RELAXATION AND REGULARITY IN THE CALCULUS OF VARIATIONS

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1 RELAXATION AND REGULARITY IN THE CALCULUS OF VARIATIONS ALESSANDRO FERRIERO Abstract. In this work we prove that, if L(t, u, ξ) is a continuous function in t and u, Borel measurable in ξ, with bounded non-convex pieces in ξ, then any absolutely continuous solution ū to the variational problem { b } min L(t, u(t), u(t))dt : u W, 0 (a, b) a is quasi-regular in the sense of Tonelli, i.e. ū is locally Lipschitz on an open set of full measure of [a, b], under the further assumption that either L is Lipschitz continuous in u, locally uniformly in ξ, but not necessarily in t, or L is invariant under a group of C transformations (as in the Noether s theroem). Without one of those further assumptions the solution could be not regular as shown by a recent example in [5]; our result is then optimal in this sense. Moreover, we improve the standard hypothesis used so far [2, 6, 7, 8, 7] which have been the Lipschitz continuity of L in u, locally uniform in ξ and t, and some growth condition in ξ. We also show that the relaxed and the original problem have the same solutions (without assuming any of the two further assumptions above). This extends a result in [6] to the non-autonomous case. Keywords: variational problems, regularity, relaxation, quasilinear differential equations. AMS: 49B05, 49A05, 49C05, 35J20.. Introduction A solution ū in the Sobolev space W, 0 (a, b) to the variational problem { } b min I(u) := L(t, u(t), u(t))dt : u W, 0 (a, b) a is said to be quasi-regular in the sense of Tonelli if ū is locally Lipschitz on an open set of full measure of [a, b], while ū is said to be regular in the sense of Tonelli if its derivative ū is extended-value continuous on [a, b]. Clearly, since ū belongs to (a, b), if ū is regular in the sense of Tonelli, then ū is quasi-regular. In this paper, we shall deal with the quasi-tonelli partial regularity after having faced in two recent works [2, 3] the Tonelli partial regularity. We shall show that, if L(t, u, ξ) is continuous in (t, u) in ([a, b] \ Σ L ) R d, for every ξ in R d, with Σ L a closed set of zero measure in [a, b], is affine minorized, has bounded non-convex pieces in ξ and either (H) L is Lipschitz continuous in u, locally uniformly in ξ, but not necessarily in t, or (H2) L is invariant under a group of C transformations, then, the solutions to the associated variational problem are quasi-regular in the sense of Tonelli. W, 0 Date: Received: date / Revised version: date.

2 2 ALESSANDRO FERRIERO The precise meaning of bounded non-convex pieces is given by the bounded intersection property of Cellina [3]: a function f : R d R has the bounded intersection property if, for every ξ in R d, there exists p in the subgradient of its convexified function f at ξ such that the set in which f is affine near ξ, i.e. {w R d : f (w) = f (ξ) + p, w ξ }, is bounded. It is clear that the bounded intersection property is an optimal condition for the quasi-tonelli partial regularity of the solutions. Indeed, the null Lagrangian L(t, u, ξ) := p, ξ, for any given vector p in R d, admits solutions which have not that regularity. Our Lipschitz condition (H) on L in u is precisely the following: for each R > 0, there exists an integrable function C R : [a, b] R + such that () L(t, u, ξ) L(t, v, ξ) C R (t) u v, for a.e. t in [a, b] and every vector u, v, ξ in R d with modulus smaller than R. This condition is optimal as shown by a recent example in [5]. Up to now [2, 6, 7, 8, 7], the Lipschitz continuity of L in u, locally uniformly in t and ξ, and the superlinearity in ξ have been the weakest conditions assumed for proving the quasi-tonelli partial regularity. In two recent works [2, 3], we deal with the convex case proving a stronger regularity than the one presented here. The main theorems in [2, 3] states that, if L is strictly convex in ξ, then any absolutely continuous solution ū is regular in the sense of Tonelli. In [3], we assume hypothesis (H) meanwhile in [2] we assume (H2). The condition (H2) on the symmetries of L is particularly meaningful because, as established in the fundamental theorem of E. Noether [4], each invariant for L is strictly related to a first integral of the system. As a corollary, we obtain, for the autonomous case, i.e. L(t, u, ξ) = L(u, ξ), the quasi-tonelli partial regularity result without assuming any regularity of L in u (more than continuity). We present also two new results concerning the relaxation of variational problems which will be used to prove our main theorem. Namely, for L only continuous in t and u, and Borel measurable in ξ, we prove that any absolutely continuous solution ū is such that L(t, ū(t), ) is convex at ū(t), i.e. L(t, ū(t), ū(t)) = L (t, ū(t), ū(t)), for almost every t in (a, b), and that, if L is affine minorized and has the bounded intersection property in ξ, then the infimum of the relaxed functional, i.e. the problem with L replaced by its convexified in ξ, coincides with the infimum of the original one (see [3, 8, 0, 6] for previous results of this kind). In [6], the authors prove a similar result in the autonomous case. As last comment, we would like to point out a minor difference in the assumption on L(t, u, ξ) with respect to t between the present work and [8]. In [8], L is assumed to be Borel and locally bounded in t in [a, b], meanwhile here we assume L to be continuous in t in [a, b] minus a closed set of zero measure Σ L. Also, differently than in [8], we deal with vector-value functions u : (a, b) R d, d. The paper is organized as follows. In section 2, we show our relaxation theorem. In section 3, we present the main result concerning the quasi-tonelli partial regularity; the proof is based on the ideas contained in our works [2, 3]. 2. Relaxation Result In the present paper we shall deal with Lagrangian functions L(t, u, ξ) : ([a, b] \ Σ L ) R d R d R,

3 RELAXATION AND REGULARITY IN THE CALCULUS OF VARIATIONS 3 where Σ L is a closed set of zero Lebesgue measure in [a, b], which are continuous with respect to t and u, for a.e. ξ in R d, and Borel measurable in ξ, for every (t, u) in ([a, b] \ Σ L ) R d. The variational problem we are interested in is { } b (2) min I(u) := L(t, u(t), u(t))dt : u W, 0 (a, b), a where W, 0 (a, b) denotes as usual the Sobolev space of functions u from [a, b] to R d, d, with zero boundary conditions. The case of general Dirichlet boundary condition u(a) = A, u(b) = B, i.e. r + (a, b) as space of admissible functions, where r(t) := (t a)(b A)/(b a)+a, can be deduced by the zero boundary values case (preserving the assumptions on L) by considering the modified Lagrangian L(t, u, ξ) := L(t, u r(t), ξ ṙ(t)). For any solution ū in W, 0 (a, b) to I corresponds a solution ũ = r + ū in r + W, 0 (a, b) to the functional Ĩ associated to the Lagrangian L, and conversely. Throughout all the paper, {o n ()} denotes any sequence which converges to 0, as n goes to, and B(ξ; R) is the closed ball of R d with center ξ and radius R. We say that a function f from R d to R is convex at ξ R d if W, 0 λf(ξ ) + ( λ)f(ξ 2 ) f(ξ), for every λ in [0, ] and every ξ, ξ 2 in R d such that λξ + ( λ)ξ 2 = ξ. Lemma. If ū in W, 0 (a, b) is a solution to (2) with I(ū) finite, then L(t, ū(t), ) is convex at ū(t), for a.e. t in (a, b). Proof. Since ū in W, 0 (a, b) is a finite minimum for I, there exists a set E (a, b) of full measure such that every point t of E is a Lebesgue point for ū and for L(, ū, ū). Suppose that, contrarily to what we state, there exists a point t 0 in E such that L(t 0, ū(t 0 ), ) is not convex at ū(t 0 ). That is, there exist c > 0, d + vectors ξ,, ξ d+ in R d which have d-dimensional convex hull, and λ,, λ d+ in [0, ], such that λ + + λ d+ =, λ ξ + + λ d+ ξd+ = ū(t 0 ) and (3) c + λ L(t 0, ū(t 0 ), ξ ) + + λ d+ L(t 0, ū(t 0 ), ξ d+ ) L(t 0, ū(t 0 ), ū(t 0 )). Consider the competitor w n in W, 0 w n (t) := t a (a, b) defined by d+ ẇ n (τ)dτ, ẇ n (t) := ū(t)χ [a,b]\in (t) + j= ξ j χ I j n (t), where I n := (t 0 /n, t 0 + /n), In is the interval (t 0 /n, t 0 /n + λ n I n ), for j 2, In j is the interval [t 0 /n+[λ n + +λ n j ] I n, t 0 /n+[λ n + +λ n j ] I n ), and λ n j in [0, ] are such that λn + + λ n d+ =, λ n ξ + + λ n d+ ξ d+ = I n I n ū(t)dt (this is possible even in the case that ū(t 0 ) belongs to the boundary of the convex hull of { ξ,, ξ d+ } by replacing one of the ξ j with its symmetric with respect to the hyper-plane spanned by ξ,, ξ j, ξ j+,, ξ d+, and by possibly passing to a subsequence in n).

4 4 ALESSANDRO FERRIERO Notice that In j are contained in I n and In j = λ n j I n = λ n j 2/n. Moreover, since t 0 is a Lebesgue point for ū, the average of ū over I n converges to ū(t 0 ) and we can choose λ n j such that (4) lim n λn j = λ j, for every j =,..., d +. By definition, w n coincides with ū on [a, b]\i n and w n ū(t 0 ) L (I n) 0. Hence, since t 0 is a Lebesgue point for L(t, ū, ū), and from (4), we infer that L(t, ū(t), ū(t))dt = I n [L(t 0, ū(t 0 ), ū(t 0 )) + o n ()] I n I n [λ L(t 0, ū(t 0 ), ξ ) + + λ d+ L(t 0, ū(t 0 ), ξ d+ ) + c + o n ()] = I n [λ n L(t 0, ū(t 0 ), ξ ) + + λ n d+l(t 0, ū(t 0 ), ξ d+ ) + c + o n ()] = L(t, w n (t), ẇ n (t)) + I n [c + o n ()]. I n Let n be such that o n () < c. We conclude that I(w n ) < I(ū), in contradiction with the minimality of ū. For any t and u, let L (t, u, ) be the convexified function of L(t, u, ), i.e. the maximal convex function smaller then L(t, u, ), and by I the associated functional. If L is minorized by an affine function, then L is well-defined. We say that a function f : R d R has the bounded intersection property if, for every ξ in R d, there exists p in the subgradient of f at ξ, i.e. p f (ξ), such that the set {w R d : f (w) = f (ξ) + p, w ξ } is bounded. The above definition is due to Cellina [3]. Roughly speaking, it says that if f has the bounded intersection property, then the affine pieces of f (which part of them correspond to the pieces where f is non-convex) have bounded domain in R d. Following a standard procedure for proving relaxation results, in the following lemma we show that the value of the action functional at any w can be approximated by the value of its convexified functional computed at an appropriate modified competitor u. Lemma 2. Assume that L(t, u, ξ) is affine minorized and has the bounded intersection property in ξ, for every (t, u) in ([a, b] \ Σ L ) R d. If u in W, 0 (a, b) is such that I (u) is finite, then, for every ɛ > 0, there exists w in W, 0 (a, b) with I(w) I (u) + ɛ. Proof. Fix ɛ > 0. Since I (u) is finite, there exists a set E (a, b) of full measure such that every point t of E is a Lebesgue point for u and for L (, u, u). For every t in E, by Theorem in [3], there exist ξ (t),, ξ d+ (t) in R d which have d-dimensional convex hull and λ (t),, λ d+ (t) in [0, ] such that λ (t) + + λ d+ (t) =, L(t, u(t), ξ j (t)) = L (t, u(t), ξ j (t)), i.e. L(t, u(t), ) is convex at ξ j (t), for every j =,, d +, and (5) λ (t)ξ (t) + + λ d+ (t)ξ d+ (t) = u(t), λ (t)l(t, u(t), ξ (t)) + + λ d+ (t)l(t, u(t), ξ d+ (t)) = L (t, u(t), u(t)).

5 RELAXATION AND REGULARITY IN THE CALCULUS OF VARIATIONS 5 For any integer n, let K n be the family of open intervals I k (t) := (t /k, t+/k) where t varies in E and k n (t) is such that, for every k k n (t), I k (t) (a, b), u(τ) u(t) dτ o n (), I k (t) I k (t) (6) L (τ, u(τ), u(τ)) L (t, u(t), u(t)) dτ o n (), I k (t) I k (t) L(τ, u, ξ) L(t, u(t), ξ) o n (), (τ, u, ξ) I k (t) B(u(t); [max ξ j (t) + u(t) ]2/k) B(0; max ξ j (t) ). j j The family K n covers E in the Vitali sense [8]. Hence, by the Vitali covering lemma, there is a sequence of disjoint intervals {I ki (t i )} K n such that (7) E I ki (t i ) = 0. i= We can suppose, without loss of generality, that λ (t i ) /(d + ), for every i. Consider the competitor w n in W, 0 (a, b) defined by t d+ w n (t) := ẇ n (τ)dτ, ẇ n (t) := ξ j (t i )χ I j,n(t). i a i= j= where I,n i is the interval (t i /k i, t i /k i + λ n (t i )2/k i ), for j 2, I j,n i is the interval [t i /k i +[λ n (t i )+ +λ n j (t i)]2/k i, t i /k i +[λ n (t i )+ +λ n j (t i)]2/k i ) and λ n (t i ),, λ n d+ (t i) in [0, ] are such that λ n (t i ) + + λ n d+ (t i) =, λ n (t i )ξ (t i ) + + λ n d+(t i )ξ d+ (t i ) = I ki (t i ) I ki (t i) u(t)dt. Similarly than in the proof of Lemma, we can choose λ n j (t i) such that (8) lim n λn j (t i ) = λ j (t i ), for every j =,..., d +, i N. By (6), λ n j (t i) λ j (t i ) = o n (), uniformly in i and j. Notice that I j,n i are contained in I ki (t i ) and I j,n i = λ n j (t i) I ki (t i ) = λ n j (t i)2/k i. By (8) and (6), w n = u on I ki (t i ) and w n u(t i ) L (I ki (t i)) [max j ξ j (t i ) + u(t i ) ]2/k i, for every i. Therefore, by applying in order (7), (6) 2, (5) 2, (8) and (6) 3, we obtain I (u) = = = = i= i= I ki (t i) I ki (t i) L (τ, u(τ), u(τ))dτ L (t i, u(t i ), u(t i ))dτ + o n () d+ I ki (t i ) λ j (t i )L(t i, u(t i ), ξ j (t i )) + o n () i= d+ i= j= I j,n i j= Let n be such that o n () ɛ. proof. L(t i, u(t i ), ξ j (t i ))dτ + o n () = I(w n ) + o n (). Hence, I(w) I (u) + ɛ. This concludes the

6 6 ALESSANDRO FERRIERO Our relaxation result is a corollary of the two lemmas above. Theorem 3. Assume that L(t, u, ξ) is affine minorized and has the bounded intersection property in ξ, for every (t, u) in ([a, b] \ Σ L ) R d. If inf{i (u) : u W, 0 (a, b)}, then, inf{i (u) : u W, 0 (a, b)} = inf{i(u) : u W, 0 (a, b)}. Moreover, if ū in W, 0 (a, b) is a solution to (2), then, ū is also a solution of the convexified problem and I (ū) = I(ū). Proof. By definition of convexified function, inf{i (u) : u W, 0 (a, b)} inf{i(u) : u W,(a, b)}. If inf{i (u) : u W, 0 (a, b)} =, then the result is trivially true. Suppose thus that the infimum of I is finite. Let {u n } be a minimizing sequence for I. By Lemma 2, there exists {w n } in W, 0 (a, b) such that I(w n) I (u n )+ /n. Hence, from inf{i (u) : u W, 0 (a, b)} = lim n I (u n ) + /n lim inf I(w n) inf{i(u) : u W, n 0 (a, b)}, we infer the stated equality. Let ū in W, 0 (a, b) be a solution to (2). Then, by Lemma, I (ū) = I(ū) and since, as we have just proved, the convexified functional has the same infimum has the original one, we deduce that ū is also a solution of the convexified problem. If we assume that L is affine minorized uniformly in t and u, then inf{i (u) : u W, 0 (a, b)} =. If we know that (2) has a solution, then assuming that L is affine minorized locally uniformly in t and u is enough for having inf{i (u) : u W, 0 (a, b)}. Lemma 2 and Theorem 3 are more general than Theorem 3 in [3] and of Lemma 2.6 and Theorem 2.7 in [8]. Here we deal with Lagrangians which might be nonautonomus, without requiring any growth condition on L nor continuity on L. We replace the growth condition on L by the geometrical condition expressed by the bounded intersection property. 3. Main Result In what follows, R d denotes the one point compactification R d { } of R d and Lip c (G), G an open set of [a, b] \ Σ L, is the space of locally Lipschitz functions on G. A Lagrangian L(t, u, ξ) is affine minorized in ξ, locally uniformly in (t, u) in ([a, b] \ Σ L ) R d, if, for any compact set K of ([a, b] \ Σ L ) R d, there exist p in R d, β 0 such that L(t, u, ξ) p, ξ β, for every (t, u, ξ) in K R d. We say that L(t, u, ξ) has the bounded intersection property in ξ, locally uniformly in (t, u) in ([a, b] \ Σ L ) R d, if, for any compact set K of ([a, b] \ Σ L ) R d, for every ξ in R d, there exists p(t, u) ξ L (t, u, ξ), such that the set {A(t, u, ξ) : (t, u) K}, where A(t, u, ξ) := {w R d : L (t, u, w) = L (t, u, ξ) + p(t, u), w ξ } is the affine piece of L (t, u, ) at ξ, is bounded. Notice that this bound depends only on K and not on t, u. 0

7 RELAXATION AND REGULARITY IN THE CALCULUS OF VARIATIONS 7 In our main theorem we shall use one of the following assumptions: (H) for each R > 0, there exists an integrable function C R : [a, b] R + such that (9) L(t, u, ξ) L(t, v, ξ) C R (t) u v, for a.e. t in [a, b] and every vector u, v, ξ in R d with modulus smaller than R; (H2) L is invariant under a group of C transformations (τ x (t), φ x (u)) : [a, b] R d [a, b] R d, with x [, ], x R d, x, such that (τ 0 (t), φ 0 (u)) = (t, u) and [ x τ x (t)] x=0 + [ xi φ x i (u)] x=0 0, for every t, u and i =,..., d. That is, for arbitrary x, x, t 0 < t in [a, b] and u W, (a, b), (0) τ x (t ) τ x (t 0) L(τ, φ x (u(t x (τ))), 0 d dτ φx (u(t x (τ))))dτ = t t 0 L(t, u(t), u(t))dt, where t x is the inverse function of τ x. (Notice that the inverse function t x exists for x small enough by the C regularity of τ x in x and since τ 0 is the identity. Hence, without loss of generality, by rescaling the parametrization of τ x in x, we can assume that τ x admits inverse for every x in [, ]). Condition (0) is especially meaningful because, as established in the fundamental theorem of E. Noether [4], each invariant for L yields a first integral for the system. Nevertheless, in this work we will not make explicitly use of these underline first integrals. Two interesting examples for the invariance of L are, first, the autonomous case L(u, ξ), for which (τ x (t), φ x (u)) = (t+x, u), and the corresponding conserved quantity is the energy of the system, and, second, Lagrangians which are independent on u, that is L(t, ξ), for which (τ x (t), φ x (u)) = (t, u + x), and the corresponding conserved quantity is the total momentum. Lemma 4. If L(t, u, ξ) is affine minorized and has the bounded intersection property in ξ, locally uniformly in (t, u) in ([a, b] \ Σ L ) R d, then, L is a continuous function in t, u and ξ, affine minorized and with the bounded intersection property in ξ, locally uniformly in (t, u) in ([a, b] \ Σ L ) R d. Moreover, if L enjoys (H), then, L enjoys (H). Proof. Clearly, L is well-defined, affine minorized and with the bounded intersection property in ξ, locally uniformly in (t, u). To prove that L is continuous, let {(t n, u n )} be a sequence which converges to (t, u). By Theorem in [3], there exist ξ,, ξ d+ in R d and λ,, λ d+ in [0, ] such that λ ξ + + λ d+ ξ d+ = ξ, Then, λ L(t, u, ξ ) + + λ d+ L(t, u, ξ d+ ) = L (t, u, ξ). L (t, u, ξ) = lim n [λn L(t n, u n, ξ ) + + λ n d+l(t n, u n, ξ d+ )] lim inf n [λn L (t n, u n, ξ ) + + λ n d+l (t n, u n, ξ d+ )] lim inf n L (t n, u n, ξ). Besides, by Theorem in [3], there exists ξ n,, ξ n d+ in Rd and λ n,, λ n d+ in [0, ] (which might be different from the ones above) such that λ n ξ n + + λ n d+ξ n d+ = ξ, λ n L(t n, u n, ξ n ) + + λ n d+l(t n, u n, ξ n d+) = L (t n, u n, ξ).

8 8 ALESSANDRO FERRIERO By the local uniform bounded intersection property, the {ξ n },, {ξd+ n } are bounded. Thus, there exists ξ,, ξ d+ in R d and λ,, λ d+ in [0, ] such that λ n j λ j, ξj n ξ j, for every j =,, d +, and λ ξ + + λ d+ ξd+ = ξ. Therefore, lim inf n L (t n, u n, ξ) = λ L(t, u, ξ ) + + λ d+ L(t, u, ξ d+ ) λ L (t, u, ξ ) + + λ d+ L (t, u, ξ d+ ) L (t, u, ξ). By the two inequalities above, we obtain that L (t, u, ξ) = lim inf L (t n, u n, ξ) and, by the arbitrariness of the sequence {(t n, u n )}, we conclude that L is continuous at (t, u). Since L is convex in ξ and, hence, continuous in ξ, then L is continuous in all its variables. If L enjoys (H), then, for every t in ([a, b]\σ L ) R d and every u, v, ξ in B(0; R), L(t, v, ξ) L(t, u, ξ) C R (t) v u. Since L has the bounded intersection property in ξ, locally uniformly in (t, u), the convexified L is determined locally in ξ, and, hence, for every u, v, ξ in B(0; R), L (t, v, ξ) L (t, u, ξ) C R (t) v u. By changing u with v, we conclude that L (t, v, ξ) L (t, u, ξ) C R (t) v u. Definition 5. We say that a solution ū in W, 0 (a, b) to (2) is quasi-regular in the sense of Tonelli if there exists a zero measure closed set Z of [a, b] such that ū belongs to Lip c ([a, b] \ Z). We define the oscillation of a function w in L (a, b) at t 0 as osc t0 w := lim ɛ 0 esssup{ w(t) w(τ) : t B(t 0 ; ɛ), τ B(t 0 ; ɛ)}. Our main result is the following. Theorem 6. If L(t, u, ξ) is affine minorized, has the bounded intersection property in ξ, locally uniformly in (t, u) in ([a, b] \ Σ L ) R d and either (H) or (H2) holds, then, any solution ū in W, 0 (a, b) to the variational problem (2) is quasi-regular in the sense of Tonelli. Namely, the set t0+ɛ E := {t [a, b] \ Σ L : lim sup ɛ 0 2ɛ ū(t)dt < } is an open set of full measure such that ū is locally bounded on E and, for any t 0 in E, there exists ξ(t 0 ) in R d, ξ(t 0 ) = ū(t 0 ) at any Lebesgue point t 0 of ū, such that t 0 ɛ osc t0 ū diam A(t 0, ū(t 0 ), ξ(t 0 )). Moreover, if d 2, for any t 0 in [a, b] \ (Σ L E), lim ū(t) =, t t 0 and, if d =, there exist two disjoint closed set Z + and Z in [a, b] \ (Σ L E) such that, for any t 0 in Z ±, lim ū(t) = ±. t t 0

9 RELAXATION AND REGULARITY IN THE CALCULUS OF VARIATIONS 9 Proof. The proof is similar to the main theorems in [2] and [3]. We explain here the main differences. Let ū in W, 0 (a, b) be a solutions to (2). By Theorem 3 and Lemma 4, we can assume, by replacing L with its convexified function L, that L is also convex and, hence, continuous in ξ. Moreover, if L enjoys (H), then, L enjoys (H) too while if L enjoys (H2) then L does not enjoy (H2) anymore. Nevertheless, by Lemma and since L L, L enjoys (H2) with an inequality which is enough for proving our result. Part. Let t 0 be any point in [a, b) \ Σ L, I(ɛ) be the closed interval [t 0, t 0 + ɛ], and fix C to be a compact set of [a, b] \ Σ L big enough to contain t 0 as interior point. We claim that either lim ū(t)dt ɛ 0 ɛ = or ū(t)dt A(t 0, ū(t 0 ), ɛ ξ) + o ɛ (), I(ɛ) for a vector ξ in R d. Observe that the claim is true at a.e. point t 0 in [a, b) \ Σ L, that is at the Lebesgue points of ū. What we claim is that this is true for all points of [a, b) \ Σ L. Suppose, on the contrary, that there exist two sequences of positive numbers {ɛ n}, {ɛ 2 n}, ɛ 2 n+ < ɛ n < ɛ 2 n, for every n, which converge to 0, such that lim n lim n ɛ n I(ɛ) ū(t)dt =: ξ A(t 0, ū(t 0 ), ξ ), I(ɛ n ) ū(t)dt I(ɛ 2 n ) = or lim ū(t)dt =: ξ 2 A(t 0, ū(t 0 ), ξ 2 ), n I(ɛ 2 n ) ɛ 2 n with A(t 0, ū(t 0 ), ξ ) A(t 0, ū(t 0 ), ξ 2 ). By the continuity of the integral function M(I(ɛ)) := ū(t)dt, for ɛ in [ɛ I(ɛ) n, ɛ 2 n], I(ɛ) ɛ 2 n and by the bounded intersection property of L, we can suppose, by moving ɛ 2 n closer to ɛ n, that lim ū(t)dt =: ξ 2 A(t 0, ū(t 0 ), ξ 2 ) n I(ɛ 2 n ) with ɛ 2 n () A(t 0, ū(t 0 ), ξ ) A(t 0, ū(t 0 ), ξ 2 ). We can also suppose, by replacing L(t, u, ξ) with α[l(t, u, ξ) + p, ξ ξ ] + β, for suitable α, β 0 and p in R d, that (2) L(t, u, ξ) ξ, for any (t, u, ξ) in C B(0; ū L (a,b) + ) R d (see Lemma 7). If (H) holds, then we proceed as in the proof of Theorem 3 in [3]. Namely, one can shows that ū(i(ɛ 2 n)) \ A(t 0, ū(t 0 ), ξ 2 ) converges to 0 and then prove the three cases. If (H2) holds, then we proceed as in the proof of Theorem 3 in [2] observing that the invariant property does not holds anymore with the equality for the convexified

10 0 ALESSANDRO FERRIERO Lagrangian but, by Lemma, we have τ x (t ) τ x (t 0) L(τ, φ x (ū(t x (τ))), d dτ φx (ū(t x (τ))))dτ t t 0 L(t, ū(t), ū(t))dt, for every t 0 < t in [a, b], which yields the good inequality for the proof. In both cases, one obtains the claim at the beginning of the proof, that is, either lim ū(t)dt ɛ 0 ɛ = or ū(t)dt A(t 0, ū(t 0 ), ɛ ξ) + o ɛ (). I(ɛ) Analogously, one can prove the same for the closed left interval [t 0 ɛ, t 0 ], ɛ > 0, and, also, that the limit of the right and the left averages of ū at t 0 must be either both or both belong to the same affine piece of L, i.e. A(t 0, ū(t 0 ), ξ) = A(t 0, ū(t 0 ), ξ ). Part 2. Let {t n } [a, b] \ Σ L be a sequence of Lebesgue points of ū which converges to t 0. We claim that, either I(ɛ) lim ū(t n ) = or ū(t n ) A(t 0, ū(t 0 ), ξ) + o n () n accordingly to the behavior of the averages of ū at t 0 described in Part, where ξ is defined as in Part. Indeed, suppose that this is not true. Assume also for simplicity that t n < t 0, for every n (the other case t n > t 0, for every n, can be proved analogously). Being t n a Lebesgue point for ū, there exists ɛ n in (0, ɛ 2 n) such that (3) ɛ n tn+ɛ n t n ū(t)dt = ū(t n ) + o n (). Proceeding similarly as in Part (replacing I(ɛ n) with [t n, t n + ɛ n] and I(ɛ 2 n) with [t n, t 0 ]), one can define a competitor with smaller value of the action than the value of the action at ū and thus reach a contradiction which implies the claim. We have therefore obtained by Part and Part 2 that the set E := {t [a, b] \ Σ L : lim sup ɛ 0 2ɛ t0+ɛ t 0 ɛ ū(t)dt < } is an open set of full measure (since it contains the Lebesgue points of ū). For the same reason, for any t 0 in [a, b] \ (Σ L E), lim ū(t) =. t t 0 Moreover, ū is locally bounded on E since, for any t 0 in E, ξ = ξ(t 0 ) defined as in Part is such that ū(t) belongs A(t 0, ū(t 0 ), ξ(t 0 )), as t converges to t 0. Thus, osc t0 ū diam A(t 0, ū(t 0 ), ξ(t 0 )). This concludes the proof for d 2. If d =, using that any continuous path in R that goes from ± to is forced to pass by an element of R (which is not the case in the vectorial case), one can prove that, for any t 0 in [a, b] \ (Σ L E), either lim ū(t) = + or lim ū(t) =. t t0 t t0 Hence, setting Z + and Z in [a, b] \ (Σ L E) accordingly to the value of this limit, we obtain the stated result.

11 RELAXATION AND REGULARITY IN THE CALCULUS OF VARIATIONS From Theorem 6, the set Z := Σ L E is such that ū belongs to Lip c ([a, b] \ Z). Remark. The bounded intersection property is clearly an optimal condition for the quasi-tonelli partial regularity of the solutions to (2) as the null Lagrangian example L(t, u, ξ) := p, ξ, for any given vector p in R d, shows. We prove here a lemma used in the proof of Theorem 6. Lemma 7. If L(t, u, ξ) is convex and has the bounded intersection property in ξ, for every (t, u) in ([a, b] \ Σ L ) R d R d, then, for every (t 0, u 0 ) in ([a, b] \ Σ L ) R d there exist a compact set K := ([t 0 r, t 0 + r] [a, b]) B(u 0 ; R), α > 0, β 0 and p in R d such that for every (t, u, ξ) in K R d. L(t, u, ξ) p, ξ α ξ β, Proof. By the bounded intersection property of L, there is p in ξ L(t 0, u 0, 0) such that the set {ξ R d : L(t 0, u 0, ξ) = L(t 0, u 0, 0) + p, ξ } is bounded. Therefore, by the convexity of L in ξ, there exists ρ such that L(t 0, u 0, ξ) L(t 0, u 0, 0) p, ξ > 0, for every ξ ρ. By the continuity of L, there exist a compact set K := ([t 0 r, t 0 + r] [a, b]) B(u 0 ; R) and α > 0 such that, for every (t, u, ξ) K {ξ R d : ξ = ρ}, L(t, u, ξ) L(t, u, 0) p, ξ ρ α. Since, by the convexity of L in ξ, [L(t, u, ξ) L(t, u, 0)]/ ξ [L(t, u, ξρ/ ξ ) L(t, u, 0)]/ρ, it follows that L(t, u, ξ) L(t, u, 0) p, ξ ξ L(t, u, ξρ/ ξ ) L(t, u, 0) p, ξρ/ ξ ρ α, for every (t, u, ξ) K {ξ R d : ξ ρ}. Setting β to be the maximum of L(t, u, ξ) p, ξ α ξ + L(t, u, 0), for (t, u, ξ) K {ξ R d : ξ ρ}, we conclude that for every (t, u, ξ) K R d. L(t, u, ξ) p, ξ α ξ β, Remark. Recall that a function f : R d R has superlinear growth if there exists a continuous function θ such that θ(ξ)/ ξ, as ξ, and f(ξ) θ(ξ), for every ξ in R d. In [3, 4] it has been introduced a slower growth condition which, roughly speaking, describes all the cases in which L cannot be approximated linearly at infinity. The superlinear growth implies this slow growth condition which implies the bounded intersection property. Acknowledgement The author wishes to thank the Spanish Ministry of Science and Innovation (MICINN) for supporting the research contained in this work.

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