A Regularity Result for a Class of Anisotropic Systems

A Regularity Result for a Class of
Anisotropic Systems
Antonia Passarelli di Napoli and Francesco Siepe ()
Sommario.R - Si prova la regolarita parziale dei minimi del funzionale
I (u) =
G(Du), con G integrando convesso a crescita anisotropa.
Non si fanno ipotesi speciali sulla struttura di G.
Summary.R - We prove partial regularity of minimizers of the functional
I (u) =
G(Du), where G is a convex integrand satisfying anisotropic
growth condition. No special structure assumption is needed on G.
1. Introduction
In this paper we study the partial regularity of minimizers of integral
functionals of the type
I (u) =
Z
G(Du(x))dx
(1:1)
u : IRn ! IRN , N 1, where G is a C 2 convex integrand
satisfying the growth condition:
C j jq G( ) L(1 + j jp)
(1:2)
(*) Indirizzi degli Autori: Antonia Passarelli di Napoli: Dipartimento di
Matematica e Applicazioni \R. Caccioppoli", Universita di Napoli \Federico
II", Complesso Monte s.Angelo, Via Cintia - 80126 Napoli, e-mail: [email protected]. Francesco Siepe: Dipartimento di Matematica
\U. Dini", Universita di Firenze, Viale Morgagni 67/A - 50134 Firenze, e-mail:
[email protected].
Key words and phrases: regularity, minimizers, convexity.
This work has been supported by M.U.R.S.T. (40%).
14
A. PASSARELLI DI NAPOLI and F. SIEPE
with p > q .
Few years ago it was observed that even in the scalar case, i.e.
N = 1, minimizers of (1.1) may fail to be regular (see [M2], [G2]),
when p is too large with respect to q . On the other hand, one can
prove regularity of scalar minimizers of (1.1) if p is not too far away
from q (see e.g. [M3], [FS] and the references given in [M3]). More
precisely, in [M3] it is shown that if one writes down the Euler equation for the functional I , under suitable assumptions on p and q , the
Moser iteration argument still works, thus leading to a sup estimate
for the gradient Du of the minimizer.
Clearly this approach can not be carried on in the vector valued
case, i.e. when N > 1. As far as we know, the only regularity
results for systems are proved under special structure assumptions
(see [AF2], [M4]).
Namely, the model case covered in [AF2] is the functional
Z
jDujp +
k
X
=1
jDujp
with u : IR n ! IRN , N 1, 1 k n, 2 p < p, and p not
too far from p, while in [M4], it is proved everywhere regularity of
minimizers of (1.1) when G( ) = f (j j).
In this paper we prove that if G satises (1.2) and the strong
ellipticity assumption
hD2G(); i (1 + jj2) q jj2
2
and
2
qn
(1:3)
2 q < p < min q + 1; n 1 ;
a minimizer u 2 W 1;q (
; IRN ) of functional (1.1) is C 1; for all < 1
in an open set 0 such that meas(
n 0) = 0.
We point out that a part from condition (1.3), no special structure assumption is needed on G.
The proof of our result goes through a more or less standard
blow-up argument aimed to establish a decay estimate on the excess
function for the gradient
U (x0; r) =
Z
Br (x0 )
jDu (Du)x ;r j2 + jDu (Du)x ;rjq dx:
0
0
The essential tool in the case we consider, is a lemma due to Fonseca
and Maly (see [FM] and also Lemma 2.3 below) which makes possible
to connect in the annulus Br n Bs two W 1;q functions v and w with
a function z 2 W 1;p(Br n Bs ) if q < p < nqn1 .
A REGULARITY RESULT etc.
15
2. Statements and preliminary Lemmas
Let us consider the functional
I (u) =
Z
G(Du(x))dx
where is a bounded open set of IRn , n 2. Let G : IRnN ! IR,
N 2, satisfy the following assumptions:
G 2 C2
(H 1)
C j jq G( ) L(1 + j jp)
(H 2)
q
hD2G(); i (1 + jj2) jj2
(H 3)
2
2
2 q < p < min q + 1; nqn 1
where
It is well known that
jDG()j c(1 + jjp 1):
(H 4)
We say that u 2 W 1;q (
; IRN ) is a minimizer of I if
I (u) I (u + v )
for any v 2 u + W01;q (
; IRN ).
Remark 1. If u is a local minimizer of I and 2 C01 (
; IRN ) from
the minimality condition one has for any " > 0
0
Z
[G(Du + "D) G(Du)]dx
Z
Z1
@G (Du + "tD)D i dt
0 @i
Dividing this inequality by ", and letting " go to zero, from (H4) and
the assumption p q + 1 we get
Z
@G (Du)D i dx 0
@
i
and therefore by the arbitrariness of the usual Euler-Lagrange
="
dx
system holds:
Z
@G (Du)D i dx = 0
@
i
8 2 C01(
; IRN )
16
A. PASSARELLI DI NAPOLI and F. SIEPE
We prove the following
Theorem 2.1. Let G be as above and let u 2 W 1;q (
; IRN ) be a
minimizer of I . Then there exists an open subset 0 of such that
meas(
n 0) = 0
and
u 2 C 1;(
0; IRN ) for all < 1 :
N
1;q
RIn the following, we will denote by u a W (
; IR ) minimizer
of G(Du)dx and assume that G satises (H1), (H2), (H3). We set
for every Br (x0 ) U (x0; r) =
where
Z
Br (x0 )
Z
Br (x0 )
jDu (Du)x ;r j2 + jDu (Du)x ;rjq dx;
0
0
g = (g )x ;r = meas(B1 (x ))
Z
Br (x0 )
r 0
0
g:
The next Lemma can be found in [FM], (Lemma 2.2), in a slightly
dierent form.
Lemma 2.1. Let v 2 W 1;q (B1 (0)) and 0 < s < r < 1. There exists
a linear operator T : W 1;q (B1 (0)) ! W 1;q (B1 (0)) such that
Tv = v on (B1 n Br ) [ Bs
and for all > 0 , for all p < q n n 1
jjTvjjW ; (BrnBs) + jjTvjjW ;p(Br nBs)
12
(
1
h
C (r s) sup (t s) jjvjjW ; (BtnBs) +
1
2
t2(s;r)
12
i
+ sup (r t) jjv jjW ; (Br nBt) +
t2(s;r)
1
2
12
h
+ (r s) sup (t s) q jjv jjW ;q (BtnBs) +
t2(s;r)
1
1
+ sup (r t) q jjv jjW ;q (Br nBt )
t2(s;r)
1
i)
1
where C = C (n; p; q ) > 0, = (n) > 0 and = (n; p; q ) > 0.
A REGULARITY RESULT etc.
17
Let us recall an elementary lemma also proved in [FM].
Lemma 2.2. Let be a continuous nondecreasing function on an
interval [a; b], a < b. There exist a0 2 [a; a + 31 (b a)], b0 2 [b 13 (b
a); b] such that a a0 < b0 b and
(t) (a0) 3 (b) (a)
t a0
b a
(2.3)
(b0) (t) 3 (b) (a)
b0 t
b a
for all t 2 (a0; b0).
Finally the next result is a straightforward generalisation to our
case of Lemma 2.4 in [FM]. We give the proof here for completeness.
Lemma 2.3. Let v; w 2 W 1;q (B1 (0)) and 41 < s < r < 1. Fix
q < p < nnq1 , for all > 0 and m 2 IN there exist a function
z 2 W 1;q (B1 (0)) and 14 < s < s0 < r0 < r < 1 with r0 ; s0 depending
on v; w and , such that
(2:4)
z = v on Bs0 ; z = w on B1 n Br0 ;
r s r0 s0 r s
m
3m
and
jjzjjW ; (Br0 nBs0 ) + jjzjjW ;p(Br0 nBs0 )
hZ
(
r
s
)
C m
1 + jDv j2 + jDwj2 + jv j2 + jwj2+
Br nBs
2
(2.5)
+ m2 j(vr ws)j2 +
Z
1 + jDv jq + jDwjq + jv jq + jwjq +
+ q
Br nBs
q i
+ mq jv wjq
(r s)
where C = C (n; p; q ) > 0 and = (p; q; n) > 0.
Proof. As in Lemma 2.4 in [FM], choose m 2 IN and set
2
f = 1 + jDv j2 + jDwj2 + jv j2 + jwj2 + m2 j(vr ws)j2 +
q
+ q 1 + jDv jq + jDwjq + jv jq + jwjq + mq j(vr ws)jq :
12
1
1
2
18
A. PASSARELLI DI NAPOLI and F. SIEPE
We may nd k 2 f1; :::; mg such that
Z
Bs+ k(r s) nBs+ (k
m
r s)
m
1)(
Z
1
fdx m
Br nBs
fdx ;
r s) ; s + k(r s) ],
Set, for t 2 [s + (k 1)(
m
m
(t) =
Z
BtnBs
fdx
which is a continuous nondecreasing function. By Lemma 2.2, there
r s) ; s + k(r s) ] such that
exists [s0; r0] [s + (k 1)(
m
m
r s r0 s0 r s
m
3m
and
Z
BtnBs0
0 )m Z
(
t
s
fdx 3
r s Bs k r s nBs
m
0Z
t
s
3r s
fdx;
Br nBs
+
(
)
+
(
k
r s)
m
1)(
fdx
Z
0
fdx
fdx 3 rr st
Br nBs
Br0 nBt
Z
(2.6)
(2:7)
for all t 2 (s0 ; r0). Set
8
v (x)
>
>
>
>
>
< 0
jxj s0 )w(x)
u = > (r jxj)v(xr)+(
0 s0
>
>
>
>
: w(x)
if x 2 Bs0
if x 2 Br0 n Bs0
if x 2 B1 n Br0 .
A direct computation shows that
juj2 + jDuj2 + q (jujq + jDujq ) Cf :
If we apply Lemma 2.1 to the function u, we then nd z 2 W 1;q (B1 )
satisfying (2.4). Moreover, from (2.6) and (2.7) one readily cheks
A REGULARITY RESULT etc.
19
that
jjzjjW ; (Br0 nBs0 ) + jjzjjW ;p(Br0 nBs0 )
(
!
Z
0 s0 )
(
r
c 0 0 jBr0 n Bs0 j
f +
Br nBs
(r s )
!q )
Z
0 s0 )
(
r
jBr0 n Bs0 j q
+
f
0
0
q
Br nBs
(r s )
1
12
1
2
1
2
1
2
1
1
1
(
c (r0 s0)
Z
Br nBs
f
! 12
+ (r0 s0 )
Z
Br nBs
f
! 1q )
;
from which (2.5) follows choosing = minf; g.
}
3. Proof of Theorem 1
As usual, to get the partial regularity result stated in Theorem 1,
we need a decay estimate for the excess function U (x0; r) dened in
section 2.
Proposition 3.1. Fix M > 0. There exists a constant CM > 0
such that for every 0 < < 41 , there exists = (; M ) such that, if
j(Du)x ;r j M
0
and
U (x0 ; r) then
U (x0 ; r) CM 2 U (x0; r) :
Proof. Fix M and . We shall determine CM later.
We argue by contradiction. We assume that there exists a sequence Brh (xh ) satisfying
Brh (xh ) ;
but
Set
j(Du)xh;rh j M;
lim
U (xh ; rh) = 0;
h
U (xh ; rh) > CM 2 U (xh ; rh) :
ah = (u)xh;rh
Ah = (Du)xh;rh
2h = U (xh ; rh) :
(3:1)
20
A. PASSARELLI DI NAPOLI and F. SIEPE
[Blow up.] We rescale the function u in each Brh (xh ) to
obtain a sequence of functions on B1 (0). Set
v (y ) = 1 [u(x + r y ) a r A y ];
Step 1.
h
h
h rh
then
Clearly we have
h
h h
Dvh (y ) = 1 [Du(xh + rh y ) Ah ] :
h
(vh )0;1 = 0
Moreover,
h
Z
B1(0)
(Dvh )0;1 = 0 :
(1 + hq 2jDvh jq 2 )jDvh j2dy = 1 :
(3:2)
Passing possibly to a subsequence we may suppose that
vh * v
weakly in W 1;2(B1 ; IRN )
and, since 8h jAh j M ,
Ah ! A :
Step 2.
(3:3)
(3:4)
Now we show that
Z
@ 2G (A)D v j D i dy = 0
B (0) @i @j
1
8 2 C01(B1; IRN ) : (3:5)
Since we assume p 1 q we can write the usual Euler-Lagrange
system for u (see Remark 1). Then, rescaling in each Brh (xh ), we
get for any 2 C01 (B1; IRN ) and any 1 i N
Z
@G (A + Dv )D i dy = 0 :
h
h h B (0) @i
1
Then
Z
@G (A + Dv ) @G (A )]D i dy = 0 :
[
h B (0) @i h h h @i h 1
1
(3:6)
A REGULARITY RESULT etc.
21
Let us split
B1 = Eh+ [ Eh
= fy 2 B1 : h jDvh (y )j > 1g [ fy 2 B1 : h jDvh (y )j 1g ;
then by (3.2) we get
jEh+j Z
Eh+
2h jDvhj2dy 2h
Z
B1 (0)
jDvhj2dy c2h:
(3:7)
Now, by (H4) and Holder inequality, we observe that
1 j Z [DG(A + Dv ) DG(A )]Ddy j
h
h Eh
c jEh+j + cph
+
h
ch + c
Z
2
h
Z
Eh+
h
jDvhjp 1dy
q 2jDvh jq dy
E+ h
h
h
!p
q
1
p q
q
2
h
2
jEh+j
q p+1
q
where we used the assumption p 1 q .
From this it follows that
Z
1
[DG(Ah + h Dvh ) DG(Ah )]Ddy = 0:
lim
h h Eh+
ch
(3:8)
On Eh we have
1
Z
[DG(Ah + h Dvh ) DG(Ah )]Ddy
Eh
Z Z1
D2G(Ah + shDvh )Dvh Ddsdy
=
Eh 0
Z Z1
[D2G(Ah + sh Dvh ) D2G(Ah )]Dvh Ddsdy +
=
Eh 0
Z
+
D2G(Ah )Dvh Ddy :
Eh
that (3.7) ensures that Eh ! B1 in Lr (B1 ) for all r <
h
Note
and by (3.2) we have, passing possibly to a subsequence,
h Dvh (y ) ! 0 a:e: in B1 :
1
22
A. PASSARELLI DI NAPOLI and F. SIEPE
Then, by (3.3), (3.4) and the uniform continuity of D2 G on bounded
sets, we get
Z
lim 1
[DG(A + Dv ) DG(A )]Ddy
Rh
h
Eh
h
h
h
h
= B D2 G(A)DvDdy :
By (3.6), (3.8) and the above equality, we obtain that v satises
equation (3.5), which is elliptic by (H3). We have for any 0 < < 1
1
Z
Z
B
jDv (Dv) j2dy c 2
Moreover we have
and
Step 3.
q
B1
jDv (Dv)1j2dy c 2:
v 2 C 1 (B1 ; IRN ) :
(3:9)
(3:10)
2
1;q (B ; IRN )
h q (vh v ) * 0 weakly in Wloc
1
[Upper bound.] We set
Gh ( ) = 12 [G(Ah + h ) G(Ah) h DG(Ah) ]
and for every r < 1
h
Ih;r (w) =
Z
Br
Gh (Dw)dy :
Note that by the strong ellipticity assumption (H3) it follows that
Gh ( ) 0, for any . Fix 14 < s < 1. Passing to a subsequence we
may always assume that
lim
[I (v ) Ih;s (v )]
h h;s h
exists. We shall prove that
lim
[I (v ) Ih;s (v )] 0 :
h h;s h
(3:11)
Consider r > s and x m 2 IN . Observe thatp , since v 2 W 1;q (B1 )
and vh 2 W 1;q (B1 ), Lemma 2.3, with = h p , implies that there
exist zh 2 W 1;q (B1) and 14 < s < sh < rh < r < 1 such that
zh = v on Bsh zh = vh on B1 n Brh
2
A REGULARITY RESULT etc.
and
p
23
2
jjzhjjW ; (Brh nBsZh ) + h p jjzhjjW ;p(Brh nBsh )
h
C (r ms)
(1 + jDv j2 + jDvh j2 + jv j2 + jvh j2 +
Br nBs
2
+ m2 jv vh j2 )+
(3.12)
(r s)
p qZ
+ h p
(1 + jDv jq + jDvh jq + jv jq + jvh jq +
Br nBs
q i
+ mq j(vr vsh)jq )
Since by (3.10), Dv is locally bounded on B1 we get
Ih;s (vh ) Ih;s (v )
Ih;rh (vh) Ih;rh (v) + IZh;rh (v) Ih;s(v)
= Ih;rh (vh ) Ih;rh (v ) +
Gh (Dv )
(3.13)
Brh nBs
Ih;r
(
z
)
I
(
v
)
+
c
(
r
s
)
h
h;r
h
Zh
c
[Gh (Dzh ) Gh (Dv )] + c(r s) :
12
1
2
1
2
Brh nBsh
where we used the minimality of vh . As jGh ( )j c(j j2 + ph 2 j jp)
(see [AF], Lemma II.3), we get by (3.12)
Ih;rh Z(zh ) Ih;rh (v )
c
jDzhj2 + hp 2jDzhjp
Brh nBsh Z
2 h
(
(1 + jDv j2 + jDv j2 + jv j2 + jv j2+
C r s)
m2
h
Br nBs
h
2 ip
+ m2 j(vr vsh)j2 ) +
(1 + jDv jq + jDvh jq + jv jq + jvh jq +
2
2 h p 2 qZ
Br nBs
+ C (r m2s) h p
q ip
+ mq j(vr vsh)jq )
2
= Jh;1 + Jh;2 :
Since vh ! v in L2 (B1; IRN ) we have, using (3.2)
lim sup Jh;1 Cm 2 :
h!1
24
A. PASSARELLI DI NAPOLI and F. SIEPE
Moreover, since
q(p 2) Z
p
h
and
2(
B1
q(p 2) Z
p
h
jDvhjq = h
B1
Z
p q)
p q 2
2(
B1
jDvhjq Ch
B1
jDvhjq ch
h
q(p 2) Z
p
jvh vjq ch
we have
2(
p q)
p
p q)
p
lim
J = 0:
h h;2
Hence we conclude letting rst m ! 1 and then r ! s in (3.13).
1 < r < 1,
Step 4. [Lower bound.] We shall prove that, for a.e.
4
2
if t < r then
lim sup
h
Z
Bt
jDv Dvhj2(1 + hq 2jDv Dvhjq 2)
lim
[I (v ) Ih;r (v )] :
h h;r h
For any Borel set A B1 , let us dene
Z
h (A) = (jvh j2 + jDvhj2)dx :
A
Passing possibly to a subsequence, since h (B1 ) c, we may suppose
h * weakly in the sense of measures;
where is a Borel measure over B1 . Then for a.e. r < 1
(@Br ) = 0
and let us choose such a radius r. Consider 41 < t < s < r, also such
that (@Bs ) = 0, and x m 2 IN . Observe that , as vh 2 W 1;q (B1 )
Lemmas 2.3 implies that there exist zh 2 W 1;q (B1 ) and 14 < s <
sh < rh < r < 1 such that
zh = vh on Bsh zh = vh on B1 n Brh
rh sh r3ms
A REGULARITY RESULT etc.
and
p
25
2
jjzhjjW ; (Brh nBsh ) + h p jjzhjjW ;p(Brh nBsh )
hZ
C (r ms)
(1 + jDvh j2 + jvh j2)+
B
n
B
r s
p qZ
i
p
(1 + jDvh jq + jvh jq )
+ h
1
12
(
2)
(3.14)
1
2
Br nBs
Passing possibly to a subsequence, we may suppose that
weakly in W 1;2(B1 ) :
zh * vr;s
and
vr;s = v in (B1 n Br ) [ Bs
Moreover from (3.14) it is clear that
Z
q
2
h
jDzhjq
B1
c
(3:15)
Consider h 2 C01 (Brh ) such that 0 h 1, h = 1 on Bsh and
jDhj rhCsh and set
h = h (zh
);
vr;s
= ?vr;s , and is the usual sequence of molliers. Now,
where vr;s
setting v = ? v , we observe that
Ih;rh (vh ) Ih;rh (v )
+ )+
= Ih;rh (vh ) Ih;rh (zh ) + Ih;rh (zh ) Ih;rh (vr;s
h
(3.16)
+ Ih;rh ( h + vr;s ) Ih;rh (vr;s ) Ih;rh ( h )+
Ih;rh (vr;s) Ih;rh (v ) + Ih;rh ( h )
= Rh;1 + Rh;2 + Rh;3 + Rh;4 + Rh;5
To bound Rh;1 we observe that
Ih;rh (vh) Ih;rh (zh ) =
Z
Brh nBsh
Z
Gh (Dvh)
Brh nBsh
Gh(Dzh)
Z
Brh nBsh
Gh (Dzh ) +
26
A. PASSARELLI DI NAPOLI and F. SIEPE
on the other hand we have
Z
Brh nBsh
Gh (Dzh ) Z
jDzhj2 + hp 2jDzhjp
Brh nBsh
hZ
cm 2
1 + jDvh j2 + jvh j2 +
Br nBs
p 2 qZ
ip
1 + jDvh jq + jvh jq 2
+ h p
Br nBs
and then arguing as we did in Step 3 to bound Jh;1 we get
lim sup
Z
Brh nBsh
h
Gh(Dzh) Cm
2
hence, letting h ! 1 we get
limhinf Rh;1 Cm 2
(3:17)
We obtain that
Rh;2 =
Z
BZrh nBsh
c
)
Gh(Dzh) Gh (D h + Dvr;s
j2 + p 2jD + Dv jp
jD h + Dvr;s
h
r;s
h
ZBrh nBsh j2 +
jDzhj2 + ph 2jDzhjp + jDvr;s
Z
jz v j2
c
m2 (hr sr;s)2 +
Br nBs
jp ! h h
j
z
v
h
+ mp ph 2 (r sr;s)p
= Sh;1 Sh;2
c
Brh nBsh
jp
+ hp 2 jDvr;s
(3.18)
where we used the bound rh sh r3ms . By (3.15), since p < q , we
get
Z
p 2jzh jp
B1 h
cph 2
cph 2
Z
B1
(Z
jzh (zh)0;1
B1
jzh
jp + j(z
(zh )0;1jq
jp
h)0;1
qp Z
+
B1
jzhj
p )
A REGULARITY RESULT etc.
chp 2
2(
ch
(Z
B1
jDzh
qp
q
j
Z
+
pq
Z
p q) q
q 2
jDzhjq
R
1
h
B1
B1
27
jzh
j2
p2 )
+ cph 2:
where we used (3.14) to bound B jzh j2 .Therefore
2
1
2
lim sup Sh;2 c (r m s)2
Z
h!1
B1
j2 :
jvr;s vr;s
2
To bound Sh;1 , observe that for every h
Z
Brh nBsh
Z
c
j2
jDvr;s
Br nBs
limjinf c
= c limj inf
jDvr;s
j2 + c
Z
Z
Br nBs
jDzj
Z
j2
jDvr;s Dvr;s
B1
2
j2 + c
Z
B1
j2
jDvr;s Dvr;s
2
jDvj j2 +
(Br nBs)n(Brj nBsj )
Z
Z
jDzj j2 + c
+ c lim sup
B1
Brj nBsj
j
j2
jDvr;s Dvr;s
2
We control the second integral as usual using Lemma 2.3, while the
rst is less or equal than c(Br n Bs ).
Moreover we can estimate
Z
Brh nBsh
jDzhj2 + ph 2jDzhjp
as we did in Step 3 to bound Jh;1 . Hence
limhinf Rh;2 cm 2 c(Br n Bs)+
Z
j2 +
c jDvr;s Dvr;s
B1
cm2 Z jv v j2
(r s)2 B r;s r;s
2
1
2
(3.19)
28
A. PASSARELLI DI NAPOLI and F. SIEPE
To bound Rh;3 we observe that
Gh (A + B) Gh (A) Gh (B) =
and
Z 1Z 1
0
0
D2Gh (sA + tB )ABdsdt
+ tD ) = D2 G(A + s Dv + t D )
D2 Gh (sDvr;s
h
h r;s
h h
h
is bounded and converges to D2 G(A) a.e.. Since
Z
Rh;3 =
Br h
dx
Z
[0;1][0;1]
+ th D
and we may suppose that
Z
B1
jD j2 +
D2G(Ah + sh Dvr;s
h )Dvr;s D h dsdt
1;2
h * weakly in W (B1), where
2 Z
j2+
c (r m s)2 jvr;s vr;s
B1
Z
2
(3.20)
j2
+ c jDvr;s Dvr;s
B1
2
we get easily
jj
lim sup jRh;3j c(M )jjDvr;s
L (B ) jjD jjL (B ) :
2
h
To bound Rh;4 we observe that
nRh;4 =
Then
Z
Brh nBsh
Z
1
2
2
1
2
) G (Dv )]
[Gh (Dvr;s
h
Gh (Dv )
Brh nBs cjBr n Bs j :
limhinf Rh;4 cjBr n Bs j :
Moreover (H3) implies
jRh;5j = ZIh;rh ( h )
=
Gh (D h )
BZrh
(3:21)
(1 + hq 2 jDv Bt
Dvh jq 2)jDv
(3:22)
Dvh j2
(3.23)
A REGULARITY RESULT etc.
29
for small enough.
Passing to a subsequence we may suppose that
lim sup Rh;5 = lim
R :
h h;5
h
Therefore returning to the (3.16), from (3.17), (3.19), (3.21), (3.22)
and (3.23) we get
limhinf [Ih;r (vh ) Ih;r (v )]
Z
lim sup (1 + hq 2jDv Dvhjq 2)jDv Dvh j2 +
Bs
h
jj
cjBr n Bs j c(Br n Bs ) cjjDvr;s
L (B ) jjD jjL (B ) +
Z
2 Z
m
2
2
j2 :
cm
jDvr;s Dvr;sj c (r s)2 jvr;s vr;s
B
B
2
1
2
2
1
2
1
2
1
2
Passing to the limit as ! 0+ we get easily
limhinf [Ih;r (vh ) Ih;r (v )]
Z
lim sup (1 + hq 2jDv Dvh jq 2)jDv Dvhj2 +
Bs
h
cjBr n Bs j c(Br n Bs ) cm 2
then passing to the limit as m ! 1 and s ! r we get
lim sup
Z
h
Br
jDv Dvhj2(1+q 2jDv Dvhjq ) lim
[I (v ) Ih;r (v )]:
h h;r h
[Conclusion.] From the two previous steps we conclude
that, for any B , with 0 < < 41
Step 5.
lim
h
Z
B
jDv Dvhj2(1 + q 2jDv Dvhjq ) = 0 :
Now, from this equality and by (3.9) we get
lim U (xh ; rh)
h
2h Z
1
= lim
h 2
h Brh (xh )
(jDu (Du)rh j2 + jDu (Du)rh jq )dx
30
A. PASSARELLI DI NAPOLI and F. SIEPE
= lim
h
=
Z
Z
B
(jDu (Du) j2 + hq 2 jDu (Du) jq )dy
(jDv (Dv ) j2)dy
B
CM 2
which contradicts (3.1) if we choose CM = 2CM .
}
The proof of Theorem 1 follows by proposition 3.1 by a standard
iteration argument, see [G1].
Remark 2. Notice that the proof of Proposition
3.1 and of Theorem
nq
1 still works if, beside assuming p < n 1 , we have p q + 1.
References
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p; q
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31
[G1] Giaquinta M., Multiple integrals in the calculus of variations and
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p; q
Pervenuto in Redazione il 3 Febbraio 1996.

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