Corollary 3.1 to obtain that [ ξ L ( t, ˆx(t), ˆx (t) ), ( θn p ) (t) + x L ( t, ˆx(t), ˆx (t) ),θn p (t) ] dt = 0. t follows that ξ L ( t, ˆx(t), ˆx (t) ) p+1 dt = 2 ξ L p p (b a) + k C B p n ξ L ( t, ˆx(t), ˆx (t) ),vn p / vn p dt x L ( t, ˆx(t), ˆx (t) ),θ p n (t) dt ξ L ( t, ˆx(t), ˆx (t) ) p dt ξ L ( t, ˆx(t), ˆx (t) ) p dt ξ L ( t, ˆx(t), ˆx (t) ) p dt, x L ( t, ˆx(t), ˆx (t) ) dt where C is independent of n p (suppose C 1). The sequence of maps ( ξ L ( t, ˆx(t), ˆx (t) ) p+1 χ m n=2 (t)) m

11 366 A. Ferriero, E.M. Marchini / J. Math. Anal. Appl. 304 (2005) converges monotonically to ξ L(t, ˆx(t), ˆx (t)) p+1 χ { n>1 }(t), each integral ξ L(t, ˆx(t), ˆx (t)) p+1 χ { mn=2 }(t) dt is bounded by the same constant C n=2 ξ L ( t, ˆx(t), ˆx (t) ) p dt = C \C 1 ξ L ( t, ˆx(t), ˆx (t) ) p dt. Hence, by the monotone convergence theorem, ξ L ( t, ˆx(t), ˆx (t) ) p+1 dt C ξ L ( t, ˆx(t), ˆx (t) ) p dt < +. \C 1 \C 1 On C 1, ξ L(t, ˆx(t), ˆx (t)) <k 1, proving that ξ L(, ˆx( ), ˆx ( )) L p+1 (). Moreover, we have also obtained that ξ L ( t, ˆx(t), ˆx (t) ) p+1 dt C p ξ L ( t, ˆx(t), ˆx (t) ) dt, \C 1 so that ( \C 1 \C 1 ξ L ( t, ˆx(t), ˆx (t) ) p+1 dt) 1/(p+1) CS, where S = max{1, \C 1 ξ L(t, ˆx(t), ˆx (t)) dt}. Setting T = max{1,m(c 1 )} we have that, for all p N, ( ξ L (p+1) k (p+1) 1 m(c 1 ) + ξ L ( t, ˆx(t), ˆx (t) ) ) 1/(p+1) p+1 dt k 1 m(c 1 ) 1/(p+1) + \C 1 ( ξ L ( t, ˆx(t), ˆx (t) ) ) 1/(p+1) p+1 dt k 1 T + CS = K. \C 1 Corollary 3.3. Under the same conditions as in Theorem 2.1, for every variation η, η(a) = 0, η(b) = 0 η L 1 (), we have [ ξ L ( t, ˆx(t), ˆx (t) ),η (t) + x L ( t, ˆx(t), ˆx (t) ),η(t) ] dt = 0. As a consequence, t ξ L(t, ˆx(t), ˆx (t)) is absolutely continuous. Proof. We shall prove that, for every η in AC(), such that η(a) = η(b) = 0, we have [ ξ L ( t, ˆx(t), ˆx (t) ),η (t) + x L ( t, ˆx(t), ˆx (t) ),η(t) ] dt 0.

12 A. Ferriero, E.M. Marchini / J. Math. Anal. Appl. 304 (2005) (1) Fix η. Through the same steps as in point (1) of the proof of Theorem 2.1, for every n N we can define a closed set C n such that on it η is continuous, ˆx is continuous with values in G, ξ L is continuous in C n R s G lim n + m( \ C n ) = 0. n particular, it follows that there are constants k n c n > 0 such that, for all t C n, ξ L ( t, ˆx(t), ˆx (t) ) <k n η (t) <c n. Define v n, B n, η n η n as in the proof of Corollary 3.1. Since, for all t, η n (t) 1 + η (t), it follows that η n 1 (b a) + η 1. Moreover, η n η n 1 (b a) + η 1. (2) As in point (3) of the proof of Theorem 2.1, there exists δ n > 0 such that for (t,x,ξ) C n R s G, with d((t, x, ξ), (t, ˆx(t), ˆx (t))) < δ n,wehave ξ L(t, x, ξ) < k n + 1. (3) For < min{1/((b a)+ η 1 ), δ n /((b a)+ η 1 ), δ n /(c n + 1)}, consider the integrals L t, ˆx(t) + ηn (t), ˆx (t) + η n (t)) L ( t, ˆx(t), ˆx (t) )] dt = = + C n L t, ˆx(t) + ηn (t), ˆx (t) + η n (t)) L ( t, ˆx(t) + η n (t), ˆx (t) )] dt L t, ˆx(t) + ηn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt L t, ˆx(t) + ηn (t), ˆx (t) + η n (t)) L ( t, ˆx(t) + η n (t), ˆx (t) )] dt + L t, ˆx(t) + ηn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt. For every t C n, there exists ζ (t) (0,)such that L t, ˆx(t) + ηn (t), ˆx (t) + η n (t)) L ( t, ˆx(t) + η n (t), ˆx (t) )] = ξ L ( t, ˆx(t) + η n (t), ˆx (t) + ζ (t)η n (t)),η n (t) ξ L ( t, ˆx(t) + η n (t), ˆx (t) + ζ (t)η n (t)) c n <(k n + 1)c n. Hence, we can apply the dominated convergence theorem to obtain that lim L t, ˆx(t) + ηn (t), ˆx (t) + η n 0 (t)) L ( t, ˆx(t) + η n (t), ˆx (t) )] dt = ξ L ( t, ˆx(t), ˆx (t) ),η n (t) dt = ξ L ( t, ˆx(t), ˆx (t) ),η n (t) dt. C n (4) Following the point (5) of Theorem 2.1, we obtain that

13 368 A. Ferriero, E.M. Marchini / J. Math. Anal. Appl. 304 (2005) lim sup 0 L t, ˆx(t) + ηn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt x L ( t, ˆx(t), ˆx (t) ),η n (t) dt, since ˆx is a minimizer, we have 0 ξ L ( t, ˆx(t), ˆx (t) ),η n (t) + lim sup L t, ˆx(t) + ηn (t), ˆx (t) ) L ( t, ˆx(t), ˆx (t) )] dt 0 [ ξ L ( t, ˆx(t), ˆx (t) ),η n (t) + x L ( t, ˆx(t), ˆx (t) ),η n (t) ] dt. (5) Finally we have lim n + η n (t) = η (t) ξ L ( t, ˆx(t), ˆx (t) ) η n (t) ξ L (, ˆx( ), ˆx ( ) ) ( 1 + η (t) ), lim n + η n(t) = η(t) x L ( t, ˆx(t), ˆx (t) ) η n (t) x L ( t, ˆx(t), ˆx (t) ) ( (b a) + η 1 ), so that, by dominated convergence, we obtain lim ξ L ( t, ˆx(t), ˆx (t) ),η n (t) dt = ξ L ( t, ˆx(t), ˆx (t) ),η (t) dt n + lim n + x L ( t, ˆx(t), ˆx (t) ),η n (t) dt = x L ( t, ˆx(t), ˆx (t) ),η(t) dt. Hence, it follows that [ ξ L ( t, ˆx(t), ˆx (t) ),η (t) + x L ( t, ˆx(t), ˆx (t) ),η(t) ] dt 0. The absolute continuity of t ξ L(t, ˆx(t), ˆx (t)) is classical (e.g., [2]).

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