A relaxation result for autonomous integral functionals with discontinuous non-coercive integrand

April 2004, Vol. 10, 201–210 DOI: 10.1051/cocv:2004004

A RELAXATION RESULT FOR AUTONOMOUS INTEGRAL FUNCTIONALS WITH DISCONTINUOUS NON-COERCIVE INTEGRAND

Carlo Mariconda

and Giulia Treu

Abstract. LetL:R^N×R^N→Rbe a Borelian function and consider the following problems inf

F(y) =

_b

a L(y(t), y(t)) dt: y∈AC([a, b],R^N), y(a) =A, y(b) =B

(P) inf

F^∗∗(y) = _b

a L^∗∗(y(t), y(t)) dt: y∈AC([a, b],R^N), y(a) =A, y(b) =B

· (P^∗∗) We give a sufficient condition, weaker then superlinearity, under which infF = infF^∗∗ if L is just continuous inx. We then extend a result of Cellina on the Lipschitz regularity of the minima of (P) whenLis not superlinear.

Mathematics Subject Classification. 37N35.

Received June 27, 2003.

1. Introduction

We consider the relationships between the problems inf

F(y) =

_b

a

L(y(t), y(t)) dt: y∈AC

[a, b],R^N

, y(a) =A, y(b) =B

(P)

inf

F^∗∗(y) = _b

a

L^∗∗(y(t), y(t)) dt: y∈AC

[a, b],R^N

, y(a) =A, y(b) =B

· (P^∗∗)

It is well known that infF = infF^∗∗ ifL is super-linear and continuous. Recently Cellina in [5] proved that the same conclusion holds true assuming, instead of superlinearity, a weaker growth condition that we will call (GA). Roughly, a convex functionL(x, ξ) satisfies (GA) if the intersection of the supporting hyperplane to its epigraph at (ξ, L(x, ξ)) with the ordinate axis tends to−∞as |ξ|tends to +∞, uniformly with respect to xin compact sets. This condition implies, but is not equivalent to, a sort of conical growth: we say that L

Keywords and phrases. Lipschitz, regularity, non-coercive, discontinuous, calculus of variations.

1 Dipartimento di Matematica pura e applicata, Universit`a di Padova, 7 via Belzoni, 35131 Padova, Italy;

e-mail:[email protected]; [email protected]

c EDP Sciences, SMAI 2004

satisfies (CGA) if for everyξ₀ there existε, R >0 such that, for every|ξ| ≥R,

L(x, ξ)≥L(x, ξ₀) +p(x, ξ₀)·(ξ−ξ₀) +ε|ξ|+ const. (CGA) wheneverxbelongs to a prescribed compact set andp(x, ξ₀) belongs to the subdifferential ofξ→L(x, ξ) inξ₀. We weaken here the continuity assumption ofLin both variables and we prove that, ifL(x, ξ) is just continuous in xand satisfies (CGA), then infF = infF^∗∗.

The proof of the result is based on Theorem 3.2, a uniform approximation of the bipolar of a (discontinuous) function L(ξ) satisfying (CGA) in terms of the convex hull of the graph of L; this kind of result is classical whenLis supposed to be lower semi-continuous and superlinear inξ[7].

In the last part of the paper we are concerned with an application to the Lipschitz regularity of the minima of (P). It is well known that, ifL(x, ξ) is superlinear and convex inξ, then every minimizer of (P) is Lipschitz.

The same result was obtained recently by dropping some of the assumptions: no continuity and no convexity but superlinearity is assumed in [6], continuity, no convexity and assumption (GA) instead of superlinearity is assumed in [5], no continuity and no convexity but the requirement that every section λ→L(x, λu) (λ≥0,

|u|= 1) satisfies (GA) in [8], extending [6].

As a consequence of our relaxation result we prove that the minima of (P) are Lipschitz if L(x, ξ) is just continuous inxand satisfies (GA), thus extending the main result in [5].

We point out that there are several results concerning the representation of the lower semi-continuous envelope of integral functionals; we just mention [2, 4] for some recent results and references. Here we are interested in comparing the values of the infima of problems (P) and (P^∗∗) instead of establishing a representation formula.

2. Notation and preliminary results

In this paper|·|is the Euclidean norm and “·” the scalar product inR^N. For a functionL(x, ξ) :R^N×R^N →R we denote by L^∗∗(x, ξ) (resp. ∂L^∗∗(x, ξ)) the bipolar (resp. the subdifferential of the bipolar) of ξ→L(x, ξ).

Finally,AC([a, b],R^N) is the space of absolutely continuous functions on [a, b] with values in R^N.

HereL:R^N ×R^N −→Ris just a Borelian function. We assume moreover thatL^∗∗(x, ξ)=−∞for everyx andξ; this is the case, for instance, ifLis bounded below by an affine function of ξ.

The following growth condition will be assumed in the main result.

Conical growth assumption(CGA).For every compact subsetCofR^N andR₀≥0 there existε >0,R >0 andc∈Rsuch that

∀ξ∈R^N |ξ| ≥R L^∗∗(x, ξ)≥L^∗∗(x, ξ₀) +p(x, ξ₀)·(ξ−ξ₀) +ε|ξ|+c for everyx∈C,|ξ₀| ≤R₀ andp(x, ξ₀) in∂L^∗∗(x, ξ₀).

The following growth assumption was introduced by Cellina in [5] in the case whereL is continuous.

Growth assumption (GA).

We say thatLsatisfies (GA) if there existp(x, ξ) in∂L^∗∗(x, ξ) such that

|ξ|→+∞lim p(x, ξ)·ξ−L^∗∗(x, ξ) = +∞ (2.1) uniformly forxin a compact set.

Remark 2.1.

i) We point out that, in [5], the definition of (GA) is slightly different: it is formulated in an equivalent way in terms of the polar ofLin (x, p(x, ξ)); moreover the uniformity with respect to the first variable is not required since it is a consequence of the continuity ofL. We use it here since we drop the continuity assumption.

ii) Assumption (GA) is fulfilled if, for instance, L(x, ξ) is superlinear with respect toξ; the proof can be easily done following the lines of [5].

We refer to [8] for a survey on the properties of the functions that satisfy (GA).

Theorem 2.2. [8, Cor 4.4] Assume that L is bounded on compact sets and satisfies the Growth Assump- tion (GA). ThenL satisfies(CGA).

3. Relaxation

It is well known that if L : R^N → R is a function whose bipolar is finite, then, for every ε >0 and ξ in R^N, there exists ξ₁, ..., ξ_m (m ≤N + 1) in R^N and coefficients of a convex combination α₁, ..., α_m such that

iα_iL(ξ_i)≤L^∗∗(ξ) +εand

iα_iξ_i =ξ. We prove in the next Theorem 3.2 that ifLsatisfies (CGA) then, allowing m ≤ 2N + 2, the points ξ_i may be bounded uniformly with respect to ξ in compact sets. For this purpose we first quote, in a more general setting, a powerful consequence of (CGA) that was established in [5]

in the continuous case. For every (x, ξ) we set

L(x, ξ) = lim inf

η→ξ L(x, η),

i.e. L(x, ξ) denotes the lower semi-continuous envelope of the mapη→L(x, η). The proof of the following result is based on the fact that iff :R^N →Ris convex and satisfies (CGA) then the intersection of its epigraph with any supporting hyperplane is bounded. This condition is referred in [5] as theBounded Intersection Property.

Theorem 3.1. Assume thatL satisfies(CGA)and letp(x, ξ)∈∂L^∗∗(x, ξ). Then givenR₀>0and a compact subset C of R^N there existsR >0 (depending only on R₀ andC) such that for everyx∈C; for every ξ, with

|ξ| ≤R₀, there exist at most ν ≤N+1 points ξ_i, with|ξ_i| ≤R, and coefficients of a convex combination α_i, such that

ξ L^∗∗(x, ξ)

= ν i=1

α_i ξ_i L(x, ξ_i)

andL^∗∗(x, ξ) =L(x, ξ_i) =L^∗∗(x, ξ) +p(x, ξ)·(ξ−ξ_i).

Proof. It is enough to remark that Theorem 1 in [5] holds for functions that are lower semi-continuous instead of continuous and that the bipolar of a function coincides with the bipolar of its lower semi-continuous envelope.

We are now ready to state a version of Theorem 3.1 that does not involve the lower semi-continuous envelope ofL.

Theorem 3.2. Assume thatL(x, ξ)satisfies (CGA) and thatL is bounded on the compact sets. Then given R₀>0 and a compact subsetC of R^N, there existsR >0 (depending only on R₀ and C) such that for every x in C, for every ξ, with |ξ| ≤R₀ and ε > 0, there exist at most m≤ 2N + 2 points ξ_i, with |ξ_i| ≤R, and coefficients of a convex combinationλ_i, such that

ξ=_m

i=1λ_iξ_{i m}

i=1λ_iL(x, ξ_i)≤L^∗∗(x, ξ) +ε.

The proof of the result needs several preliminary steps. For the convenience of the reader we first give a sketch of the proof in the case whereLdoes not depend onx.

By Theorem 3.1, for|ξ| ≤R₀, the point (ξ, L^∗∗(ξ)) can be written as a convex combination of points (ζ_i, L(ζ_i)) of the epigraph of the lower semi-continuous envelope ofL(·); moreover theζ_i are uniformly bounded, so that they all lie in a simplex generated byN+ 1 affinely independent pointsη₁, . . . , η_N+1. Now each valueL(ζ_j) can be approximated withL(η^j) for someη^j arbitrarily near toζ_j; actually it turns out that forε >0, if|η^j−ζ_j|is sufficiently small, then there is a convex combination of (η^j, L(η^j)) andN points among the (η_i, L(η_i))’s whose

projection onR^N isζ_j and whose last coordinate is less thanL(ζ_j) +ε. The conclusion follows by writingξas a convex combinations of the pointsη_i and theη^j constructed as above.

We first need two technical lemmas. Let, ifSis a subset ofR^N, intSdenote its interior and convSits convex hull.

Lemma 3.3. Letη₁, . . . , η_N+1beN+1affinely independent points ofR^N andηinint (conv{η₁, . . . , η_N+1}), the interior of the simplex whose vertices are η₁, . . . , η_N+1. Then:

i) for everyI⊂ {1, . . . , N+1}of cardinality|I| ≤N the set of points{η, η_i: i∈I}is affinely independent;

ii) for everyξ∈int (conv{η₁, . . . , η_N+1})there exists a subsetI of{1, . . . , N+1} of cardinalityN such that ξ∈conv{η, η_i: i∈I}·

Proof of Lemma 3.3. i) It is not restrictive to assume thatI={1, . . . , N}. Let

η =

N+1

j=1

λ_jη_j λ_j>0

N+1

j=1

λ_j= 1.

For everyi∈ {1, . . . , N}we have

η−η_i=

j=i

λ_jη_j+ (λ_i−1)η_i

=

j=i

λ_j(η_j−η_N+1) + (λ_i−1)(η_i−η_N+1)

so that, in a matrix notation,

[η−η₁, . . . , η−η_N] = [η₁−η_N+1, . . . , η_N−η_N+1](Λ−I) whereI is the identity and

Λ =



λ₁ . . . λ_N . . . . λ₁ . . . λ_N



.

Now det(Λ−I)= 0 since the eigenvalues of Λ areλ₁, . . . , λ_N andλ_i<1 for everyi, proving i).

Proof of ii). Let

ξ=α₁η₁+· · ·+α_N+1η_N+1 η=µ₁η₁+· · ·+µ_N+1η_N+1

and we may assume thatα_N+1/µ_N+1= min{α_i/µ_i: i= 1, . . . , N+1}(notice that all theµ_iare strictly positive).

Setc_N+1=α_N+1/µ_N+1and, fori∈ {1, . . . , N},c_i=α_i−µ_ic_N+1: then, for everyi,c_i ≥0; moreover

N+1

i=1

c_i= N i=1

α_i−c_N+1 N i=1

µ_i+c_N+1

= 1−α_N+1−(1−µ_N+1)c_N+1+c_N+1

= 1−α_N+1+µ_N+1c_N+1= 1

and

N i=1

c_iη_i+c_N+1η= N i=1

(α_i−µ_ic_N+1)η_i+c_N+1

N+1

i=1

µ_iη_i

= N i=1

(α_i−µ_ic_N+1+µ_ic_N+1)η_i+c_N+1µ_N+1η_N+1

= N i=1

α_iη_i+α_N+1η_N+1=ξ

so thatξ∈conv{η, η_i: i∈ {1, . . . , N}}·

Lemma 3.4. Let η₁, . . . , η_N+1 be N+ 1affinely independent points of R^N and lety₁, . . . , y_N+1 be real numbers and K > 0. For every ε > 0 there exists δ > 0 such that for every η, ξ ∈ int (conv{η₁, . . . , η_N+1}) and y, β in [−K, K], with |η−ξ| < δ and |y−β| < δ, there exists a subset I of {1, . . . , N+ 1} of cardinality N and coefficientsλ, λ_i (i∈I)of a convex combination satisfying

ξ=λη+

i∈Iλ_iη_i β+ε > λy+

i∈Iλ_iy_i.

Remark 3.5. Geometrically Lemma 3.4 states that givenN+1 points (η_i, y_i) ofR^N×Rand (ξ, β) inR^N×R such that ξ lies in the interior of the convex hull Λ of the η_is then, given a positive ε, for every point (η, y) that is sufficiently near to (ξ, β) withη∈Λ there existN points among the (η_i, y_i)s which, together with (η, y), generate a N- dimensional simplex inR^N ×Rwhose projection in R^N containsξ and such that (ξ, β+ε) lies above it.

Proof of Lemma 3.4. For everyI ⊂ {1, . . . , N+ 1}, |I| =N, y ∈ R and η ∈ Λ := int conv{η₁, . . . , η_N+1} by Lemma 3.3i) there exists a unique hyperplane z=a^I(η, y)·ξ+b^I(η, y) containing the points (η, y) and (η_i, y_i) (i∈I). Moreover the coefficientsa^I(η, y),b^I(η, y) are continuous functions of (η, y); in fact from the equations

a^I(η, y)·η+b^I(η, y) =y

a^I(η, y)·η_i+b^I(η, y) =y_i(i∈I) we deduce that the vectora^I(η, y) solves the system

a^I(η, y)·(η−η_i) =y−y_i (i∈I);

again by Lemma 3.3i) the vectors η −η_i (i ∈ I) are independent so that the latter system has a unique solutiona^I(η, y) given by Cramer’s rule which is a continuous function ofη andy; the continuity ofb^I follows from the equalityb^I(η, y) =y−a^I(η, y)·η.

Set, for everyI⊂ {1, . . . , N+1},

ϕ^I(η, y;ζ) =a^I(η, y)·ζ+b^I(η, y);

we point out that, by construction, for a fixed (η, y) the point (ζ, ϕ^I(η, y;ζ)) belongs to the (unique) hyperplane containing the points (η, y) and (η_i, y_i) (i∈I). Since, for every ζ∈Λ andβ∈R,

ϕ^I(ζ, β;ζ) =β,

then, by the uniform continuity of ϕ^I on Λ×[−K, K]×Λ, for everyε > 0 there exists δ >0 such that, for every subset Iof{1, . . . , N+1}of cardinalityN,

ϕ^I(η, y;ζ)< β+εwheneverη, ζ ∈Λ |η−ζ|< δand y, β∈[−K, K] |y−β|< δ. (3.1) Now fixξin Λ andε >0. LetI⊂ {1, . . . , N+1}be such thatξbelongs to conv{η, η_i:i∈I}; such a set exists by Lemma 3.3ii). Letδ be such as in (3.1) and |η−ξ| < δ, |y−β|< δ so that ϕ^I(η, y;ξ)< β+ε. Then, if we set

ξ=λη+

i∈I

λ_iη_i

for some coefficientsλ, λ_i (i∈I) of a convex combination, the linearity ofϕ^I in the third variable yields λϕ^I(η, y;η) +

i∈I

λ_iϕ^I(η, y;η_i)< β+ε,

proving the claim sinceϕ^I(η, y;η) =y andϕ^I(η, y;η_i) =y_i. We are now ready to prove Theorem 3.2.

Proof of Theorem 3.2. Fix x in C. By Theorem 3.1 there exists R₁ > 0 (depending only on R₀ and C), ζ₁, . . . , ζ_ν (ν ≤N+1), with|ζ_j| ≤R₁, and coefficientsα_j of a convex combination satisfying

ξ=_ν

j=1α_jζ_j L^∗∗(x, ξ) =_ν

j=1α_jL(x, ζ_j);

where L denotes as usual the lower semi-continuous envelope of L(x,·). It is not restrictive at this stage to assume thatL(x, ξ) =L(ξ). Letη₁, . . . , η_N+1 be such that

ζ∈R^N : |ζ| ≤R₁

⊂int (conv{η₁, . . . , η_N+1}) and set

y_i=L(η_i), i= 1, . . . , N+1, K= sup{L(ζ) : |ζ| ≤R₁}·

Fixε >0 andjin{1, . . . , ν}; setβ =L(ζ_j). Correspondingly, letδ >0 satisfy the property stated in Lemma 3.4.

By the definition ofLthere existη^j∈int (conv{η₁, . . . , η_N+1}) such that

|η^j−ζ_j|< δ and L(η^j)≤L(ζ_j) +δ.

We apply Lemma 3.4 withη=η^j,ξ=ζ_jandy=L(η^j): there exists a subsetI_jof{1, . . . , N+1}of cardinalityN and coefficientsλ^j, λ^j_i, (i∈I_j), such that

ζ_j=λ^jη^j+

i∈Ijλ^j_iη_i λ^jL(η^j) +

i∈Ijλ^j_iL(η_i)≤L(ζ_j) +ε.

Therefore we obtain that

ξ= ν j=1

α_jζ_j= ν j=1

α_j



λ^jη^j+

i∈Ij

λ^j_iη_i





= ν j=1

α_jλ^jη^j+

i∈Ij



^ν

j=1

α_jλ^j_i



η_i

and moreover

L^∗∗(ξ) +ε= ν j=1

α_j

L(ζ_j) +ε

≥ ν j=1

α_j



λ^jL(η^j) +

i∈Ij

λ^j_iL(η_i)





= ν j=1

α_jλ^jL(η^j) +

i∈Ij



^ν

j=1

α_jλ^j_i



L(η_i).

If we set λ_i=α_iλⁱ ifi∈ {1, . . . , ν} λ_i=

jα_jλ^j_i−ν ifi∈ {ν+ 1, . . . , ν+ (N+1)}

the above formulae can be rewritten as ξ=

i≤νλ_iηⁱ+

i>νλ_iη_i

i≤νλ_iL(ηⁱ) +

i>νλ_iL(η_i)≥L^∗∗(ξ) +ε.

Moreover |ηⁱ| ≤ R and |η_i| ≤ R, where R = max{|η_i| : i = 1, . . . , N+ 1} (which depends only onR₁ and

therefore only onR₀ andC); proving the claim.

We consider here the problems inf

F(y) =

_b

a L(y(t), y(t)) dt: y∈AC([a, b],R^N), y(a) =A, y(b) =B

(P)

inf

F^∗∗(y) = _b

a

L^∗∗(y(t), y(t)) dt: y∈AC([a, b],R^N), y(a) =A, y(b) =B

· (P^∗∗)

It is well known that infF = infF^∗∗ if L is continuous and superlinear ([7], Th. IX.3.1); actually in this caseF^∗∗ is the relaxed functional of F. In [5] Cellina proved that infF = infF^∗∗ ifL is just continuous and satisfies (GA). We examine here the case whereLis just continuous in the first variable, focusing our attention on the infima of the functionalsF andF^∗∗ instead on the relaxed functional ofF.

Theorem 3.6. Assume thatLis bounded on compact sets and thatx→L(x, ξ)is continuous for everyξ∈R^N. If Lsatisfies(CGA)theninfF = infF^∗∗.

Proof. We follow the lines of the proof of the analogous result ([5], Th. 3) in the case where L is continuous in both variables, but instead of Theorem 3.1 we use Theorem 3.2. Letx∈ AC([a, b],R^N) and ε >0. From Theorem 2.4 and Remark 2.8 of [1] applied toL^∗∗ there exists a Lipschitz functionx_R0 of Lipschitz constantR₀ satisfying the boundary conditions and such that

_b

a

L^∗∗

x_R0(t), x_R0(t) dt≤

_b

a

L^∗∗(x(t), x(t)) dt+ε/3.

SetC ={x_R0(t) : t ∈[a, b]}. Since |x_R

0(t)| ≤R₀ for a.e. t then, by Theorem 3.2, there existsR (depending only onR₀ andC),m≤2N+ 2 coefficientsλ_i(t) of a convex combination and vectorsy_i(t) (i= 1, . . . , m) with

|y_i(t)| ≤Rsuch that



 x_R

0(t) =_m

i=1λ_i(t)y_i(t) _m

i=1λ_i(t)L(x_R0(t), y_i(t))≤L^∗∗(x_R0(t), x_R

0(t)) + ε 3(b−a)·

By a standard selection argument, we may assume that the maps y_i andλ_i are measurable. Fix an integer k and consider the intervalsI_j = [t_j, t_j+1], wheret_j =a+j^b−a_k (j = 0, . . . , k−1) and callχ_Ij their characteristic function. By Lyapunov’s Theorem on the range of vector measures [9] there exists a partition of [a, b] intom measurable subsetsE_i, with characteristic functionsχ_Ei, such that, forj = 0, . . . , k−1, one has

Ij

m i=1

λ_i(t)y_i(t) dt=

Ij

m i=1

χ_Ei(t)y_i(t) dt

Ij

m i=1

λ_i(t)L(x_R0(t), y_i(t)) dt=

Ij

m i=1

χ_Ei(t)L(x_R0(t), y_i(t)) dt.

Denote byx_k the absolutely continuous defined byx_k(a) =Aand x_k(t) =

_t

a

i,j

y_i(s)χ_Ij∩Ei(s) ds;

in particular for everykand everyj= 1, . . . , k, we have

Ij

x_R

0(t) dt=

Ij

x_k(t) dt,

so that the functionsx_R0 andx_k coincide at each pointt_j. Since L(x_R0(t), x_k(t)) =

i,j

χ_Ij∩Ei(t)L(x_R0(t), y_i(t))

we also have that _b

a

L(x_R0(t), x_k(t)) dt= _b

a

m i=1

λ_i(t)L(x_R0(t), y_i(t)) dt;

so that, from≈, it follows that _b

a

L(x_R0(t), x_k(t)) dt≤ _b

a

L^∗∗(x_R0(t), x_R0(t)) dt+ε/3.

Now _b

a

L(x_R0(t), x_k(t)) dt= _b

a

L(x_k(t), x_k(t)) dt+ _b

a

L(x_R0(t), x_k(t))−L(x_k(t), x_k(t)) dt;

moreover,x_R0 is uniformly continuous, the functionsx_k are equi-Lipschitz,x_k(t_j) =x_R0(t_j) (j= 0, . . . , k−1).

Hence, ift∈[a, b] andt_j≤t≤t_j+1,

|x_k(t)−x_R0(t)| ≤ |x_k(t)−x_k(t_j)|+|x_k(t_j)−x_R0(t_j)|+|x_R0(t_j)−x_R0(t)|

=|x_k(t)−x_k(t_j)|+|x_R0(t_j)−x_R0(t)| ≤(R+R₀)(b−a)/k

so thatx_kconverges uniformly tox_R0 asktends to +∞. By our assumption the functionL(x_R0, x_k)−L(x_k, x_k) is bounded a.e. by a constant that does not depend onk. The continuity ofLwith respect to the first variable together with the dominated convergence theorem imply that

k→+∞lim _b

a

L(x_R0(t), x_k(t))−L(x_k(t), x_k(t)) dt= 0.

It follows that for ksufficiently large,

L(x_k(t), x_k(t)) dt≤ _b

a

L(x_R0(t), x_k(t)) dt+ε/3≤ _b

a

L(x(t), x(t)) dt+ε

proving that infF ≤infF^∗∗.

We point out that, under the assumptions of Theorem 3.6, the functionalF^∗∗ is not in general the relaxed functional of F; we refer to [2] for some recent results in this direction. This is the case in the forthcoming example, where we also show that the conclusion of Theorem 3.6 does not hold ifLis not continuous inx.

Example 3.7. Let g be the characteristic function of R\ {0} and h(ξ) = ξ² if ξ = 0, h(0) = 1 and set L(x, ξ) =g(x) +h(ξ). Let (P), (P^∗∗) be the problems

inf

F(y) = ₁

0

L(y(t), y(t)) dt; y(0) = 0, y(1) = 0, y∈AC([0,1],R)

(P) inf

F^∗∗(y) = ₁

0

L^∗∗(y(t), y(t)) dt; y(0) = 0, y(1) = 0, y∈AC([0,1],R)

· (P^∗∗) For every xinR we haveL^∗∗(x, ξ) =g(x) +ξ², so that the minimum of the problem (P^∗∗) is equal to 0 and it is obviously assumed fory(t) = 0. However infF ≥1; in fact lety ∈AC([0,1],R) and set E ={t∈[0,1] : y(t) = 0}, theny(t) = 0 a.e. onE, so that

₁

0

L(y(t), y(t)) dt=

E

L(0,0) dt+

[0,1]\E

L(y(t), y(t)) dt

≥

E1 dt+

[0,1]\Eg(y(t)) dt

≥ |E|+|[0,1]\E|= 1.

Notice that nevertheless, from [3], the minima ofF are Lipschitz.

4. Lipschitz regularity of the minima of (P )

In this section we apply our result to the problem of the Lipschitz regularity of the minima of (P). It is well known that if L(x, ξ) is continuous, convex and superlinear inξ then every minimum of (P) is Lipschitz.

In some recent papers the same conclusion is proved under weaker assumptions; we just mention [5, 6, 8]. Our result is in the same spirit of the last two that we recall here.

Theorem 4.1. [5] Assume thatL(x, ξ)is continuous in both variables and satisfies(GA). Then every minimizer of (P)in AC([a, b],R^N)is Lipschitz.

Theorem 4.2. [8] Assume that L(x, ξ) is convex in ξ and satisfies (GA). Then every minimizer of (P) in AC([a, b],R^N)is Lipschitz.

The following theorem weakens the continuity assumption of Theorem 4.1.

Theorem 4.3. Assume that x → L(x, ξ) is continuous for every ξ and that L satisfies (GA). Then every minimizer of (P) inAC([a, b],R^N)is Lipschitz.

Proof. By Theorem 3.6, infF = infF^∗∗; therefore every minimum of F is a minimum of F^∗∗. The func- tionL^∗∗(x, ξ) is convex inξ and satisfies (GA): Theorem 4.2 yields the conclusion.

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