Relaxation of degenerate variational integrals

Nonlinear Anolysrs, Theory, Methods & Appl~cufrons, Vol. 22, No. 4, pp. 409-424, 1994 CopyrIght 8 1994 Elsevier Science Ud Printed m Great Britain. All rights reserved 0362-546X/94 $6.00+ .OO

Pergamon

RELAXATION

OF DEGENERATE

VALERIA CHIAD~ P1AT-f

VARIATIONAL

INTEGRALS

and FRANCESCO SERRA CASSANO$

t Dipartimentodi Matematica, Politecnico di Torino, C.so Duca degli Abruzzi 24, I-10129, Torino; and $ Dipartimento di Matematica, Universita degli Studi di Trento, via Sommarive 14, I-38050, Povo (TN), Italy (Received

12 October

Key words and phrases:

1992; received for publication

Degenerate

variational

integrals,

2 June 1993) relaxation

methods.

1. INTRODUCTION LET~:R”xR”-+

[0, +a[

be a Caratheodory

function

f(x, *) is convex Then the functional u ++

such that

for a.e. x E R".

(1.1)

s

j-(x, Du) dx

(1.2)

D

is naturally defined for every open set Sz c R" and for every u E C’(n). Under suitable assumptions it can be proved that, for every bounded open set a, the functional considered in (1.2) is lower semicontinuous on e’(Q) for the weak convergence in IV” ‘(a) (see [l]). Moreover, it can be extended to wider function spaces, preserving its lower semicontinuity property. A possible way to extend it, preserving also the integral form, is to set F(i-2, 24)=

s

f(x, Du) dx

for every u E IV’,‘(Q).

(1.3)

a

If one is interested in minimum problems related to the functionals arises about the comparison between the corresponding infima,

(1.2), (1.3), a natural question i.e. for instance, whether

or not, for a given function u. E L’(Q). When this happens we say that the Lavrentiev phenomenon occurs (see [2, 31). On the other hand, the Lavrentiev phenomenon is forestalled if one extends the functional in (1.2) in a different way, by setting F(Q, u) = inf for every bounded easy to see that F semicontinuous on that it gives rise to

Then it is natural

f(x, Du,) dw: uh E e’(a),

liEi_nf (

uh - u weakly in W’,‘(a)

.i a

(1.4) 1

open set Q and every u E W’*‘(Q) (see, for example, [4]). In this case it is coincides with the functional in (1.2) for every u E e’(a), that it is lower the space W “‘(L2) with respect to the weak convergence in W “‘(Cl), and a minimum problem with the same infimum, i.e.

to study the cases when F and F coincide. 409

V. CHIAL~ PIAT and F. SERRA CASSANO

410

Generally, the type

this problem J.(x)lV

has been

studied

under

5 f(x, 0 5 A@)(1 + IV)

growth

and

coerciveness

conditions

for a.e. x E Rn, v lj E R”,

of (1.3

with p > 1, A and A positive measurable functions in Lo,,,, A-l/@-l) E L,&(R”). Under these assumptions the weak convergence in W’*‘(Q) can be replaced by the strong convergence in L’(Q), and the infimum of the “relaxed” problem concerning the functional F over W1pl(Q) turns out to be a true minimum. The first result in this context is probably due to Meyers and Serrin [5] who proved the density of C?‘(a) in W1vp(sZ) (H = W) from which the identity F = F follows when f satisfies (1.5) with A and A positive constants (see remark 3.6 below). Moreover, in another of Serrin’s papers [4] the same result is proved under less restrictive conditions on f,but assuming in particular the global regularity off. In the special case n = 1 the identity between F and F is true under the only assumption (1.1) and a weak coercivity condition on F (see [3, 61). When f(x, -) is just a nonnegative quadratic form, an interesting approach to the problem within the framework of Dirichlet forms can be found in [7]. A situation in which F # F is presented in [6, remark 1.51, where an explicit two-dimensional example is constructed in the case when f satisfies (1.5) with A E 1, A E Lp(R2), and 1 < p < 2. In the present paper we give a first positive answer to this problem when A/1 E L”(R”) and I is a weight in the Muckenhoupt class A, (see theorem 3.3) by the classical techniques of approximation by convolution in a suitable weighted Sobolev space. On the other hand, we construct a weight A that is not in the A,-class, for which the approximation by convolution does not converge in the right way (see proposition 4.3). This seems to suggest that these techniques are not appropriate to treat the problem of the identity between F and F for more general A. The plan of the paper is the following. Section 2 is devoted to notation and preliminary results concerning weighted Sobolev spaces and the Muckenhoupt class A,. In Section 3 we introduce two integral functionals F,(C2, u) and F2(Q, u) defined as in (1.2) for u E C?‘(Q) and u E C?‘(R”), respectively, and we prove that in both cases the corresponding “relaxed” functionals F, and F, coincide with F if A/I E L”(R”) and ,r, E A, (theorem 3.3). Finally, Section 4 contains a counterexample (proposition 4.3 and corollary 4.5) to the approximation by convolution in weighted Sobolev spaces with a weight E, $ A,.

2. NOTATIONS

AND

PRELIMINARY

Let us fix a real number p > 1. Given a measurable p: A + R we say that p is a p-weight on A if p > 0 Given

a.e. on A,

an open set n c R” and a p-weight

RESULTS

set A c R” and a measurable

p,,u--“@-‘I

E L;,,(A).

p on R”, we denote

by P(sZ, p) the space

function

(2.1)

411

Degenerate variational integrals

endowed

with the topology

induced

by the norm

IIUIILP(a,P)= (1. Moreover,

we indicate

by Lf,,(Q

IUP- tiy’p.

p) the set

L~,,@-2,p) = (2.4E L:“,(sz): v a’ c n Finally,

we introduce

the weighted

w’~P(C2, p) =

Sobolev

U E w,;;‘(sz)

with the topology

induced

(2.3)

space IDulPp dx < +a,

: b+L

i endowed

.?4E LfyQ’,p)).

(2.4)

13

by the norm

LEMMA 2.1. If p > 1 and p is a p-weight embedded in W1p’(C2).

on R”, then

W1’p(S& p) is a reflexive

Banach

space

Proof. It is easy to verify that W ““(Cl, p) is a Banach space embedded in W ‘, ‘(Cl). Let us prove that it is reflexive. Given a bounded sequence (u& in Wlsp(C2, p), since W’*‘(a) is compactly embedded in Lo,, and Lp(Q,u) is itself reflexive, there exists a function u E L,!,,(Q) with DU E (Lp(Q p))“, such that, up to a subsequence u,, + u in L&,(l2) and Du, - DU weakly in (Lp(Cl, p))“. Then, since (u~)~ is actually bounded in L’(Q), the result follows by the monotone convergence theorem. n In the following, we shall denote by Q any (open or closed) cube of R” with sides parallel to the coordinate axes, and by Q, any cube Q centered at x E R”. Moreover, we shall use the notation

to denote the average of a functionf E L’(A), where A c R” is bounded and IA I is its Lebesgue measure. We set B,(x) = (y E R” : Ix - yl < rf for every x E R”, r > 0. Finally we denote by xs the characteristic function of a measurable set S of R”. Definition 2.2. Let p > 1, k > 1 and let R be a rectangle of R” (or R” itself). p-weight Iz on R” is in the Muckenhoupt class A,(R, k) if (~~“~~(s,“-l/‘p-“dx)yl for every cube Q contained

in R. We set A,(R)

Definition 2.3. Given f E L&(R”),

(MfW is called the maximal function

off.

the function

= sup

I k, =

(2.5)

Uk, 1A,(R, k).

Mf defined

B(x) If( r>O f ,

We say that a

dv

by x E R”

(2.6)

412

V. CHIAD~

PIATandF.SERRACASSANO

We remark that for everyf E L&,(R"), the maximal function AJ_f is defined a.e. on R". The following theorem shows how it is related to the class A,(R), introduced in definition 2.2. THEOREM 2.4 (Muckenhoupt [8]). Let R be a rectangle of R", or R" itself, p > 1, and ,D be a p-weight on R". Then the following conditions are equivalent: (i) there exists c > 0 such that >

lMflPp dx

5

c

.R / (ii) JI E A,(R).

e

vf

R Ifl”P~

E

Lp(R,~1;

(2.7)

A simple proof of theorem 2.4 when R = R" can be found in [9]. The following partial extension result holds for the A,-weighted Sobolev

spaces.

THEOREM 2.5.Let Sz be a bounded open set of R" with Lipschitz boundary and let I E A,(R"). Then there exists an integer N such that for every i = 0, 1, . . . , N there exist a p-weight Ai E A,(R"),a linear and continuous operator P;.: W1*p(fi, A) + W""(R", pi),and an open rectangle Ri of R" centered on aa, such that: (i) lJy= 1 Ri 2 Xl; (ii) I, = 1 in R", li= A in Sz fl Ri (i = 1, . . ., N); (iii) CfT_O~~~~in~,spt(P,u)~~,spt(~~)CRi(i= l,...,N)foreveryu~W’*“(Q,A). The proof

of theorem

2.5 relies on the following

three lemmas.

LEMMA 2.6.Let p > 1, k > 1, and let R be a rectangle of R" (open or closed). Suppose that I E A,(R, k). Then there exists an extension x to the whole of R" such that 1 E A,(R",ii) for a suitable positive constant k = k(n,p, A, k). In the sequel we shall use the following J = l-1, J+ = l-1, and if

notation

I[“, Jo = l-1, l[n-’

l[‘-l

x (O],

x IO, I[, J- = l-1,

f: J+ + R, we shall denote by f: J -+ R the even extension f(x’,x,)

=

f(x’T x,) f(x', -4J

(2.8)

l[nP1 x ]-l,O[, off,

i.e.

if x E Jf, ifxE

J-

(2.9)

where x = (xi , . . . , x,_, , x,) = (x’, x,). LEMMA 2.7.Let p > 1, k > 1, and J, be a p-weight on J+, of class A,(J+, function defined in (2.9) with f = A. Then 1 is a p-weight on J of class A,(J, constant 2 = k(n,p, k, ,I). The proofs

of lemmas

2.6 and 2.7 are just technical

and so they are omitted

k). Let i; be the I?) for a suitable

(see also [lo]).

Degenerate variational integrals

413

LEMMA 2.8. Letp > 1, k > 1. Given ap-weight A E Ap(Ji, k) and a function u E W1~p(J+, A), let 1 and U be the even extensions defined in (2.9) with f = A and f = u, respectively. Then u E W’*P(J, X),

D~(X’, X,) =

Di”(x’,

Xn)

ifxEJ+,O
DiU(X’,

-X,)

ifxeJ-,O5iln-

-D,

1,

(2.10)

if x E J-,

U(X’, -x,)

and (2.11)

Ilfi IIFV’.P(J,K)5 2llnll w’qJ+,x).

Proof. By the extension theorem for classical Sobolev spaces (see, for instance, [ 11, theorem 1X.71) it follows that U E W’,‘(J) and that (2.10) holds. To conclude the proof it is sufficient to show that IDU 1 is summable in J. By (2.9) and (2.10)

px

ID+‘, -x,)l”A(x’, = 2

-x,)

dx’

lDulpl dx.

(2.12)

,i .I+ Then,

by (2.12) we have that U E Wlsp(J, 2) and that (2.11) holds.

Proof of theorem 2.5. Let J, J’, and Lipschitz-continuous (see [lo, rectangles Ri (i = 1, 2, . . . , N) of Qi: Ri + J with Lipschitz inverse,

H

Jo be the sets defined proposition R” centered such that

~i(Ri n ~2) = J+,

in (2.8). Then, since a0 is bounded 1 .l]), there exist a finite number N of open on aQ verifying (i), and N Lipschitz maps

Qi(Rj

n asz) = Jo

(2.13)

for every i = 1,2, . . ., N. Let R, = R”\aQ and let (Bi]y= o be a partition of unity associated to the open cover of R” (Ri]yEo, i.e. for every i = 0 , . . . , N tli E C?“(R”), 0 I Bi I 1, and the following conditions are satisfied spt e. c R”\ac&

Spt Bi c

R;

;&El

ifi=

1,2 ,...,

N,

in R”.

(2.14) (2.15)

i=O

Let us set

(2.16)

Ui = 8iU, for every u E W’sp(CA, I) and every i = 0, 1, . . . . N. We first extend Co(x) =

UO

ifxEQ

0

if x E R”\SZ.

u. by defining

414

V. CHIAD~ PIAT and F. SERRA CASSANO

If we rename A = A,, by (2.154, the weighted Poincare inequality (see, for instance, [lo, theorem 1.71) and the fact that 8, E C’(R”) Cl L”(R”), De, E L”(R”), and DC, = &,Du + uD8, in 0, it follows that t7, E W1*P(R”, A,). Moreover, there exists a positive constant c0 such that llfi0 IIFV'.P(R",X,,) s cOiiullW'J'(n,X) for every u E W’Sp(fi, A). For ui(Q;‘(y)) for y E J+. Then, lemma 1.7 in [12], pi E A,,(J+, defined in (2.9) with f = pi, f by lemma 2.8 that Ci E W”D(J, Let us set

(2.17)

every i = I,2 , . . ., N let us set ,~~(y) = A(@,;‘(y)) and vi(y) = by the weighted Poincare inequality 2); E W’*P(Jf,pi) and, by ki) (for a suitable constant ki > 1). Let ,Uj, pi be the functions = Ui, respectively. By lemma 2.7 we have that pi E A,(J, ki) and pi). Moreover, by lemma 2.6 we can suppose that ,& E A,(R”).

IziCx>

=

(2.18)

XER,,

Di(@iCx))

X E

Wj(X) = &($(x))

Ri,

(2.19)

for every i = 1,2, . . . . N. By lemma 1.7 in [ 121, ili turns out to be of class A,(Ri) ki) for a suitable constant ki > 1; on the other hand, by lemma 2.6, we can suppose that iii E A,(R”). By (2.19) we have that Wi E W”“(Ri, 13i), wi = Ui in s2 n Ri and that there exists a positive constant Ci > 0 such that

IIwiI/ for every u E W1~p(Q, A). Then,

Poincare

inequality, Ui=ui

Finally, if we define the operator (ii) and (iii) follow. H

6i Cx) wi

o

cillull

W’J’(flnR,,X)

(2.20)

Cx)

if

X E

Ri

ifxER”\R;

(2.21)

we have that iii E W1,P(R”, Ai) and inQflRi,

spt tii c R;.

(2.22)

4: W1*p(C& A) --f W’,p (R”, Ai) as P;.u = pi, by (2.16)-(2.21),

Finally, we recall that, given a functional that F is L’(Q)-lower semicontinuous if

for every Moreover, defined as be proved

5

if we define

t7i(X) = by the weighted

W’,P(R,,X,)

F: @,;‘(R”)

+ R and an open set C2of R”, we say

u E II&’ and for every sequence (Us)* c II&’ converging to u in L’(a). we denote by SC-(L’(Q))F the L’(Q)-lower semicontinuous envelope of F, which is the greatest L l(Q)-lower semicontinuous functional smaller than or equal to F. It can that it can be characterized as follows SC-(L’(fi))F(u)

(see, for instance,

[l]).

= min

lim inf h-+W

: (~4~)~ c

W,A;‘(R”), u,, + u in L’(Q)

(2.23)

415

Degenerate variational integrals 3. SETTING

Given function

a real number

OF THE PROBLEM

p > 1, let A be p-weight

is measurable

f(x, *I MrlP We denote

is convex

of all bounded

constant.

for every < E R”

(3.2)

for a.e. x E R”, v l E R”.

open subsets

We fix a

(3.1)

for a.e. x E R”

If&, 0 5 cW(l + lW

by C? the family

RESULTS

on R” and c be a positive

f: R” x R” -+ [0, +oo[, such that f(-7 0

AND MAIN

of R” and we define

(3.3)

the functionals

F, F, , F, : Q x W,A;‘(R”) -+ [O, + 001 as F(M, u) = R

F,(Q u) =

f(x, Du)ti

i’

F2P2,~1 =

n

f (x, Da) dx

(3.4)

if u E C!‘(a) (3.5)

+oO

otherwise

1

if u E C’(R”)

+m

otherwise.

,R

f (x, Da) dx

(3.6)

It is easy to see that F(CJ, *) is L’(Q)-lower semicontinuous on W,A;‘(R”), for every C2 E a. Our problem is to understand under which conditions the functional F(Q *) coincides with the L ‘(a)-lower semicontinuous envelope of F;(CJ, a) (i = 1, 2), that we shall indicate by E(Q (see (2.23)). In this section ensure the identity between

*) = SC_(L ‘(sz))4(sz,

we present a situation e (i = 1,2) and F.

0)

(3.7)

in which suitable

assumptions

on A and ~2

Remark 3.1. Let us observe that, since F(Q u) I F,(Q u) I F2(Q u) by the definition

of E and by the L’(Q)-lower

F(Q, u) I F,(Q u) 5 &Cl, u)

for Q E a, u E w,&‘(R”), semicontinuity

(3.8)

of F, we have

for C2 E a, u E vA;‘(R”).

(3.9)

Remark 3.2. Let us define Q(Q)

= (u

E

W,:,‘(R”) :e(sZ,

U) < +m)

(3.10)

for i = 1,2 and C2 E @.. Then e(Q, U) = F(C2, u) for every u E oi(n). In fact, let A, = Sz and A, = R”; then, if u E Di(sZ) (i = 1,2) we can suppose that there exists a sequence (u& in WIPP(C& A) tl C?‘(Ai) converging to u weakly in WITp(Q, A). On the other hand (see, for instance, corollary V.3.14 in [13]), we can construct a sequence (z&)* where each term tih is a convex combination of elements of (u& , converging to u strongly in WLPp(sZ, A). In particular

416

V. CHIAD~

(tl&, is in W’PP(Q

A) n e’(&).

and we can conclude

the proof

SERRA CASSANO

PIAT and F.

Th en, by the dominated

convergence

theorem

we have that

by (3.9).

3.3. Let f: R” x R” + [0, +a[ be a function satisfying (3.1)-(3.3) with A a p-weight of class A,(R”) (see definition 2.2). Let F, F;, E: a x W,A;‘(R”) + [0, +m] be the functional defined in (3.4)-(3.7). Then: (i) &(a, U) = F(Cl, u) v Q E Q, u E W,A;‘(R”); (ii) I’,(Q, U) = F(Q, u) v Q E Q. with Lipschitz boundary, u E W,&‘(R”). THEOREM

Before proving

theorem

3.3, we state some preliminary

results that are needed

in the proof.

LEMMA 3.4. Let (ph)* be a sequence of mollifiers, i.e. ph@) = h”p(hx), where p E C?;@?,(O)), p I 0, and jR” p(u) dr = 1. Then there exists a positive constant c such that for every u E Lo,, and for every h E N I@ * PhW

where u * ph is the usual convolution, u*P,(x) and Mu is the maximal For a proof

function

see, for instance

5 Gfw

a.e. on R”,

(3.11)

i.e. =

R”

of u defined [12, lemma

(3.12)

u(x - Y)P~(Y) dy in (2.6).

1.61.

PROPOSITION 3.5. Let p > 1 and let A be a p-weight

of R” with Lipschitz

boundary.

Then

of class AJR”). Let 0 be an open subset C?‘(R”) is dense in W’.p(C& A).

Proof. For every i = 0, 1, . . . , N let ;li, R;, Pi be as in theorem given u E W1*p(Q, A). If we set

2.5, and let fii = Piu for a

N t?=

C

(3.13)

pi,

i=O

then 1? E W,~;‘(R”), as tii E W,&‘(R”), for every i = 0, 1, . . ., N. By (iii) of theorem that spt z& c a,

spt z& c R; fi=u

Given a sequence usual convolution

in s1.

(3.14) (3.15)

of mollifiers (ph)* as in lemma 3.4, we set uh = ti * p,,, where * denotes as in (3.12). Since fi E W,A;‘(R”), by (3.15) we have at once that vh+ti=u

Then,

(i = 0, 1, . . . , N),

2.5 it follows

in order to get the thesis,

it is enough

Dv,,=Dii*p,+Dii=Du

in L’(Q). to prove that in (P(Q

A))“,

the

417

Degenerate variational integrals

or, equivalently, by virtue of the pointwise convergence of (Dv& to Dtl, that for every E > 0 there exist 6 > 0 and i; E N such that, for every measurable set S c Q with IS 1 < 6, we have (3.16) for every h 1 i;. By (3.13) it follows

that Dv~ = ~ Dki*ph;

(3.17)

i=O

moreover,

for every h verifying i < dist(spt t?i ) R” \Ri),

i < dist(spt Co, R”\Cl),

(i = 1, . . ..N)

(3.18)

(3.14) yields spt(u”o * Ph) c Q

Then, by (i) of theorem 2.5, (3.17), (3.19), and c = c(p, N) such that, for every h verifying (3.18)

IC

s

(i = 1, . . ..N).

sPt(ai * oh) C Ri 9 (3.11)

there

bWb’~o~)Ip~o dx + f

exists

a positive

JM(ID2lil)l'li

i=l

(3.19)

.

dx

SnR;

constant

>

By (ii) of theorem 2.5 and the previous inequality, we have that, since M(ID17il) E Lp(R”, Ai) for every i = 1, . . . . N (see theorem 2.4), (3.16) holds for h satisfying (3.18) and (~1 small enough. n We can now give the proof

of theorem

3.3.

Proof of theorem 3.3. (i) Since by (3.9) F(CJ, U) I F,(Q u), it is enough to prove the reverse inequality. To this aim, it is not restrictive to suppose that F(L2, U) < +a~; hence, by (3.3) u E Wisp(L2, A). Then, in order to obtain the thesis, it is sufficient to show that

c el(i-2)n Wqi-2, n)

3 (u~)~

such that uh -+ u in W1~p(Q A).

In fact, from (3.3) we have that 0 I j-(x, D+(x)) and by the dominated

F,(Q 4)

convergence

= F,Pz, u*) =

I cA.(x)(l + IDuh(x)lp) theorem

a.e. in Q

and (3.20)

f(x, Dud dx -+

R

f(x, Du) dx = F(!i2, u).

(3.20)

V. CHIAD~ PIAT and F. SERRA CASSANO

418

By the L’(Q)-lower

semicontinuity

of F(Q, a) we obtain

F,(Q, U) 5 limmf F,(Q - m

u,) =

that

f(x, Du) dx = F(S& u). in

To conclude the proof it remains to show (3.20). To this aim, it is not restrictive to assume that u E L”(a) fl W’.P(L2, A). In fact, every function u E W1*p(Q, A) can be approximated in W1,p(CJ, A) by the sequence of its truncated functions uk, defined by

Uk =

k

ifu>k

u

if-k
i -k

if u < -k.

This can easily be seen by noticing that u,(x) + u(x) and Du,(x) Moreover, by the definition of uk, we have that l~,(x>l 5 Iu(x)l,

D&)l

-+ Du(x)

5 Du(x)l

for a.e. x E CJ. Then the result follows by the dominated convergence we observe that, if u E L”(Q) n W1*p(C& A), we have that $24 E W”p(Q

for a.e. x E Sz.

theorem.

In particular,

A)

(3.21)

for every 4 E C?;(n). Now, (3.20) can be proven by using, with suitable modifications, the same techniques of Meyers-Serrin’s theorem (see [5, 141) to which we address the reader for details. The key point in the proof is to approximate functions such as $u by convolution in W’~p(CJ, A). If we denote by u the extension of $u to the whole of R” by the constant 0, then u E W’.“(R”, A), u * ph E C?‘(R”), and it can easily be seen that u * ph + u in W’sP(R”, A). This follows (again through the dominated convergence theorem) from the fact that u * Ph -+ u and that, by inequality (3.11) (see theorem 2.4). (ii) Since by (3.9) F(Q, u) I it is not restrictive to assume proposition 3.5, there exists a (Qh Then,

arguing

as before,

in L’(G),

a.e. in M,

with u = 1~~1, IDV * phi is dominated

by M(~DZJ~) E P(R”,

A)

F,(0, u), it is enough to prove the reverse inequality. To this aim that F(fi, u) < +a~; hence, by (3.3) u E W’2p(sZ, A). Then, by sequence (u~)~ such that uh + u in W1,p(C2,A).

C e’(R”),

we obtain

(3.22)

that

F,(Q, Uh) = F(Q, u/J = .i n and we can conclude

Du(x) *ph + Du(x)

m, ml)

by using the L*(Q)-lower

dx +

i

j-(x, Du) dx = F(Q, u) I1

semicontinuity

of &(Q

m). w

Remark 3.6. (i) When II = Lo with &, positive constant, theorem 3.3 is a well-known result. In fact, for instance, by Meyers-Serrin’s theorem (see [5]), if D,(Q) is as in (3.10), then Di(fi) = W’,“(Q) and by remark 3.2 we can obtain at once condition (i) of theorem 3.3. Analogously, by the classical extension theorem for W ‘,p(Cl), L&(Q) = W1sp(sZ)and, therefore, condition (ii) also follows directly.

Degenerate

variational

integrals

419

(ii) In the case p = 2, A E & and f(x,
In this section we want to make some comments and to underline some open problems on this subject. First of all, we remark that we have not been able to treat the general case of arbitrary degeneracy, i.e. for instance to see whether the identity between F and pi (see (3.4) and (3.5)) holds or not, with Ax, 0 = ~(x)lrlP, under the only assumption for A to be ap-weight on R” (see (2.1)). We have the impression that, in general, the techniques that we use in the proof of (i) in theorem 3.3 (and, more generally, of Meyers-Serrin’s theorem), i.e. the approximation by convolution, may fail in the general case where A is only a p-weight, as shown by the counterexample given in proposition 4.3. and by stating some preliminary results. Let Let us start by giving some definitions, ~1: [0, +m[ -+ [0, +oo[ be the function defined by vp > 0,

V(P) = ~~OY)h(Y) where v),,:[0,+a[ -+ [0,+co[are the functions

V)h(P)

=

(4.1)

given by

(4.2)

2fih

0

otherwise,

with 01,/I E R, /I I a. LEMMA4.1. Let S, t be real numbers.

Then

s 1

P,“(P)P’~P < +a~*

0

The proof as follows

of lemma

4.1 is left to the reader.

fP(lxlj

A(x) =

L

Let us now define the functions

IXln-l

if 0 < 1x1 < 1

1

if 1x1 2 1,

My

u(x) =

CYS-t-l<0

AxI)

(4.3)

/?-t-l<<.

A, U: R” + R

(4.4)

if 0 < 1x1 < 1 (4.5)

i 0

if Ix] L 1,

where y E R, y 2 0. The main properties of I, and u are summarized in the following where, for simplicity, we have denoted by B the unit ball of R”, B,(O).

lemma,

V. CHIAD~) PIAT and F. SERRA CASSANO

420

LEMMA 4.2. Let p, q E [l, +oo], r > 1. Then the following

conditions

hold:

rEL’I(B)amax(ol,P)
(4.6)

H max{a,jI]

I

1 - n,

(4.7)

A-’ E Lq(B) ti min(cr, j?) > 1 - n - i, 1 + ry

u E L'(R", A)timin(a, /3) > l-r

The proof of lemma 4.2 follows easily from the definitions Let us now come to the counterexample. PROPOSITION

u E

4.3. For every n 2 1 and p > 1 there

(4.8)

a

(4.9)

of A and U, and from lemma 4.1.

exist a p-weight

L and a radial

function

Lp(R", A)such that IU *p/JPl

lim h-+w

(4.10)

dX = +a,

s Bl/h(O)

Mu $ Lp(R”, A).

(4.11)

Proof. Let us consider the functions A and U, as defined by (4.4) and (4.5). We shall prove that for every n 2 1 andp > 1 there exists a suitable choice of the parameters (Y,p, y, such that u E Lp(R”, A) with 1 p-weight, and (4.10) and (4.11) are satisfied. Let us start by showing (4. lo), whose proof will be divided into five steps. Step 1. There exists a constant

c = c(n, 7) > 0 such that

v t

u(x) dx 2 CtY+“+@ B,(O)

Proof of step 1. By performing

the change of variable

E

(4.12)

IO, 11.

1x1 = o in the left-hand

side of (4.12),

we have t u(x)

dx

=

da,

s 0 P(G)

s MO) where no,_, denotes the Lebesgue of q into account, and by noticing

$+n-l -

no,_,

measure of the unit sphere of R”. By taking the definition that for every t E IO, l] there exists h, E N such that 1

i,t,-

2h,-1

2ht

7

we have t

ay+n-l

-da? s 0 cp(o)

l/z*f s0

y+n-1

y+n-1

0 ----do> P(d

do = 2-oh,

L s4

PC4

D~+~-’ da. Ih4

Degenerate

variational

integrals

421

Since t/2 I 1/2hr < t, we can conclude that

for every t E IO, 11, that is (4.12). Step 2. There exists a constant c = c(n, y) > 0 such that r u(y) dr 1 c(;f, - Ixly+‘+’ JB mm \a& /

vx E Br,&%

(4.13)

where k > 0. Proof of step 2. Let k > 0 and let x E B1,2k(0). Then B,,,,_~,~(O) c B,,,,(x) hence, (4.13) follows directly from (4.12), with t = 1/2k - 1x1.

c B and,

Step 3. Let p E C?F(B,(O))be such that c = inf(p(t) : t E B&O)] > 0 and JR”p dy = 1, and set pk(x) = k”p(kx), for every k E N. Then (u *P/c)(X) 2 c

U(Y)dy f Bl/Zk@)

Proof of step 3. By the definition of convolution @ * P/m

=

vx~R”,vk~N.

(4.14)

we have

I~,~o~u(x-~~p,(v)dy~csa,,*~(oi~(y)dy

u(x-y)MAd~= R”

which is (4.14) Step 4. Let p, pk be defined as in step 3. Then there exists a constant c = c(n, y, p, p) > 0 such that B,,ZrC(0) (u * /-#A h 2 ck

(4.15)

for every k E N. Proof of step 4. It is a direct consequence of steps l-3. Step 5. Assume that CY- 1 - p(y f 8) > 0.

(4.16)

Then (4.10) holds. Proof of step 5. For every k E N let us set sk = min(s E N : 2” 2 k), tk = 2Sk; we remark that sk tend to +a~ together with k. In order to prove (4.10), we can estimate the right-hand side

V. CHIAD~) PIAT and F. SERRA CASSANO

422

2 no,_,

1 = - no,_, 4

L

1

_

-+a0

no,_,

C

4

2’a-

1 =

-no,-,

1 _

4

The constant is a p-weight,

~W-‘-P~-P~-P@

h=sk+2 1 -Py-Pn-pB)(sx

+2) =

2(a-l-Py-pn-pJ3)

cd9

- 1 -P-l-Pn-Po)sk

c is positive if cz - 1 < p(y + n + fi); but y + n + fi 2 0 by assumption, hence, at least L’(Q), which means CY< 1 (see (4.6)). Then we have (u * P~)~,I du 2 ckp” /_ 1 Bl/ZdO) ,- &Pn 2-Pk

(& 2’“-

Since k > 2skP1, the last term in the preceding Q- 1 -p(~+&>O.Actually,ifwetakecu~Rsuchthatl

-

lM)p(iin+P)i

and ;i

h

1 pPYmPb)Sk

inequality

1 - 2n - np < /I < min

(4.17)

tends to -np
+co as k ++w 1,/3
if

,

(Y,y i

I

and y 1 0 such that max

i

p-1 P

P,-n-P


then, by (4.6), (4.9) and (4.16) it follows that ,I is a p-weight on R”, u E LP(R”, A), and that (4.10) holds. Let us prove now (4.11). By (4.13) we have that there exists a positive constant cr such that u(y) dy)’

1 cr/rp’(&

-

Ix~>‘“~+“‘”

B,/,,(x)

for every x E B,,,,(x),

k E N, and, therefore,

by (2.6) we obtain (r+n+mP

(4.18)

Degenerate

variational

423

integrals

for every x E B,,,,(x), k E N. Let (._v~)~be the sequence defined in step 5; then, (4.17) it follows that there exists a positive constant c2 such that I&fu(x)IPA(x)

&

by (4.18) and

C2kPn2-Pnsk2(01-‘-PY~PP)Sk

1

(4.19)

~1/2*@) for every k E N. Since k > 2Sr-1, by (4.16) and (4.19) we obtain IMUlPi du = +oo

lim k-+m

and so (4.11) holds.

J B,/,,(O)

W

Remark 4.4. It is easy to see that (4.10) still holds if we replace Bi,,,(O) with the half-ball B,,h(0) tl (X E R" :x,2 0),or more generally, with sets obtained by intersecting B,,,(O) with a cone of vertex 0. This is due to the fact that, by (4.14) and (4.13) the integrand in (4.10) can be estimated from below by the radial function (1/2k - Ix~)‘+~+~, whose integrals on these sets differ just by a constant factor. 4.5. Let n = 1 and u E W1*P(R,A) such that

COROLLARY

p >

lim k-+m

1. Then

there

exist

a p-weight

A and

a function

a IDo * PklPA dx = +m so

for every a > 0. Proof. Let I, u be the functions defined by (4.4), (4.Q a suitable choice of CX,/3, and y. Let us define

with n = 1, and satisfying

(4.10) for

1x1 p

v(x) =

-

dt

for x E R.

0 v(t) and Dv(x) = U(X) for x 2 0. Here, u E W,;;'(R), w-e = ~l~lY~~~l~l~~~~~I~l~~ W'*P(R,A)and the thesis follows from (4.10) and remark 4.4. W

Then

v E

Remark 4.6. Let f be as in (3.1)-(3.3), and let F,, F, be as in (3.5), (3.6); then it is well known that, if n = 1 (see, for instance, [3,6]), E(a, U) = F(Q u) for every u E W,~;.'(R), and for every interval 0. In particular this holds for u = v as in corollary 4.5, and at the same time lim

k-+m i =

fi(Qz, u *pk) = +to,

1,2.

Acknowledgements-We We also wish to thank this work was done.

are grateful the hospitality

to Professor Giuseppe Buttazzo for many interesting of the Laboratoire d’Analyse NumCrique, Universite

discussions on the subject. de Paris VI, where part of

424

V. CHIAD~ PIAT and F. SERRA CASSANO REFERENCES

Relaxation, and Integral Representation in the Calculus of Variations. Longman, 1. BUTTAZZO G., Semicontinuity, Harlow (1989). 2. LAVRENTIEV M., Sur quelques problemes du calcul des variations, Ann. Mat. pura appl. 4, 107-124 (1926). of the Lavrentiev phenomenon by relaxation, J. funct. Analysis 3. BUTTAZZO G. & MIZEL V. J., Interpretation (to appear). 4. SERRIN J., On the definition and properties of certain variational integrals, Trans. Am. math. Sot. 101, 139-167 (1961). 5. MEYERS N. G. & SERRIN J., H = W, Proc. natn Acad. Sci. USA 51, 105551056 (1964). 6. DE ARCANGELIS R., Some remarks on the identity between a variational integral and its relaxed function, Ann. Univ. Ferrara 35, 135-145 (1989). media and asymptotic Dirichlet forms, Preprint Univ. Roma “La Sapienza” (1992). 7. Mosco U., Composite Trans. Am. math. Sot. 8. MUCKENHOUPT B., Weighted norm inequalities for the Hardy maximal function,

207-226 (1972). 9. COIFMANR. R. & FEFFERMANC., Weighted Math. 51, 241-250 (1974).

norm inequalities

10. MODICA G., Quasiminimi di alcuni funzionali degeneri, Masson, Paris (1983). 11. BREZIS H., Anafyse fonctionefle. della G-convergenza 12. SERRA CASSANO F., Un’estensione

Ann.

for maximal

functions

and singular integral,

165,

Studia

Mat. pura appl. 142, 121-143 (1985).

alla classe degli operatori

ellittici degeneri,

Rc. Mat.

38,

167-197 (1989). Part one: General Theory. Wiley Interscience, New York 13. DUNFORD N. & SCHWARTZ J. T., Linear Operators. (1958). 14. ADAMS R. A., Sobolev Spaces. Academic Press, New York (1975). of functionals of quadratic type, Ann. Mat. pura appl. 129, 15. Fusco N. & MOSCARIELLOG., Lz-lower semicontinuity 305-326 (1981).