Antonio Esposito , Francesco Leonetti and Pier Vincenzo Petricca

Absence of Lavrentiev gap for non-autonomous functionals with (p,q)-growth

De Gruyter | Published online: December 20, 2016

Abstract

We consider non-autonomous functionals of the form ( u , Ω ) = Ω f ( x , D u ( x ) ) 𝑑 x , where u : Ω N , Ω n . We assume that f ( x , z ) grows at least as | z | p and at most as | z | q . Moreover, f ( x , z ) is Hölder continuous with respect to x and convex with respect to z. In this setting, we give a sufficient condition on the density f ( x , z ) that ensures the absence of a Lavrentiev gap.

MSC 2010: 49N60

1 Introduction

We consider variational integrals of the form

(1.1) ( u , Ω ) = Ω f ( x , D u ( x ) ) 𝑑 x

where u : Ω n N , n 2 , N 1 , f : Ω × n N is a Caratheodory function, and Ω is bounded and open. Moreover, we assume that, for exponents 1 < p q and constants ν , L ( 0 , + ) , c [ 0 , + ) , we have

(1.2) ν | z | p - c f ( x , z ) L ( 1 + | z | q ) .

In the scalar case N = 1 , when p = q , the local minimizers u W 1 , p ( Ω ) of (1.1) are locally Hölder continuous, see [13] and [19, p. 361]. If p < q and q is far from p, then the local minimizers might be unbounded, see [12, 17, 18, 15] and [19, Section 5]. We are concerned with higher integrability for the gradient of minimizers. More precisely, assume that u : Ω n N makes the energy (1.1) finite. Then the left-hand side of (1.2) implies that D u L p . In addition to finite energy, we assume that u is a minimizer of (1.1) and we ask: Does the minimality of u boost the integrability of the gradient Du from L p to L q ? The answer is given in [9]: We assume that (1.2) holds with ν = 1 , c = 0 , we require that z f ( x , z ) C 1 ( n N ) and, for constants μ [ 0 , 1 ] and α ( 0 , 1 ] , we assume that the following hold:

(1.3) L - 1 ( μ 2 + | z 1 | 2 + | z 2 | 2 ) p - 2 2 | z 1 - z 2 | 2 f z ( x , z 1 ) - f z ( x , z 2 ) ; z 1 - z 2 ,
(1.4) | f z ( x , z ) - f z ( y , z ) | L | x - y | α ( 1 + | z | q - 1 ) .

The exponents p and q must be close enough, i.e.,

(1.5) 1 < p q < p ( n + α n ) .

We recall that n is the dimension of the space where x lives, i.e., x Ω n . Let us remark that (1.5) asserts that the smaller α is, the closer p and q must be. In addition, we assume that the Lavrentiev gap on u is zero:

(1.6) ( u , B R ) = 0

for every ball B R Ω . Such a Lavrentiev gap will be defined in the next section. Under (1.2)–(1.6), the local minimizers u of (1.1) enjoy higher integrability, i.e., D u L loc q ( Ω ) . Checking (1.6) is not easy for non-autonomous densities f ( x , z ) ; it has been done in [9] for some model functionals using some arguments due to [22]. The aim of the present paper is to give a sufficient condition on the density f ( x , z ) that ensures the vanishing of the Lavrentiev gap (1.6), see Theorem 3.1(4).

2 Preliminaries

In the following Ω will be an open, bounded subset of n , n 2 , and we will denote

B R B R ( x 0 ) := { x n : | x - x 0 | < R } ,

where, unless differently specified, all the balls considered will have the same center. We assume that f : Ω × n N is a Caratheodory function verifying the ( p , q ) -growth (1.2) with ν = 1 and c = 0 . Moreover, we assume that z f ( x , z ) is convex. Due to the non-standard growth behavior of f, we shall adopt the following notion of local minimizer.

Definition 2.1.

A function u W loc 1 , 1 ( Ω ; N ) is a local minimizer of if and only if x f ( x , D u ( x ) ) L loc 1 ( Ω ) and

supp ϕ f ( x , D u ( x ) ) 𝑑 x supp ϕ f ( x , D u ( x ) + D ϕ ( x ) ) 𝑑 x ,

for any ϕ W 1 , 1 ( Ω ; N ) with supp ϕ Ω .

Let us now explain the Lavrentiev gap. We adopt the viewpoint of [4], see also [3]. Let us set

X = W 1 , p ( B R ; N ) , Y = W loc 1 , q ( B R ; N ) W 1 , p ( B R ; N ) .

We consider functionals 𝒢 : X [ 0 , + ] that are sequentially weakly lower semicontinuous (s.w.l.s.c.) on X, and we set

¯ X = sup { 𝒢 : X [ 0 , + ] 𝒢  s.w.l.s.c.,  𝒢  on  X } ,
¯ Y = sup { 𝒢 : X [ 0 , + ] 𝒢  s.w.l.s.c.,  𝒢  on  Y } .

We have ¯ X ¯ Y , and we define the Lavrentiev gap as follows:

( v , B R ) = ¯ Y ( v ) - ¯ X ( v ) for every  v X ,

when ¯ X ( v ) < + and ( v , B R ) = 0 if ¯ X ( v ) = + . Since f ( x , z ) is convex with respect to z, standard weak lower semicontinuity results give ¯ X = (see, for instance, [14, Chapter 4]).

The following lemma will be used in the proof of the main theorem (see [4]).

Lemma 2.2.

Let u W 1 , p ( B R ; R N ) be a function such that F ( u , B R ) < + . Then L ( u , B R ) = 0 if and only if there exists a sequence { u m } m N W loc 1 , q ( B R ; R N ) W 1 , p ( B R ; R N ) such that

u m u weakly in  W 1 , p ( B R ; N )

and

( u m , B R ) ( u , B R ) .

3 Main section

Theorem 3.1.

Let f : Ω × R n N R be a function satisfying the following conditions:

  1. (1)

    | z | p f ( x , z ) L ( 1 + | z | q ) , 1 < p < q < + ,

  2. (2)

    | f ( x , z ) - f ( x ~ , z ) | H | x - x ~ | α ( 1 + | z | q ) , 0 < α 1 ,

  3. (3)

    z f ( x , z ) is convex for all x,

  4. (4)

    for B R Ω , ε 0 ( 0 , 1 ] such that B R + 2 ε 0 Ω , x B R and ε ( 0 , ε 0 ) , there exists y ~ = y ~ ( x , ε ) B ( x , ε ) ¯ such that for z n N and y B ( x , ε ) ¯ , we have f ( y ~ , z ) f ( y , z ) .

Let u W loc 1 , p ( Ω ; R N ) such that x f ( x , D u ( x ) ) L loc 1 ( Ω ) and assume that

(3.1) q p ( n + α n ) .

Then L ( u , B R ) = 0 for all B R Ω .

Remark 3.2.

Let us now explain condition (4). For every fixed z, y f ( y , z ) is continuous, so the minimization of f ( y , z ) when y B ( x , ε ) ¯ gives a minimizer y depending on x , ε and z. Condition (4) asks for independence on z, i.e., there exists a minimizer y ~ that works for every z. We will first give the proof of Theorem 3.1, and then we will show examples of densities f ( x , z ) satisfying condition (4).

Remark 3.3.

Let us compare (1.5) with (3.1). When proving the absence of the Lavrentiev gap ( u , B R ) = 0 , the borderline case q = p ( n + α n ) is allowed but we need strict inequality (1.5) when proving higher integrability of minimizers, see [9, p. 32].

Proof.

Consider 0 < ε < ε 0 1 as in hypothesis (4), then u W 1 , p ( B R + 2 ε 0 ; N ) and

( u , B R + 2 ε 0 ) = B R + 2 ε 0 f ( x , D u ( x ) ) 𝑑 x < + .

Let us denote u ε ( x ) := ( u * ϕ ε ) ( x ) , the usual mollification, where x B R , and define

(3.2) f ε ( x , z ) = min y B ( x , ε ) ¯ f ( y , z ) .

By definition, it follows that

| D u ε ( x ) | ( B R + 2 ε 0 | D u ( y ) | p d y ) 1 p ( n | ϕ ε ( y ) | p ) 1 p C ε - n p ,

where C = C ( D u L p ) > 1 . Moreover, by the Hölder continuity hypothesis (i.e., hypothesis 2), we have

(3.3) f ε ( x , z ) f ( x , z ) - H ε α ( 1 + | z | q ) .

Note that the left-hand side of hypothesis (1) gives

(3.4) | z | p f ε ( x , z ) .

Now we observe that is possible to find K = K ( p , q , D u L p , H ) < + such that

(3.5) f ( x , z ) K f ε ( x , z ) + H , x B R , | z | C ε - n p .

Indeed, let us fix δ ( 0 , 1 ) and observe that, using (3.3), (3.4) and | z | C ε - n p , we get

f ε ( x , z ) = δ f ε ( x , z ) + ( 1 - δ ) f ε ( x , z )
δ f ( x , z ) - δ H ε α ( 1 + | z | q ) + ( 1 - δ ) | z | p
= δ f ( x , z ) - δ H ε α | z | q + ( 1 - δ ) | z | p - δ H ε α
= δ f ( x , z ) - δ H ε α | z | p | z | q - p + ( 1 - δ ) | z | p - δ H ε α
δ f ( x , z ) - δ C q - p H ε α + ( p - q p ) n | z | p + ( 1 - δ ) | z | p - δ H ε α
δ f ( x , z ) + ( 1 - δ - δ C q - p H ) | z | p - δ H ,

where the last estimate relies on the fact that q p n + α n , 0 < α 1 and 0 < ε < 1 . Then (3.5) follows choosing K = 1 δ = 1 + C q - p H . Now, using hypothesis (4), Jensen’s inequality and (3.2), we obtain

f ε ( x , D u ε ( x ) ) = f ( y ~ , D u ε ( x ) )
B ( x , ε ) f ( y ~ , D u ( y ) ) ϕ ε ( x - y ) 𝑑 y
B ( x , ε ) f ( y , D u ( y ) ) ϕ ε ( x - y ) 𝑑 y
= ( f ( , D u ( ) ) * ϕ ε ) ( x )
(3.6) = : f ( , D u ( ) ) ε ( x ) .

Therefore, using (3.5), we have

f ( x , D u ε ( x ) ) K f ( , D u ( ) ) ε ( x ) + H .

Finally, since f ( , D u ( ) ) ε ( x ) f ( x , D u ( x ) ) strongly in L 1 ( B R ) , by recalling that u ε u in W 1 , p ( B R ; N ) , and by using a well-known variant of Lebesgue’s dominated convergence theorem and Lemma 2.2, the proof is completed. ∎

Remark 3.4.

We note that our assumption (4) is very close to [22, assumption (2.3)]. Our proof is inspired by the one of [9, Lemma 13], which, in turn, is based on some arguments used in [22].

Remark 3.5.

Now we give some examples of functions for which Theorem 3.1 is valid.

  1. (1)

    f ( x , z ) = b ( z ) + a ( x ) c ( z ) with the following conditions:

    1. (1)(i)

      a C 0 , α ( Ω ¯ ) and a ( x ) 0 for all x,

    2. (1)(ii)

      b and c are convex functions such that

      | z | p b ( z ) H ( | z | q + 1 ) for  H 1 , 0 c ( z ) L ( | z | q + 1 ) for  L 1 .

    For instance, we can consider the following functions:

    • f ( x , z ) = b ( z ) , independent of x.

    • f ( x , z ) = | z | p + a ( x ) | z | q . This example has been already dealt with in [9]; see also [22, 10, 8, 7, 1, 6].

    • f ( x , z ) = | z | p + a ( x ) | z | p ln ( e + | z | ) . This example is taken from [2, 1].

    • f ( x , z ) = | z | 2 + a ( x ) [ max { z n , 0 } ] q , where q > 2 . This example is inspired by [21].

    • f ( x , z ) = | z | p + a ( x ) [ | z 1 - z 2 | q + | z 1 | q ] , where 1 < p < q < + . This example is inspired by [5].

  2. (2)

    f ( x , z ) = i = 1 k [ b i ( z ) + a i ( x ) c i ( z ) ] , where k and a i , b i , c i verify the corresponding conditions of the previous example for all i { 1 , 2 , , k } .

  3. (3)

    f ( x , z ) = h ( i = 1 k [ b i ( z ) + a i ( x ) c i ( z ) ] ) , where, in addition to the previous conditions, h is increasing, convex, Lipschitz and such that s h ( s ) α s + β .

  4. (4)

    f ( x , z ) = h ( a ( x ) , z ) with the following conditions:

    1. (4)(i)

      t h ( t , z ) is increasing,

    2. (4)(ii)

      h is convex with respect to the second variable,

    3. (4)(iii)

      a C ( Ω ¯ ) ,

    4. (4)(iv)

      f verifies assumptions (1) and (2) of Theorem 3.1.

    For example,

    f ( x , z ) = | z | p + ( e + a ~ ( x ) | z | ) a + b sin ( ln ( ln ( e + a ~ ( x ) | z | ) ) ) ,

    where a ~ C 0 , α ( Ω ¯ ) , a ~ ( x ) 0 for all x, a 1 + 2 b 2 and b > 0 . In order to satisfy the non-standard ( p , q ) -growth condition, we can consider 1 < p < a + b q . This example is inspired by [11, 20], see also [16].

Remark 3.6.

Hypothesis (4) was used during the proof of Theorem 3.1 in order to obtain the second increase in (3.6). Now we want to show an example of a function for which hypothesis (4) fails. Let us consider Ω = B ( 0 , 1 ) 2 and the function f : B ( 0 , 1 ) × 2 N such that

f ( x , z ) = | z | p + a ( x ) ( | z | q - 1 ) + 1 ,

where, for x = ( x 1 , x 2 ) ,

a ( x ) = { x 2 if  x 2 > 0 , 0 if  x 2 0 .

In this case the minimum point of the function changes depending on the choice of z. Indeed, let us consider B R = B ( 0 , 1 2 ) , ε 0 = 1 8 , x = 0 . Then we deal with the two cases: | z | = 0 and | z | = 2 .

When | z | = 0 , we have

f ( y , z ) = { - y 2 + 1 if  y 2 > 0 , 1 if  y 2 0 ,

and then the minimum value in B ( 0 , ε ) ¯ is reached for y ~ = ( 0 , ε ) .

If | z | = 2 , then

f ( y , z ) = { 2 p + y 2 ( 2 q - 1 ) + 1 if  y 2 > 0 , 2 p + 1 if  y 2 0 ,

and therefore, in this situation, y ~ is any point ( y 1 , y 2 ) such that y 2 0 .

Corollary 3.7.

Let h : Ω × R n N R be a function verifying Theorem 3.1 and let f : Ω × R n N R be a function such that

h ( x , z ) - c 1 f ( x , z ) h ( x , z ) + c 2 ,

where c 1 , c 2 0 . We consider u W loc 1 , p ( Ω ; R N ) such that x f ( x , D u ( x ) ) L loc 1 ( Ω ) and assume q p ( n + α n ) . Then L ( u , B R ) = 0 for all B R Ω .

Proof.

We follow the proof of [9, Theorem 6]. By the proof of Theorem 3.1, if u W loc 1 , p ( Ω ; N ) is such that h ( x , D u ( x ) ) L loc 1 ( Ω ) , then there exists a sequence { u m } m W 1 , q ( B R ; N ) such that u m u strongly in W 1 , p ( B R ; N ) , D u m ( x ) D u ( x ) a.e., h ( x , D u m ( x ) ) h ( x , D u ( x ) ) a.e., and B R h ( x , D u m ( x ) ) B R h ( x , D u ( x ) ) . Using a well-known variant of Lebesgue’s dominated convergence theorem, we have that

B R f ( x , D u m ( x ) ) B R f ( x , D u ( x ) ) ,

and then, by Lemma 2.2, ( u , B R ) = 0 . ∎

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Received: 2016-09-10
Accepted: 2016-09-27
Published Online: 2016-12-20

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