Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2017: 1.89

Open Access
Online
ISSN
2191-950X
See all formats and pricing
More options …

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

Antonio Esposito / Francesco Leonetti / Pier Vincenzo Petricca
Published Online: 2016-12-20 | DOI: https://doi.org/10.1515/anona-2016-0198

Abstract

We consider non-autonomous functionals of the form (u,Ω)=Ωf(x,Du(x))𝑑x, where u:ΩN, Ωn. 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.

Keywords: Variational integrals; non-standard growth; regularity; Lavrentiev gap

MSC 2010: 49N60

1 Introduction

We consider variational integrals of the form

(u,Ω)=Ωf(x,Du(x))𝑑x(1.1)

where u:ΩnN, n2, N1, f:Ω×nN is a Caratheodory function, and Ω is bounded and open. Moreover, we assume that, for exponents 1<pq and constants ν,L(0,+), c[0,+), we have

ν|z|p-cf(x,z)L(1+|z|q).(1.2)

In the scalar case N=1, when p=q, the local minimizers uW1,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:ΩnN makes the energy (1.1) finite. Then the left-hand side of (1.2) implies that DuLp. 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 Lp to Lq? The answer is given in [9]: We assume that (1.2) holds with ν=1, c=0, we require that zf(x,z)C1(nN) and, for constants μ[0,1] and α(0,1], we assume that the following hold:

L-1(μ2+|z1|2+|z2|2)p-22|z1-z2|2fz(x,z1)-fz(x,z2);z1-z2,(1.3)|fz(x,z)-fz(y,z)|L|x-y|α(1+|z|q-1).(1.4)

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

1<pq<p(n+αn).(1.5)

We recall that n is the dimension of the space where x lives, i.e., xΩn. 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:

(u,BR)=0(1.6)

for every ball BRΩ. 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., DuLlocq(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, n2, and we will denote

BRBR(x0):={xn:|x-x0|<R},

where, unless differently specified, all the balls considered will have the same center. We assume that f:Ω×nN is a Caratheodory function verifying the (p,q)-growth (1.2) with ν=1 and c=0. Moreover, we assume that zf(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 uWloc1,1(Ω;N) is a local minimizer of if and only if xf(x,Du(x))Lloc1(Ω) and

suppϕf(x,Du(x))𝑑xsuppϕf(x,Du(x)+Dϕ(x))𝑑x,

for any ϕW1,1(Ω;N) with suppϕΩ.

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

X=W1,p(BR;N),Y=Wloc1,q(BR;N)W1,p(BR;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,BR)=¯Y(v)-¯X(v)for every vX,

when ¯X(v)<+ and (v,BR)=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 uW1,p(BR;RN) be a function such that F(u,BR)<+. Then L(u,BR)=0 if and only if there exists a sequence {um}mNWloc1,q(BR;RN)W1,p(BR;RN) such that

umuweakly in W1,p(BR;N)

and

(um,BR)(u,BR).

3 Main section

Theorem 3.1.

Let f:Ω×RnNR be a function satisfying the following conditions:

  • (1)

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

  • (2)

    |f(x,z)-f(x~,z)|H|x-x~|α(1+|z|q), 0<α1,

  • (3)

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

  • (4)

    for BRΩ, ε0(0,1] such that BR+2ε0Ω, xBR and ε(0,ε0) , there exists y~=y~(x,ε)B(x,ε)¯ such that for znN and yB(x,ε)¯ , we have f(y~,z)f(y,z).

Let uWloc1,p(Ω;RN) such that xf(x,Du(x))Lloc1(Ω) and assume that

qp(n+αn).(3.1)

Then L(u,BR)=0 for all BRΩ.

Remark 3.2.

Let us now explain condition (4). For every fixed z, yf(y,z) is continuous, so the minimization of f(y,z) when yB(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,BR)=0, the borderline case q=p(n+αn) is allowed but we need strict inequality (1.5) when proving higher integrability of minimizers, see [9, p. 32].

Proof.

Consider 0<ε<ε01 as in hypothesis (4), then uW1,p(BR+2ε0;N) and

(u,BR+2ε0)=BR+2ε0f(x,Du(x))𝑑x<+.

Let us denote uε(x):=(u*ϕε)(x), the usual mollification, where xBR, and define

fε(x,z)=minyB(x,ε)¯f(y,z).(3.2)

By definition, it follows that

|Duε(x)|(BR+2ε0|Du(y)|pdy)1p(n|ϕε(y)|p)1pCε-np,

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

fε(x,z)f(x,z)-Hεα(1+|z|q).(3.3)

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

|z|pfε(x,z).(3.4)

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

f(x,z)Kfε(x,z)+H,xBR,|z|Cε-np.(3.5)

Indeed, let us fix δ(0,1) and observe that, using (3.3), (3.4) and |z|Cε-np, we get

fε(x,z)=δfε(x,z)+(1-δ)fε(x,z)δf(x,z)-δHεα(1+|z|q)+(1-δ)|z|p=δf(x,z)-δHεα|z|q+(1-δ)|z|p-δHεα=δf(x,z)-δHεα|z|p|z|q-p+(1-δ)|z|p-δHεαδf(x,z)-δCq-pHεα+(p-qp)n|z|p+(1-δ)|z|p-δHεαδf(x,z)+(1-δ-δCq-pH)|z|p-δH,

where the last estimate relies on the fact that qpn+αn, 0<α1 and 0<ε<1. Then (3.5) follows choosing K=1δ=1+Cq-pH. Now, using hypothesis (4), Jensen’s inequality and (3.2), we obtain

fε(x,Duε(x))=f(y~,Duε(x))B(x,ε)f(y~,Du(y))ϕε(x-y)𝑑yB(x,ε)f(y,Du(y))ϕε(x-y)𝑑y=(f(,Du())*ϕε)(x)=:f(,Du())ε(x).(3.6)

Therefore, using (3.5), we have

f(x,Duε(x))Kf(,Du())ε(x)+H.

Finally, since f(,Du())ε(x)f(x,Du(x)) strongly in L1(BR), by recalling that uεu in W1,p(BR;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)

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

    • (1)(i)

      aC0,α(Ω¯) and a(x)0 for all x,

    • (1)(ii)

      b and c are convex functions such that

      |z|pb(z)H(|z|q+1)for H1,0c(z)L(|z|q+1)for L1.

    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|pln(e+|z|). This example is taken from [2, 1].

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

    • f(x,z)=|z|p+a(x)[|z1-z2|q+|z1|q], where 1<p<q<+. This example is inspired by [5].

  • (2)

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

  • (3)

    f(x,z)=h(i=1k[bi(z)+ai(x)ci(z)]), where, in addition to the previous conditions, h is increasing, convex, Lipschitz and such that sh(s)αs+β.

  • (4)

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

    • (4)(i)

      th(t,z) is increasing,

    • (4)(ii)

      h is convex with respect to the second variable,

    • (4)(iii)

      aC(Ω¯),

    • (4)(iv)

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

    For example,

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

    where a~C0,α(Ω¯), a~(x)0 for all x, a1+2b2 and b>0. In order to satisfy the non-standard (p,q)-growth condition, we can consider 1<p<a+bq. 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)×2N such that

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

where, for x=(x1,x2),

a(x)={x2if x2>0,0if x20.

In this case the minimum point of the function changes depending on the choice of z. Indeed, let us consider BR=B(0,12), ε0=18, x=0. Then we deal with the two cases: |z|=0 and |z|=2.

When |z|=0, we have

f(y,z)={-y2+1if y2>0,1if y20,

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

If |z|=2, then

f(y,z)={2p+y2(2q-1)+1if y2>0,2p+1if y20,

and therefore, in this situation, y~ is any point (y1,y2) such that y20.

Corollary 3.7.

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

h(x,z)-c1f(x,z)h(x,z)+c2,

where c1,c20. We consider uWloc1,p(Ω;RN) such that xf(x,Du(x))Lloc1(Ω) and assume qp(n+αn). Then L(u,BR)=0 for all BRΩ.

Proof.

We follow the proof of [9, Theorem 6]. By the proof of Theorem 3.1, if uWloc1,p(Ω;N) is such that h(x,Du(x))Lloc1(Ω), then there exists a sequence {um}mW1,q(BR;N) such that umu strongly in W1,p(BR;N), Dum(x)Du(x) a.e., h(x,Dum(x))h(x,Du(x)) a.e., and BRh(x,Dum(x))BRh(x,Du(x)). Using a well-known variant of Lebesgue’s dominated convergence theorem, we have that

BRf(x,Dum(x))BRf(x,Du(x)),

and then, by Lemma 2.2, (u,BR)=0. ∎

References

  • [1]

    P. Baroni, M. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. 121 (2015), 206–222.  CrossrefWeb of ScienceGoogle Scholar

  • [2]

    P. Baroni, M. Colombo and G. Mingione, Non-autonomous functionals, borderline cases and related functional classes, St. Petersburg Math. J. 27 (2016), 347–379.  CrossrefGoogle Scholar

  • [3]

    M. Belloni and G. Buttazzo, A survey of old and recent results about the gap phenomenon in the calculus of variations, Recent Developements in Well-Posed Variational Problems, Math. Appl. 331, Kluwer Academic, Dordrecht (1995), 1–27.  Google Scholar

  • [4]

    G. Buttazzo and V. J. Mizel, Interpretation of the Lavrentiev phenomenon by relaxation, J. Funct. Anal. 110 (1992), no. 2, 434–460.  CrossrefGoogle Scholar

  • [5]

    A. Cianchi, Local boundedness of minimizers of anisotropic functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 2, 147–168.  CrossrefGoogle Scholar

  • [6]

    F. Colasuonno and M. Squassina, Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl. (4) 195 (2016), no. 6, 1917–1959.  CrossrefWeb of ScienceGoogle Scholar

  • [7]

    M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), 219–273.  CrossrefWeb of ScienceGoogle Scholar

  • [8]

    M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015), 443–496.  CrossrefWeb of ScienceGoogle Scholar

  • [9]

    L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with (p,q)-growth, J. Differential Equations 204 (2004), no. 1, 5–55.  Google Scholar

  • [10]

    I. Fonseca, G. Mingione and I. Maly, Scalar minimizers with fractal singular sets, Arch. Ration. Mech. Anal. 172 (2004), 295–307.  CrossrefGoogle Scholar

  • [11]

    N. Fusco and C. Sbordone, Higher integrability of the gradient of minimizers of functionals with non standard growth conditions, Comm. Pure Appl. Math. 43 (1990), 673–683.  CrossrefGoogle Scholar

  • [12]

    M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math. 59 (1987), 245–248.  CrossrefGoogle Scholar

  • [13]

    M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, Acta Math. 148 (1982), 31–46.  CrossrefGoogle Scholar

  • [14]

    E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003.  Google Scholar

  • [15]

    M. C. Hong, Some remarks on the minimizers of variational integrals with non standard growth conditions, Boll. Unione Mat. Ital. VII. Ser. A 6 (1992), 91–101.  Google Scholar

  • [16]

    F. Leonetti, Weak differentiability for solutions to elliptic systems with p,q-growth conditions, Ann. Mat. Pura Appl. (4) 162 (1992), 349–366.  Google Scholar

  • [17]

    P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Ration. Mech. Anal. 105 (1989), 267–284.  CrossrefGoogle Scholar

  • [18]

    P. Marcellini, Regularity and existence of solutions of elliptic equations with p,q-growth conditions, J. Differential Equations 90 (1991), 1–30.  Google Scholar

  • [19]

    G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations, Appl. Math. 51 (2006), 355–426.  CrossrefGoogle Scholar

  • [20]

    G. Talenti, Buondedness of minimizers, Hokkaido Math. J. 19 (1990), 259–279.  CrossrefGoogle Scholar

  • [21]

    Q. Tang, Regularity of minimizers of nonisotropic integrals in the calculus of variations, Ann. Mat. Pura Appl. (4) 164 (1993), 77–87.  CrossrefGoogle Scholar

  • [22]

    V. V. Zhikov, On Lavrentiev’s phenomenon, Russ. J. Math. Phys. 3 (1995), 249–269.  Google Scholar

About the article

Received: 2016-09-10

Accepted: 2016-09-27

Published Online: 2016-12-20


The authors thank GNAMPA and UNIVAQ for the support.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 73–78, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0198.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

Comments (0)

Please log in or register to comment.
Log in