Differentiability for bounded minimizers of some anisotropic integrals

᎐650 2001 doi:10.1006rjmaa.2000.7186, available online at http:rrwww.idealibrary.com on

Differentiability for Bounded Minimizers of Some

Anisotropic Integrals

A. Canale

Dipartimento di Ingegneria dell’Informazione e Matematica Applicata, Uni¨ersita di Salerno, Facolta’ di Scienze, Via S. Allende,`

( )

84081 Baronissi Salerno , Italy

A. D’Ottavio and F. Leonetti

Dipartimento di Matematica Pura ed Applicata, Uni¨ersita di L’Aquila,` 67100 L’Aquila, Italy

and M. Longobardi

Dipartimento di Ingegneria dell’Informazione e Matematica Applicata, Uni¨ersita di Salerno, Facolta’ di Scienze, Via S. Allende,`

( )

84081 Baronissi Salerno , Italy Submitted by Arrigo Cellina

Received February 9, 1999

We prove the existence of second weak derivatives for bounded minimizers u: n N _{Ž< <}2 _{< <}q

. Ž .

⍀ ; R ª R of the integral H Du q D u⍀ n dx, when 2- q F 2 n y 1 r Žny 3 , n G 4. This allows us to improve on the Hausdorff dimension of the. singular set of u. 䊚 2001 Academic Press

1. INTRODUCTION Let us consider the integral functional

I u

Ž .

s F Du x

_H

Ž

Ž .

.

dx ,

Ž

1.1

.

⍀

where ⍀ ; Rn _{is a bounded open set, u: ⍀ ª R}N_{, and F verifies the}

growth condition

< <p _{< <}q

c z1 y c F F z F c z q c ,2

Ž .

3 4

Ž

1.2

.

640 0022-247Xr01 $35.00

Copyright䊚 2001 by Academic Press All rights of reproduction in any form reserved.

for some positive constants c , c , c , c , p, and q with 1_{1 2 3 4} - p F q.

Regu-Ž .

larity properties of minimizers of 1.1 have been deeply studied when

w x

ps q; see 13, 16 . When p - q we say that F has ‘‘non standard’’ growth

w x

or ‘‘ p, q growth,’’ following Marcellini 21, 22 . Model functionals with ‘‘non standard’’ growth are

ny1 2 q < < < < I u1

Ž .

s

_H

ž

a

_Ý

D ui q b D un

/

dx ,

Ž

1.3

.

⍀ is1 < <2 _{< <}q I u2

Ž .

s

_H

_Ž

Du q D un

_.

dx ,

Ž

1.4

.

⍀

where a, b, q are positive constants with 2- q, D u sj ⭸ ur⭸ x , Du sj

ŽD u, . . . , D u . In these functionals the last component of the gradient1 n . D u has a different exponent q: because of the different behaviour withn

respect to the x axis, we say that I and I are anisotropic integrals; seen 1 2 w x29 for connection with some reinforced materials. Let us recall that ‘‘non

Ž .

standard’’ growth condition 1.2 allows minimizers to be singular, if p and

ny 1

Ž .

q are too far apart. More precisely, for the integral 1.3 , if 2_{ny 3}- q, then

w x

minimizers may be unbounded 14, 17, 20 . On the other hand, when

ny 1 _{w x}

2- q F 2ny 3, scalar minimizers u: ⍀ ª R are bounded 12 . Related

w x

results are in 5, 8᎐10, 19, 23, 24, 27, 28 . Let us explicitly mention the w x

maximum principle of 11 which applies also to vector-valued mappings u:

N _{Ž . Ž .}

⍀ ª R minimizing 1.3 or 1.4 : if u is bounded on ⭸ ⍀, then it is bounded in ⍀ too. Existence of second weak derivatives has been proved

2 n 2 n ny 1

w x

in 3, 18 when 2- q -_{ny 2}. Note that _{ny 2}- 2_{ny 3} for nG 4. In this paper we show that bounded minimizers have second weak derivatives also

2 n ny 1

when _{ny 2}F q F 2_{ny 3}, nG 4. This higher differentiability property allows us to estimate the Hausdorff dimension of the singular set. More precisely,

ny 1

N _{Ž .} 0,␥_{Ž .}

if u: ⍀ ª R minimizes 1.4 with 2 - q - 2ny 3, then Dug Cloc ⍀ ,0

n_{Ž .} w x s

for some␥ ) 0 and some open set ⍀ ; ⍀ with HH ⍀ _ ⍀ s 0 1 . HH0 0 is the s-dimensional Hausdorff measure. The existence of second deriva-tives allows us to improve on the estimate of the singular set ⍀ _ ⍀ of0

ny2q⑀_{Ž .}

bounded minimizers: HH ⍀ _ ⍀ s 0, for every₀ ⑀ ) 0. In the frame-w x work of partial regularity under ‘‘ p, q growth,’’ let us also mention 26 . In

Ž .

our paper we deal with functionals whose model is 1.3 , 2- q. The case

w x

q- 2 has been studied in 4, 7 .

2. NOTATIONS AND RESULTS

Let ⍀ be a bounded open set of Rn_{, nG 3, u: ⍀ ª R}N_{, NG 1. Let us}

consider variational integrals

I u

Ž .

s F Du x

_H

Ž

Ž .

.

dx ,

Ž

2.1

.

n N 2_Ž n N_.

where F: R ª R, F g C R and for some positive constants

c, m, M, q, ny1 2 q <F z

Ž .

<F c 1 q

ž

_Ý

< <zj q z< <n

/

,

Ž

2.2

.

js1 1y1r2 ny1 ⭸ F _{< <}2 _{< <}q z F c 1 q z q z if is 1, . . . , n y 1, 2.3

Ž .

_Ý

j n

Ž

.

␣

ž

/

⭸ zi js1 1y1rq ny1 ⭸F _{< <}2 _{< <}q F c 1 q

_Ý

zj q zn ,

Ž

2.4

.

␣

ž

/

⭸ z zn

Ž .

js1 ny1 n N _{⭸ F}2 2 qy2 2 _{␣ ␤} < < < < < < m

ž

_Ý

␭ q zi n ␭n

/

F

_Ý

_Ý

_{␤ ␣}

Ž .

z ␭ ␭i j ⭸ z ⭸ zj i is1 i , js1 ␣ , ␤s1 < <2 _{< <}qy2_{< <}2 F M

_Ž

␭ q zn ␭n

_.

Ž

2.5

.

n N _ ␣_{4 } ␣₄ < <2

for any ␭, z g R , ␣ s 1, . . . , N, where ␭ s ␭ , z s z , ␭ si i N < ␣<2

Ý_␣s1 ␭ . About q we assume thati

2- q - q ,n

Ž

2.6

.

where q is defined as follows:_n

¡

20q 8 6

'

if ns 3

'

6q 4 2 if ns 4

'

28q 8 10

~

qns _{if n}s 5

Ž

2.7

.

9 2 ny 4 if nG 6.

¢

ny 4 Ž . Ž . Ž . Ž .

We remark that the integrands of 1.3 and 1.4 satisfy 2.2᎐ 2.5 . We say

N

Ž .

that u:⍀ ª R minimizes the integral 2.1 if

1 , 1 < <qr2 2

ug W

Ž

⍀

.

D u, . . . , D1 ny1u, D un g L ⍀ ,

Ž

.

Ž

2.8

.

and

I u

Ž .

F I u q

Ž

␾

.

N 1, 1_{Ž .} < <qr2

for any ␾: ⍀ ª R with ␾ g W0 ⍀ , D1␾, . . . , Dny1␾, D ␾n g

2_{Ž .}

We prove the following higher integrability result:

⬁ _{Ž . Ž . Ž .}

THEOREM 1. If ug L_loc ⍀ minimizes the integral 2.1 under 2.2 ᎐ Ž2.7 , then.

< <qr2 4Ž qy1.r q

D u, . . . , D1 ny1u, D un g Lloc

Ž

⍀ .

.

Ž

2.9

.

Remark 1. Note that

qy 1

2- 4 - q.

Ž

2.10

.

q

This higher integrability result allows us to get higher differentiability. THEOREM2. Under the assumptions of Theorem 1 we ha¨e

< <Ž qy2.r2 1 , 2

D u, . . . , D1 ny1u, D un D un g Wloc

Ž

⍀ .

.

Ž

2.11

.

2 _{Ž .}

In order to prove existence of D D u_{n n} g L_loc ⍀ we need no

degenera-Ž .

tion, with respect to z , in the left hand side of 2.5 . More precisely we_n suppose that, for some positive constant m,

_˜

n N _{⭸ F}2 2 _{␣ ␤} < < m

˜

␭ F

_Ý

_Ý

_{␤ ␣}

Ž .

z ␭ ␭i j

Ž

2.12

.

⭸ z ⭸ zj i i , js1 ␣ , ␤s1

for any␭, z g Rn N_{. We prove}

Ž .

THEOREM 3. Under the hypotheses of Theorem 1, if in addition 2.12 holds, then

D ug W1 , 2 _{⍀ . 2.13}

Ž

.

Ž

.

n loc

Ž . Ž .

We remark that the integrand F of 1.4 satisfies 2.12 but the one in Ž1.3 does not. Boundedness of minimizers of variational integrals 2.1 has. Ž .

w x

been proved in 11, 12 under additional assumptions, so these and our results merge into the following corollaries.

N _{Ž .}

COROLLARY 1. Let u: ⍀ ª R minimize the ¨ariational integral 1.3 where 2- q - q . Ifn ny 1 q Ns 1, qF 2 , ug Lloc

Ž

⍀

.

ny 3 or NG 1, u is bounded on⭸ ⍀,

then ug L⬁loc

Ž

⍀

.

and < <Ž qy2.r2 4Ž qy1.r q 1 , 2 D u, . . . , D1 ny1u, D un D un g Lloc

Ž

⍀ l W

.

loc

Ž

⍀ .

.

ny 1 2 n ny 1

We agree that 2_{ny 3}s q⬁ when n s 3. Please note that _{ny 2}- 2_{ny 3} -q for n_n G 4.

Ž .

COROLLARY 2. Under the assumptions of Corollary 1 with 1.3 replaced

Ž .

by 1.4 , we ha¨e

ug L⬁loc

Ž

⍀

.

and

< <Ž qy2.r2 4Ž qy1.r q 1 , 2 Du, D un D un g Lloc

Ž

⍀ l W

.

loc

Ž

⍀ .

.

Ž . w x 0,␥Ž .

For minimizers u of 1.4 , it has been shown in 1 that Dug C_loc ⍀ ,₀

n_{Ž .} s

for some␥ ) 0 and some open ⍀ ; ⍀ with HH ⍀ _ ⍀ s 0, where HH_{0 0} is the s-dimensional Hausdorff measure. The existence of second deriva-tives in L2 _{is very important in order to estimate the Hausdorff dimension}

w x w x

of the singular set ⍀ _ ⍀ ; see 13, 15 . This has been done in 18 only for0

2 n 2 n

q- _{ny 2}. Here we are able to achieve the result also for _{ny 2}F q

-ny 1

 4

min 2ny 3, qn when Ns 1 or when N ) 1 and u is bounded on ⭸ ⍀, so w1, 15 and our results merge into the followingx

COROLLARY3. Under the assumptions of Corollary 2, if ny 1

2- q - min 2

½

, q_n

5

, ny 3 then

< <Ž qy2.r2 4Ž qy1.r q 1 , 2 Du, D un D un g Lloc

Ž

⍀ l W

.

loc

Ž

⍀

.

and

Dug Cloc0 ,␥

Ž

⍀0

.

for some␥ ) 0 and for some open set ⍀ ; ⍀ such that₀ H

3. KNOWN RESULTS

For a vector-valued function f, we define the difference ␶ f x s f x q he y f xs , h

Ž .

Ž

s

.

Ž .

where hg R, e is the unit vector in the x direction, and s s 1, 2, . . . , n.s s

n _{Ž .}

For x0g R , let B x be the ball centered at x with radius R. WeR 0 0 will often suppress x whenever there is no danger of confusion.0

We now recall some lemmas that are crucial to our work. In the following f :⍀ ª RN, NG 1.

Ž .

LEMMA 3.1 how to control differences by derivatives . If f, D f_s g

t Ž . L B3 R , with 1F t - q⬁, then < <t _{< <}t _{< <}t ␶ f x

Ž .

dxF h D f x

Ž .

dx

H

s , h

H

s BR B2 R < < Ž w x . for any h, with h - R. See 13 .

Ž . tŽ .

LEMMA3.2 how to get derivatives from differences . Let fg L B_{2 R} , 1- t - q⬁. If there exists a positi_¨e constant C such that

< <t _{< <}t ␶ f x

Ž .

dxF C h

H

s, h BR < < t Ž . Ž w x .

for any h, with h - R, then there exists D f g L B . See 13 ._{s R} Ž

LEMMA3.3 how to get higher integrability from differences by fractional

. 2Ž . Ž .

Sobolev spaces . If fg L B_{3 R} and for some dg 0, 1 and C ) 0

n 2 2 d <␶ f x

Ž .

< dxF C h< <

Ý

H

s, h BR ss1 < < r_{Ž . Ž . Ž}

for any h, with h - R, then f g L BRr4 for any r- 2nr n y 2 d . See w x .2 .

Ž .

LEMMA 3.4 how to control translations . For any t, with 1F t - q⬁, there exists a positi¨e constant C such that

t t < < < < < < 1q f x q

Ž .

␶ f x

Ž .

dxF C

Ž

1q f x

Ž .

.

dx

Ž

.

H

s , h

H

BR B2 R t_{Ž .} < <

LEMMA3.5. For any qG 2, G: B ª RN_{, we ha¨e} 2 R 2 Ž qy2.r2 < < ␶s , h

_Ž

G x

Ž .

G x

Ž .

_.

2 q 1 _{qy2 2} 3 < < < < F N

ž /

_H

G x

Ž .

q t␶ G xs , h

Ž .

␶ G xs, h

Ž .

dt 2 0 < <

for any h, with h - R, for any s s 1, . . . , n and for any x g B .R

4. PROOF OF THEOREM 1

Ž . Ž .

Under growth conditions 2.2 ᎐ 2.6 , the function u, which minimizes

Ž .

the integral 2.1 , solves Euler’s equation

n N _{⭸ F} ␣ Du x

Ž .

D␾

Ž .

x dxs 0

Ž

4.1

.

Ž

.

Ý Ý

H

_␣ i ⭸␰ ⍀is1␣s1 i N 1, 1_{Ž .}

for all functions_qr2 ␾: ⍀ ª R , with ␾ g W0 ⍀ , D1␾, . . . , Dny1␾, 2

<Dn␾< g L ⍀ . Let R ) 0 be such that B ; ⍀, let B and B beŽ . 4 R ␳ R

concentric balls, 0-␳ - R, and let ␩: Rnª R be a ‘‘cut off’’ function in ⬁_{Ž .}

C B0 R with

< <

␩ ' 1 on B , 0 F ␩ F 1_␳ and D␩ F 2r R y ␳ .

Ž

.

Ž 2 . Ž .

We use the test function ␾ s ␶_s,yh␩ ␶ u in 2.1 ; thus, using the_{s, h} hypotheses and Lemma 3.5, in a standard way, we get Caccioppoli’s estimate ny1 2 Ž qy2.r2 2 <␶ D u q ␶< <

_Ž

<D u< D u

_.

< dx

Ý

H

ž

s , h i s, h n n

/

B_{␳ is1} < <qy2 _{< <}qy2 _{< <}2 F c0

H

Ž

1q D un q␶ D us, h n

.

␶ u dx,s , h

Ž

4.2

.

BR Ž .

where c₀s c m, M, q, N,₀ ␳, R is a positive constant. Let us assume that qy 1

qr2 ␴

< <

D u, . . . , D1 ny1u, D un g Lloc

Ž

⍀

.

for some 2F␴ - 4 . q

4.3

q␴ q␴

By Holder’s inequality with exponents

_¨

2 qŽ y 2.,q␴ y 2 q y 2Ž ., we have < <qy2 _{< <}qy2 _{< <}2 1q D u q␶ D u ␶ u dx

Ž

.

H

n s, h n s , h BR Ž . 2 qy2 rq␴ q␴ r2 q␴ r2 < < < < F c₁

_ž

_H

_Ž

1q D u_n q␶ D u_{s, h n}

_.

_/

BR Žq␴y2 qy2 rq␴Ž .. 2 q␴ rŽq␴y2Žqy2.. < < =

ž

_H

␶ us , h

/

BR s c I II .1

Ž . Ž

.

Ž

4.4

.

Ž .

Using Lemma 3.4 and hypothesis 4.3 we get Ž . 2 qy2 rq␴ q␴ r2 < < I F c 1q D u dx - q⬁. 4.5

Ž .

2

ž

_H

_Ž

n

_.

/

Ž

.

B2 R Since 2 q␴ )␴ q␴ y 2 q y 2

Ž

.

Ž .

and u is locally bounded, by Lemma 3.1 and assumption 4.3 Žq␴y2 qy2 rq␴Ž .. 2 q␴ rŽq␴y2Žqy2..y␴ ␴ 5 5 < < II F 2 u ␶ u

Ž

.

_{L Ž B}⬁ _.

H

_{s , h} 2 R

ž

BR

/

Žq␴y2 qy2 rq␴Ž .. ␴ ␴ Ž q␴y2Žqy2..r q < < < < < < F c h3

ž

_H

D us dx

/

F c h4 ,

Ž

4.6

.

B2 R q␴ y 2 q y 2Ž . Ž . Ž . Ž . Ž .

where 0- _q␴ - 2. From 4.2 , 4.4 , 4.5 , and 4.6 we obtain

n ny1 2 Ž qy2.r2 2 Ž q␴y2Žqy2..r q <␶ D u q ␶< <

_Ž

<D u< D u

_.

< dxF c h< < .

Ý

H

ž

Ý

s , h i s, h n n

/

5 B_␳ ss1 is1

Applying Lemma 3.3 we get

2 n qr2 r < < D u, . . . , D1 ny1u, D un g Lloc

Ž

⍀

.

᭙r - _{q␴ y 2 q y 2} .

Ž

.

ny q 4.7

Ž

.

Ž . Ž .

The hypothesis 2.6 implies that there exists ␦ s ␦ n, q ) 0 such that

2 n qy 1 y␴ G ␦ ᭙␴ g 2, 4 .

Ž

4.8

.

q␴ y 2 q y 2

Ž

.

q ny q qy 1 w w

Let us summarize as follows: if we know that for some ␴ g 2, 4 _q we < <qr2 ␴ _{Ž .}

have D u, . . . , D1 ny1u, D un g Lloc ⍀ , then we gain a small amount ␦r2 of integrability independent of ␴ ; namely,

qy 1 qr2 ␴ < < D u, . . . , D1 ny1u, D un g Lloc

Ž

⍀ ,

.

2F␴ - 4 q y < <qr2 _{␴q␦ r2} D u, . . . , D1 ny1u, D un g Lloc

Ž

⍀ .

.

The iterative scheme

␴ s 2 q i␦r2,i iG 0, achieves our goal.

5. PROOF OF THEOREM 2

Ž .

We argue as in the proof of Theorem 1 until we get 4.5 . Now, using

Ž . Ž .

higher integrability 2.9 , 4.3 is fulfilled with qy 1

␴ s 4 .

q For such a value of ␴ we get

2 q␴

s␴ ; q␴ y 2 q y 2

Ž

.

Ž . Ž .

then II in 4.4 can be estimated by Lemma 3.1 in the following way: 2r␴ ␴ 2 < < < < II F D u dx h . 5.1

Ž

.

ž

_H

s

/

Ž

.

B2 R

Ž . Ž . Ž . Ž . Using 4.4 , 4.5 , 5.1 in 4.2 , we get n ny1 2 Ž qy2.r2 2 2 <␶ D u q ␶< <

_Ž

<D u< D u

_.

< dxF c h . 5.2< <

Ž

.

Ý

H

ž

Ý

s , h i s , h n n

/

6 B_␳ ss1 is1 Ž . Ž .

Finally 5.2 and Lemma 3.2 imply 2.11 .

6. PROOF OF THEOREM 3

Ž .

We choose in Euler’s equation 4.1 the same test function ␾ as in the proof of Theorem 1. We get the following Caccioppoli’s estimate using Ž2.12 instead of the left hand side in 2.5 :. Ž .

< <2 _{< <}qy2 _{< <}qy2 _{< <}2

␶ Du F c

_Ž

1q D u q␶ D u

_.

␶ u dx.

H

s , h 7

H

n s, h n s , h

B_␳ BR

As in the proof of Theorem 2 we obtain <_{␶ Du F c h .}<2 _{< <}2 6.1

Ž

.

H

s , h 8 B_␳ Ž .

Inequality 6.1 and Lemma 3.2 give the assertion.

ACKNOWLEDGMENTS

Ž .

We acknowledge the support of M.U.R.S.T. 60%, 40% and G.N.A.F.A._᎐C.N.R.

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