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volume 7, issue 3, article 88, 2006.

Received 31 March, 2006;

accepted 06 April, 2006.

Communicated by:A. Fiorenza

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Journal of Inequalities in Pure and Applied Mathematics

REGULARITY FOR VECTOR VALUED MINIMIZERS OF SOME ANISOTROPIC INTEGRAL FUNCTIONALS

FRANCESCO LEONETTI AND PIER VINCENZO PETRICCA

Universita’ di L’Aquila

Dipartimento di Matematica Pura ed Applicata Via Vetoio, Coppito

67100 L’Aquila Italy.

EMail:leonetti@univaq.it Via Sant’Amasio 18 03039 Sora Italy.

2000c Victoria University ISSN (electronic): 1443-5756 099-06

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Regularity for Vector Valued Minimizers of Some Anisotropic

Integral Functionals

Francesco Leonetti and Pier Vincenzo Petricca

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J. Ineq. Pure and Appl. Math. 7(3) Art. 88, 2006

Abstract We deal with anisotropic integral functionalsR

Ωf(x, Du(x))dxdefined on vector valued mappingsu : Ω⊂Rⁿ →R^N. We show that a suitable "monotonicity"

inequality, on the densityf, guarantees global pointwise bounds for minimizers u.

2000 Mathematics Subject Classification:49N60, 35J60.

Key words: Anisotropic, Integral, Functional, Regularity, Minimizer.

1. Introduction

We consider the integral functional

(1.1) F(u) =

Z

Ω

f(x, Du(x))dx

where u : Ω ⊂ Rⁿ → R^N and Ω is a bounded open set. When N = 1 we are dealing with scalar functions u : Ω → R; on the contrary, vector valued mappings u : Ω → R^N appear when N ≥ 2. Local and global pointwise bounds for scalar minimizers of (1.1) have been proved in [2], [7], [5], [4]. A model functional for these results is

(1.2)

Z

Ω n

X

i,j=1

aij(x)Dju(x)Diu(x)

!^p₂ dx,

where coefficients a_ij are measurable, bounded and elliptic. Previous results for scalar minimizers are no longer true in the vector valued caseN ≥2as De Giorgi’s counterexample shows, [3]. Some years later, attention has been paid to anisotropic functionals whose model is

(1.3)

Z

Ω

(|D1u(x)|^p¹ +|D2u(x)|^p²+· · ·+|Dnu(x)|^pⁿ)dx,

where each component Diu of the gradientDu = (D1u, D2u, . . . , Dnu) may have a (possibly) different exponent p_i: this seems useful when dealing with some reinforced materials, [9]; see also [6, Example 1.7.1, page 169]. In the framework of anisotropic functionals, global pointwise bounds have been

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proved for scalar minimizers in [1] and [8]. If no additional conditions are assumed, these bounds are false in the vectorial case, as the above mentioned counterexample shows, [3]. The aim of this paper is to present a “monotonicity” assumption ensuring boundedness of vector valued minimizers. In order to do that, we recall that u : Ω ⊂ Rⁿ → R^N thus Du(x)is a matrix withN rows and n columns; the densityf(x, A)in (1.1) is assumed to be measurable with respect tox, continuous with respect toAandf : Ω×R^N×n →[0,+∞).

Every matrixA={A^α_i} ∈R^N×nwill haveN rowsA¹, . . . , A^N andncolumns A₁, . . . , A_n. In this paper we will show that the following “monotonicity” inequality guarantees global pointwise bounds for vector valued minimizers of (1.1):

(1.4) f(x,A) +˜ µ

n

X

i=1

A˜i−Ai

pi

≤f(x, A) +M(x)

for every pair of matricesA, A˜ ∈R^N×nsuch that there exists a rowβwithA˜^β = 0and for every remaining rowα6=β we haveA˜^α =A^α. In (1.4)µ, p1, . . . , pn

are positive constants with p_i > 1 andM : Ω → [0,+∞)with M ∈ L^r(Ω), r ≥ 1. If we keep in mind that A = Du(x), then the left hand side of (1.4) showsPn

i=1|A˜i−Diu(x)|^pⁱ, thus each componentDiuof the gradientDumay have a possibly different exponent p_i, so we are in the anisotropic framework:

u ∈ W^1,1(Ω,R^N) with D_iu ∈ L^pⁱ(Ω,R^N). In this case the harmonic mean p=

1 n

Pn i=1

1 pi

−1

comes into play. In Section2we will prove the following Theorem 1.1. We consider the functional (1.1) under the “monotonicity” in-

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equality (1.4) with

(1.5) p^∗

p

1− 1 r

>1

where p^∗ is the Sobolev exponent of p < n. We consideru = (u¹, . . . , u^N) ∈ W^1,1(Ω,R^N), withD_iu∈L^pⁱ(Ω,R^N)∀i∈ {1, . . . , n}, such that

F(u)<+∞

and

(1.6) F(u)≤ F(v)

for everyv ∈u+W₀^1,1(Ω,R^N)withD_iv ∈L^pⁱ(Ω,R^N)∀i∈ {1, . . . , n}. Then, for every componentu^β, we have

(1.7) inf

∂Ωu^β −c∗ ≤u^β(x)≤sup

∂Ω

u^β+c∗

for almost everyx∈Ω, where c∗ =c

kMk_L^r_(Ω) µ

¹_p

|Ω|

h(¹⁻¹_r)^p_p^∗⁻¹ⁱ_p¹∗

2(¹⁻¹_r)^p_p^∗^h(¹⁻¹_r)^p_p^∗⁻¹ⁱ⁻¹ , c=c(n, p₁, . . . , p_n)>0and|Ω|is the Lebesgue measure ofΩ.

A model density f for the “monotonicity” inequality (1.4) is given in the following.

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Lemma 1.2. For every i = 1, . . . , n, let us consider p_i ∈ [2,+∞) anda_i ∈ (0,+∞); we takem: Ω→[0,+∞)andh:R→Rwith−∞<inf_Rh. Let us considerf : Ω×R^n×n →Rdefined as follows:

(1.8) f(x, A) =

n

X

i=1

a_i|A_i|^pⁱ+m(x)h(detA).

Then the “monotonicity” inequality (1.4) holds true with µ = min_ja_j and M(x) = m(x)[h(0)−inf_Rh]. Moreover, ifh≥0, thenf ≥0too.

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2. Proofs

In order to prove Theorem1.1, we need the following

Lemma 2.1. Let us consider the functional (1.1) under the “monotonicity”

assumption (1.4). Then, for every v = (v¹, . . . , v^N) ∈ W^1,1(Ω,R^N) with D_iv ∈ L^pⁱ(Ω,R^N)∀i ∈ {1, . . . , n}, for any β ∈ {1, . . . , N}, for all t ∈ R, it results that

(2.1) F(Iβ,t(v)) +µ

n

X

i=1

Z

Ω

|Di(Iβ,t(v(x)))−Div(x)|^pⁱdx

≤ F(v) + Z

{v^β>t}

M(x)dx whereI_β,t:R^N →R^N is defined as follows:

∀y= (y¹, . . . , y^N)∈R^N, I_β,t(y) = (I_β,t¹ (y), . . . , I_β,t^N(y)) with

(2.2) I_β,t^α (y) =

( y^α if α6=β y^β∧t= min{y^β, t} if α=β.

Proof. For everyv ∈W^1,1(Ω,R^N), withD_iv ∈L^pⁱ(Ω,R^N)∀i∈ {1, . . . , n}, it results thatI_β,t(v)∈W^1,1(Ω,R^N); moreover

(2.3) D_i(I_β,t^α (v)) =

( D_iv^α if α 6=β 1{v^β≤t}Div^β if α =β,

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where1_B is the characteristic function of the setB, that is,1_B(x) = 1ifx∈B and1B(x) = 0ifx /∈B. ThereforeDi(Iβ,t(v))∈L^pⁱ(Ω,R^N)∀i∈ {1, . . . , n}.

On{x∈Ω :v^β(x)> t}we haveD(I_β,t^β (v)) = 0and, forα6=β,D(I_β,t^α (v)) = Dv^α; so we can apply (1.4) withA˜=D(I_β,t(v))andA=Dv; we obtain (2.4) f(x, D(I_β,t(v(x)))) +µ

n

X

i=1

|D_i(I_β,t(v(x)))−D_iv(x)|^pⁱ

≤f(x, Dv(x)) +M(x) forx∈ {v^β > t}. On{x∈Ω :v^β(x)≤t}D(I_β,t(v)) =Dv, thus

(2.5) f(x, D(I_β,t(v(x)))) +µ

n

X

i=1

|D_i(I_β,t(v(x)))−D_iv(x)|^pⁱ =f(x, Dv(x)) forx∈ {v^β ≤t}. From (2.4) and (2.5) we have

f(x, D(Iβ,t(v(x)))) +µ

n

X

i=1

|Di(Iβ,t(v(x)))−Div(x)|^pⁱ

≤f(x, Dv(x)) +M(x)1_{vβ>t}(x) for x ∈ Ω. If x → f(x, Dv(x)) ∈ L¹(Ω), then x → f(x, D(I_β,t(v(x)))) ∈ L¹(Ω) too and, integrating the last inequality with respect to x, we get (2.1).

Whenx → f(x, Dv(x)) ∈/ L¹(Ω), we haveF(v) = +∞and (2.1) holds true.

This ends the proof of Lemma2.1.

Now we are ready to prove Theorem1.1.

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Proof. Let us fix β ∈ {1, . . . , N}. Ifsup_∂Ωu^β = +∞then the right hand side of (1.7) is satisfied. Thus we assumesup_∂Ωu^β < t0 ≤t <+∞and we note that under this assumptionI_β,t(u)∈u+W₀^1,1(Ω,R^N)andD_i(I_β,t(u))∈L^pⁱ(Ω,R^N)

∀i∈ {1, . . . , n}since

u^β∧t = min{u^β, t}=u^β−[max{u^β−t,0}] =u^β−[(u^β−t)∨0], where(u^β−t)∨0∈W₀^1,1(Ω)andD_i((u^β−t)∨0) =D_iu^β1_{uβ>t} ∈L^pⁱ(Ω)

∀i∈ {1, . . . , n}. From (1.6) and (2.1) it results that

F(u)≤ F(I_β,t(u))

≤ F(u)−µ

n

X

i=1

Z

Ω

|Di(Iβ,t(u(x)))−Diu(x)|^pⁱdx+ Z

{u^β>t}

M(x)dx, that is

(2.6) µ

n

X

i=1

Z

Ω

|D_i(I_β,t(u(x)))−D_iu(x)|^pⁱdx≤ Z

{u^β>t}

M(x)dx.

If we defineφ= (u^β −t)∨0, then we can write (2.6) as follows:

(2.7) µ

n

X

i=1

Z

Ω

|D_iφ(x)|^pⁱdx≤ Z

{u^β>t}

M(x)dx.

Ifr <+∞, we apply Hölder’s inequality toR

{u^β>t}M(x)dxand we obtain Z

{u^β>t}

M(x)dx≤ kMk_L^r_(Ω)|{u^β > t}| ¹⁻¹^r

.

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Ifr= +∞, then Z

{u^β>t}

M(x)dx ≤ kMk_L^∞_(Ω)|{u^β > t}|=kMk_L^r_(Ω)|{u^β > t}| ¹⁻¹^r

. In both cases, from (2.7) it results that

n

X

i=1

Z

Ω

|D_iφ(x)|^pⁱdx≤ kMk_L^r_(Ω)

µ |{u^β > t}| ¹⁻¹^r

in particular,∀i∈ {1, . . . , n}

Z

Ω

|Diφ(x)|^pⁱdx≤ kMk_L^r_(Ω)

µ |{u^β > t}| ¹⁻¹^r

from which Z

Ω

|D_iφ(x)|^pⁱdx _pi¹

≤

kMk_L^r_(Ω)

µ |{u^β > t}| ¹⁻^r¹

1 pi

and finally

(2.8)

" _n Y

i=1

Z

Ω

|D_iφ(x)|^pⁱdx ¹

pi

#¹_n

≤

kMk_L^r_(Ω)

µ |{u^β > t}| ¹⁻¹^r

1 p

.

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We apply the anisotropic imbedding theorem [10] and we use (2.8):

0≤ Z

{u^β>t}

u^β(x)−tp^∗

dx _p¹∗

(2.9)

=kφk_Lp∗

(Ω)

≤c

" _n Y

i=1

Z

Ω

|D_iφ(x)|^pⁱdx _pi¹#_n¹

≤c

kMk_L^r_(Ω)

µ |{u^β > t}| ¹⁻¹^r

1 p

,

wherec=c(n, p1, . . . , pn)>0. IfkMk_L^r_(Ω) = 0, then from (2.9) it results that u^β ≤ t almost everywhere in Ωand we are done. If kMk_L^r_(Ω) > 0, then for T > twe have

(T −t)^p^∗|{u^β > T}|= Z

{u^β>T}

(T −t)^p^∗dx (2.10)

≤ Z

{u^β>T}

u^β(x)−tp^∗

dx

≤ Z

{u^β>t}

u^β(x)−tp^∗

dx and from (2.9) and (2.10) we get

(2.11) |{u^β > T}| ≤c^p^∗

kMk_L^r_(Ω) µ

^p

∗

p 1

(T −t)^p^∗|{u^β > t}| ¹⁻¹^r _p∗

p

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for every T, t with T > t ≥ t₀. We set χ(t) = |{u^β > t}| and we use [7, Lemma 4.1, p. 93], that we provide below for the convenience of the reader.

Lemma 2.2. Letχ: [t₀,+∞)→[0,+∞)be decreasing. We assume that there existk, a∈(0,+∞)andb∈(1,+∞)such that

(2.12) T > t≥t₀ =⇒χ(T)≤ k

(T −t)^a(χ(t))^b. Then it results that

(2.13) χ(t₀+d) = 0 where d=

"

k(χ(t₀))^b−12^(b−1)^ab

#¹_a . We use the previous Lemma2.2and we have

(2.14) |{u^β > t₀+d}|= 0

that is

(2.15) u^β ≤t₀+d

almost everywhere inΩ, where d=c

kMk_L^r_(Ω) µ

¹_p

{u^β > t₀}

h

1−¹

r

p∗ p−1

i

1 p∗

2

1−¹

r

p∗ p

h

1−¹

r

p∗ p−1

i⁻¹ . In order to get the right hand side of (1.7), we control |{u^β > t0}| by means of|Ω|and we take a sequence{(t₀)_m}_m with(t₀)_m → sup_∂Ωu^β. Let us show how we obtain the left hand side of (1.7): we apply the right hand side of (1.7) to−u. This ends the proof of Theorem1.1.

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Now we are going to prove Lemma1.2.

Proof. We assume that A, A˜ ∈ R^n×n withA˜^β = 0 andA˜^α = A^α forα 6= β.

Then

X

α

|A^α_i|² =|A^β_i|²+X

α6=β

|A^α_i|² (2.16)

=|A^β_i −A˜^β_i|²+X

α6=β

|A˜^α_i|²

=X

α

|A^α_i −A˜^α_i|²+X

α

|A˜^α_i|² so

(2.17) |A_i|² =|A_i−A˜_i|²+|A˜_i|². Sincep_i ≥2, the previous equality gives

(2.18) |A_i|^pⁱ ≥ |A_i−A˜_i|^pⁱ +|A˜_i|^pⁱ. Moreover

(2.19) h(detA)≥inf

R

h=h(0)−[h(0)−inf

R

h] =h(det ˜A)−[h(0)−inf

R

h].

Now we are able to estimatef(x, A)andf(x,A)˜ by means of (2.18) and (2.19)

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as follows:

f(x,A) + (min˜

j a_j)

n

X

i=1

|A_i−A˜_i|^pⁱ (2.20)

≤

n

X

i=1

a_i|A˜_i|^pⁱ +m(x)h(det ˜A) +

n

X

i=1

a_i|A_i−A˜_i|^pⁱ

≤

n

X

i=1

ai|Ai|^pⁱ +m(x)h(detA) +m(x)[h(0)−inf

R

h]

=f(x, A) +m(x)[h(0)−inf

R

h]

thus the “monotonicity” inequality (1.4) holds true withµ= min_ja_jandM(x) = m(x)[h(0)−inf_Rh]. This ends the proof of Lemma1.2.

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References

[1] L. BOCCARDO, P. MARCELLINI ANDC. SBORDONE,L^∞-regularity for a variational problems with sharp non standard growth conditions, Boll.

Un. Mat. Ital., 4-A (1990), 219–226.

[2] E. DE GIORGI, Sulla differenziabilitá e l’analiticitá delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat., 3 (1957), 25–43.

[3] E. DE GIORGI, Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital., 4 (1968), 135–137.

[4] M. GIAQUINTAANDE. GIUSTI, On the regularity of the minima of vari- ational integrals, Acta. Math., 148 (1982), 31–46.

[5] O. LADYZHENSKAYAAND N. URAL’TSEVA, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.

[6] J.L. LIONS, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Gauthier - Villars, Paris, 1969.

[7] G. STAMPACCHIA, Equations Elliptiques du Second Ordre a Coeffi- cientes Discontinus, Semin. de Math. Superieures, Univ. de Montreal, 16, 1966.

[8] B. STROFFOLINI, Global boundedness of solutions of anisotropic varia- tional problems, Boll. Un. Mat. Ital., 5-A (1991), 345–352.

[9] Q. TANG, Regularity of minimizers of non-isotropic integrals of the cal- culus of variations, Ann. Mat. Pura Appl., 164 (1993), 77–87.

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[10] M. TROISI, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18 (1969), 3–24.