Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 1, Issue 2, Article 14, 2000
REGULARITY RESULTS FOR VECTOR FIELDS OF BOUNDED DISTORTION
AND APPLICATIONS
ALBERTO FIORENZA AND FLAVIA GIANNETTI
D IPARTIMENTO DI C OSTRUZIONI E M ETODI M ATEMATICI IN A RCHITETTURA ,
-80134 NAPOLI , I TALY
fiorenza@unina.it
URL: http://cds.unina.it/~fiorenza/
VIA
M ONTEOLIVETO , 3
D IPARTIMENTO DI M ATEMATICA E A PPLICAZIONI “R. C ACCIOPPOLI ”, VIA C INTIA , 80126 NAPOLI , I TALY
giannett@matna2.dma.unina.it
Received 17 January, 2000; accepted 4 April, 2000
Communicated by S. Saitoh
A BSTRACT. In this paper we prove higher integrability results for vector fields B, E, (B, E) ∈
L2− (Ω, Rn ) × L2−ε (Ω, Rn ), ε small, such that div B = 0, curl E = 0 satisfying a “reverse”
inequality of the type
1
2
2
|B| + |E| ≤ K +
hB, Ei + |F |2
K
with K ≥ 1 and F ∈ Lr (Ω, Rn ), r > 2 − ε. Applications to the theory of quasiconformal
mappings and partial differential equations are given. In particular, we prove regularity results
for very weak solutions of equations of the type
div a(x, ∇u) = div F.
2
2
If |a(x, z)| + |z| ≤ (K + 1/K) ha(x, z), zi, in the homogeneous case, our method provides a
new proof of the regularity result
1,2−ε
1,2+ε
u ∈ Wloc
(Ω) ⇒ u ∈ Wloc
(Ω)
where ε is sufficiently small. A result of higher integrability for functions verifying a reverse
integral inequality is used, and its optimality is proved.
Key words and phrases: Reverse Inequalities, Finite Distortion Vector Fields, Div-Curl Vector Fields, Elliptic Partial Differential Equations.
2000 Mathematics Subject Classification. 35J60, 26D15.
ISSN (electronic): 1443-5756
c 2000 Victoria University. All rights reserved.
We wish to thank Prof. T. Iwaniec for stimulating discussions on this subject.
This work has been performed as a part of a National Research Project supported by M.U.R.S.T..
022-99
2
A LBERTO F IORENZA AND F LAVIA G IANNETTI
1. I NTRODUCTION
1,2+ε
1,2
The usual way to establish the Wloc
(Ω), ε > 0, regularity of solutions u ∈ Wloc
(Ω) of
equations of the type
div a(x, ∇u) = div F
in
Ω,
where |a(x, z)|2 + |z|2 ≤ (K + 1/K) ha(x, z), zi, is to combine the Caccioppoli inequality
#
"Z 2
Z
Z
u
−
u
2R
|∇u|2 dx ≤ c
|F |2 dx ,
2R dx +
2Q
Q
2Q
where R is the sidelength of the cube Q ⊂ 2Q ⊂ Ω, with the Poincaré-Sobolev inequality
! 12 Z
n+2
Z 2n
u − uR 2
2n
dx
n+2
|∇u|
dx
,
≤
R Q
Q
to obtain the nonhomogeneous reverse Hölder inequality
(Z
n+2
Z
(|∇u|2 )
|∇u|2 dx ≤ c
(1.1)
Q
n
n+2
Z
n
dx
+
)
|F |2 dx .
2Q
2Q
The higher integrability result then arises by using the well-known Giaquinta-Modica technique
[3, 2].
The aim of this paper is to provide a different way to get regularity results, based on inequalities for div-curl vector fields (see Theorem 2.1, [5, 10]). Starting from these inequalities, under
the assumption of bounded distortion, we get directly a family of reverse type inequalities,
namely
Z
(1.2)
(|∇u|2 )1−ε dx
Q
Z
n+1
Z
Z
n
n
2 1−ε
2 (1−ε) n+1
≤ c1 ε (|∇u| ) dx + c2
dx
+ c3 (|F |2 )(1−ε) dx.
(|∇u| )
2Q
2Q
2Q
Notice that, even if inequality (1.2) contains an extra term, by using our method we are able
to obtain a higher integrability result also for very weak solutions of some nonlinear elliptic
equations by just assuming an integrability on the gradient below the natural exponent (see [8]).
Let us observe also that if ε = 0 the exponent n/(n + 1) in inequality (1.2) is larger than the
exponent n/(n + 2) in inequality (1.1). Actually, inequality (1.2) follows from a more general
argument about vector fields of bounded distortion, which includes an analogous result of the
theory of quasiregular mappings (with the same exponent we get in (1.2), see [7]).
After recalling known results in Section 2, we prove a higher integrability result for functions verifying a reverse-type inequality (Theorem 3.1) in Section 3. In Section 4 we give a
counterexample showing that generally the assumptions in Theorem 3.1 cannot be weakened.
In Section 5 we prove a higher integrability result for finite distortion vector fields (see Proposition 5.1), and we give some applications to the theory of quasiconformal mappings and to
the theory of regularity for very weak solutions of homogeneous nonlinear elliptic equations in
divergence form. Finally, in Section 6, we extend our method to the case of more general vector
fields in order to study the case of nonhomogeneous equations (see Theorem 6.1).
J. Inequal. Pure and Appl. Math., 1(2) Art. 14, 2000
http://jipam.vu.edu.au/
R EGULARITY R ESULTS F OR V ECTOR F IELDS
3
2. P RELIMINARY R ESULTS
In the following we will consider div-curl vector fields B = (B1 , . . . , Bn ) ∈ Lq (Rn , Rn ),
E = (E1 , . . . , En ) ∈ Lp (Rn , Rn ), 1 < p, q < ∞, p1 + 1q = 1, i.e.
∂Ei ∂Ej
curl E =
−
=0
∂xj
∂xi i,j=1,...,n
(2.1)
n
X
∂Bi
div B =
=0
∂x
i
i=1
in the sense of distributions.
The following basic estimates are established in [5] (see also [10] for the present formulation).
We denote by Q0 , Q open cubes in Rn with sides parallel to the coordinate axis, and by 2Q the
cube with the same center of Q and double side-length.
Theorem 2.1. Let 1 < p, q < ∞ be a Hölder conjugate pair, p1 + 1q = 1, and let 1 < r, s < ∞
be a Sobolev conjugate pair, 1r + 1s = 1 + n1 . Then there exists a constant cn = cn (p, s) such
that for each cube Q such that 2Q ⊂ Q0 ⊂ Rn we have
(2.2)
Z
p1 Z
Z
1q
hB, Ei (1−ε)q
(1−ε)p
|B|
dx
|E|
dx
|B|ε |E|ε dx ≤ cn ε
2Q
2Q
Q
1sZ
Z
(1−ε)s
|E|
dx
+ cn
2Q
(1−ε)r
|B|
r1
dx
,
2Q
p−1 q−1 r−1 s−1
whenever 0 ≤ 2ε ≤ min
,
,
,
and div B = 0, curl E = 0.
p
q
r
s
The following proposition by Giaquinta-Modica [3, 2] will be useful in the sequel.
Proposition 2.2. Let g ∈ Lα (Q0 ), α > 1 and f ∈ Lr (Q0 ), r > α be two non-negative functions
and suppose that for every cube Q such that 2Q ⊂ Q0 the following estimate holds
Z
α Z
Z
Z
α
α
(2.3)
g dx ≤ b
gdx +
f dx + θ
g α dx
Q
2Q
2Q
2Q
with b > 1. There exist constants θ0 = θ0 (α, n), σ0 = σ0 (b, θ, α, r, n) such that if θ < θ0 , then
g ∈ Lα+σ
loc (Q0 ) for all 0 < σ < σ0 and
(Z
)
1
1
Z
α+σ
α1 Z
α+σ
(2.4)
≤c
g α+σ dx
+
g α dx
,
f α+σ dx
Q
2Q
2Q
where c is a positive constant depending on b, θ, α, r, n.
3. R EVERSE H ÖLDER I NEQUALITIES AND H IGHER I NTEGRABILITY
This section is concerned with a variant of the result established in Proposition 2.2. We
remark that in our assumption (3.1) we will consider a family of inequalities of the type (2.3) in
which both the exponent of integrability of the function g and a coefficient in the right hand side
depend on ε. Nevertheless, even if Proposition 2.2 cannot be applied a priori, in the theorem
we will prove that we can obtain a higher integrability result for g and an estimate of the type
(2.4).
J. Inequal. Pure and Appl. Math., 1(2) Art. 14, 2000
http://jipam.vu.edu.au/
4
A LBERTO F IORENZA AND F LAVIA G IANNETTI
Theorem 3.1. Let g ∈ L2(1−ε) (Q0 ) and f ∈ Lr (Q0 ), 0 ≤ ε < 12 , r > 2(1 − ε), be nonnegative
functions such that
(Z
)
n+1
Z
Z
Z
n
n
g 2(1−ε) dx + c2
g 2(1−ε) dx ≤ c1 ε
(3.1)
f 2(1−ε) dx
g 2(1−ε) n+1 dx
+
2Q
Q
2Q
2Q
for every cube Q ⊂ 2Q ⊂ Q0 , for some constants c1 ≥ 0, c2 > 0.
Then there exist ε̄ = ε̄(c1 , n) and η̄ = η̄(c1 , c2 , r, ε, n) such that if 0 ≤ ε < ε̄, then g ∈
2(1−ε)+η
Lloc
(Q0 ), ∀ 0 ≤ η < η̄ and
)
(Z
1
1
1
2(1−ε)+η
Z
2(1−ε)+η
2(1−ε)
Z
2(1−ε)+η
2(1−ε)
2(1−ε)+η
g
dx
≤c
f
dx
,
g
dx
+
Q
2Q
2Q
where c is a positive constant depending on c2 , r, ε, n.
n
n
Proof. Since the functions gε = g 2(1−ε) n+1 , fε = f 2(1−ε) n+1 verify the inequality
(Z
n+1
Z
)
Z
Z n+1
n
n+1
n+1
n
n
fε dx
gε dx
+
gε dx ≤ c2
(3.2)
gε n dx
+ c1 ε
2Q
2Q
Q
2Q
we can apply Proposition 2.2 with α = n+1
and b = c2 . We get θ0 = θ0 (n) and σ0 =
n
σ0 (c2 , r, ε, n) such that, if (3.2) holds with c1 ε < θ20 , then gε ∈ Lα+σ
loc (Q0 ) for every 0 < σ < σ0 ,
i.e.
h
i n+1
n
+σ
n
g 2(1−ε) n+1
∈ L1loc (Q0 )
∀ 0 < σ < σ0
and
Z
n+1
+σ
n
gε
(3.3)
n+11
dx
n +σ
(Z
n+1
n
≤c
gε
n
n+1
dx
Z
fε
+
with c depending on c2 , r, ε, n.
Set
θ0
0 < ε̄ <
,
2c1
If 0 ≤ ε < ε̄ and 0 ≤ η < η̄, we have
0 < η̄ < (1 − ε̄)
ε < ε̄ < 1 − η̄
)
n
n+1
+σ
dx
2Q
2Q
Q
n+1
+σ
n
2nσ0
.
n+1
n+1
n+1
<1−η
2nσ0
2nσ0
or, equivalently,
n
n+1
2(1 − ε) + η < 2(1 − ε)
+ σ0 .
n+1
n
Therefore we get
2(1−ε)+η
g ∈ Lloc
and inequality (3.3) becomes
(Z
1
Z
2(1−ε)+η
≤c
g 2(1−ε)+η dx
Q
2Q
g 2(1−ε) dx
(Q0 )
1
2(1−ε)
Z
+
f 2(1−ε)+η dx
)
1
2(1−ε)+η
.
2Q
Let us observe that upon closer inspection of the proof of Theorem 3.1, one can note that
r
the gain of integrability given by σ0 = σ0 (c2 , r, ε, n) is actually dependent only on c2 , 2(1−ε)
, n.
Nevertheless, if f ≡ 0 a.e. in Q0 , the number σ0 , and therefore also η̄ and c, do not depend on
ε. This remark is crucial to prove the following.
J. Inequal. Pure and Appl. Math., 1(2) Art. 14, 2000
http://jipam.vu.edu.au/
R EGULARITY R ESULTS F OR V ECTOR F IELDS
5
1
2
and g ∈ L2(1−ε) (Q0 ), Q0 ⊂ Rn , be such that
Z
n+1
Z
Z
n
n
g 2(1−ε) dx ≤ c1 ε
g 2(1−ε) dx + c2
g 2(1−ε) n+1 dx
Corollary 3.2. Let 0 ≤ ε <
Q
2Q
2Q
for every cube Q ⊂ 2Q ⊂ Q0 .
Then there exists ε̄ = ε̄(c1 , n) such that if 0 ≤ ε < ε̄, then g ∈ L2+2ε
loc (Q0 ) and
1
1
2(1−ε)
2(1+ε)
Z
Z
2(1−ε)
2(1+ε)
g
dx
g
dx
≤c
(3.4)
2Q
Q
where c is a positive constant depending on c2 , n.
Proof. Let us apply Theorem 3.1 with f ≡ 0 a.e. in Q0 . If ε < min(ε̄, η̄4 ), choosing η = 4ε,
from inequality (3.1) we get g ∈ L2+2ε
loc (Q0 ) and inequality (3.4) holds.
4. A C OUNTEREXAMPLE
Let us consider f, g non-negative functions on a cube Q0 satisfying assumptions of the type
of Theorem 3.1 with c1 = 0, namely, f, g are such that g ∈ Lα (Q0 ), f ∈ Lλα (Q0 ) for some
α > 1, λ > 1 and
α1
Z
α1
Z
Z
f α dx
∀Q, 2Q ⊂ Q0 .
gdx + b
g α dx
≤a
(4.1)
2Q
2Q
Q
In this case it is known, [6], that if λ is sufficiently close to 1, g ∈ Lλα
loc (Q0 ) and
1
1
λα
λ1
Z
λα
Z
Z
λα
λ
λα
f dx
,
g dx
+ bλ
g dx
≤ aλ
(4.2)
2Q
2Q
Q
where aλ and bλ are constants depending only on n, α, a, b.
In the following we show that, even if it is still true that g ∈ Lλα
loc (Q0 ) for any λ < 1 (sufficiently small), one cannot find any λ < 1, aλ > 0, bλ > 0 such that estimate
for any
T (4.2) holds
α
α
p
g ∈ L (Q0 ), f ∈ L (Q0 ) verifying (4.1). If we consider a function f ∈ 1≤p<α L (Q0 ) such
R
that Q f α dx = +∞ ∀Q ⊂ Q0 , of course we have f ∈ Lλα (Q0 ) for λ < 1, and it is possible
to show immediately that there are no aλ , bλ > 0 such that (4.2) holds for any g ∈ Lα (Q0 ), for
any f ∈ Lλα (Q0 ) verifying (4.1).
We will proceed as follows: by a contradiction argument, we will be able to prove that there
exists λ < 1 such that any function g0 ∈ Lλα (Q0 ), g0 > 0, satisfies a certain reverse inequality,
which is generally false.
Let g ∈ C(Q0 ), g > 0. Then there exists δ > 0 such that
Z
α1
Z
α
≤2
g dx
gdx
∀Q, 2Q ⊂ Q0 , diam 2Q < δ.
2Q
2Q
In fact,
sup g(x) ≤ sup |g(x) − g(y)| + g(y) ∀y ∈ 2Q
x∈2Q
x,y∈2Q
and then, because of the uniform continuity of g, we have
sup g ≤ 2 inf g.
2Q
J. Inequal. Pure and Appl. Math., 1(2) Art. 14, 2000
2Q
http://jipam.vu.edu.au/
6
A LBERTO F IORENZA AND F LAVIA G IANNETTI
Let us divide Q0 into a finite number of disjoint cubes
Q0 =
N
[
Qj
j=1
such that
2Q ⊂ Q0 , diam 2Q ≥ δ ⇒ ∃Qj : Qj ⊂ 2Q.
Now let us point out that if f is any function in Lr (Q0 ) for every 1 ≤ r < α, but
Z
f α dx = +∞
∀j = 1, . . . , N
Qj
then there exist
k min{f, k}
fk = max
j=1,...,N
1
! 2α
Z
(min{f, k})α dx
Qj
such that
Z
2Q
1
fk
k
α α1
Z
α
≥ max g ≥
Q0
α1
∀Q, 2Q ⊂ Q0 , diam 2Q ≥ δ.
g dx
2Q
Therefore
Z
α1
α
g dx
Z
Z
≤2
1
fk
k
gdx +
2Q
2Q
2Q
α α1
∀Q, 2Q ⊂ Q0 ,
i.e. g, k1 fk satisfy inequality (4.1).
Let us suppose to the contrary that for some λ < 1
Z
g λα dx
1
λα
Z
≤ aλ
g λ dx
λ1
Z
+ bλ
2Q
Q
2Q
Letting k tend to infinity, we have the inequality
1
Z
λα
Z
λ1
λα
λ
≤ aλ
g dx
g dx
Q
1
fk
k
1
λα ! λα
∀Q, 2Q ⊂ Q0 .
∀Q, 2Q ⊂ Q0
2Q
for every continuous function g.
This inequality, by an approximation argument, extends to every function g0 ∈ Lλα (Q0 ),
g0 > 0
1
Z
λα
Z
λ1
λα
λ
≤ aλ
g0 dx
∀Q, 2Q ⊂ Q0,
g0 dx
Q
2Q
which is absurd, since this inequality implies a higher integrability for g0 .
5. H IGHER I NTEGRABILITY R ESULTS AND A PPLICATIONS
We start with the following regularity result for vector fields of bounded distortion
Proposition 5.1. Let Ω ⊂ Rn , 0 < < 1 and Φ = (E, B) ∈ L2− (Ω, Rn ) × L2−ε (Ω, Rn ) be
such that div B = 0, curl E = 0 and
1
2
2
(5.1)
|B(x)| + |E(x)| ≤ K +
hB(x), E(x)i
a.e. in Ω,
K
J. Inequal. Pure and Appl. Math., 1(2) Art. 14, 2000
http://jipam.vu.edu.au/
R EGULARITY R ESULTS F OR V ECTOR F IELDS
7
2+ε
n
n
where K ≥ 1. Then there exists ε̄ = ε̄(K, n) such that Φ ∈ L2+ε
loc (Ω, R ) × Lloc (Ω, R ) for all
0 < ε < ε̄ and
1
1
2−ε
2+ε
Z
Z
2−ε
2+ε
|Φ| dx
∀Q, 2Q ⊂ Ω,
|Φ| dx
≤c
2Q
Q
where c is a positive constant depending on K, n.
Proof. Let us fix the cube Q such that 2Q ⊂ Ω. Applying Theorem 2.1 with p = q = 2 and
2n
r = s = n+1
, from inequality (5.1) we get
Z
2
2 1−ε
(|B| +|E| )
n+1
Z
Z
n
n
2
2 (1−ε) n+1
2
2 1−ε
dx ≤ cn,K ε (|B| +|E| ) dx+cn,K
(|B| + |E| )
dx
Q
2Q
2Q
for ε sufficiently small. Substituting g 2 for |B|2 + |E|2 in the last inequality gives
n+1
Z
Z
Z
n
n
(2−2ε) n+1
2−2ε
2−2ε
g
dx
.
g
dx + cn,K
g
dx ≤ cn,K ε
2Q
2Q
Q
By Corollary 3.2 there exists ε̄ = ε̄(K, n) such that if 0 ≤ ε < ε̄, then g ∈ L2+2ε
loc (Ω) and
1
1
2−2ε
2+2ε
Z
Z
g 2−2ε dx
g 2+2ε dx
≤c
2Q
Q
and then the assertion.
Now we consider the equation on a bounded open set Ω ⊂ Rn ,
Au = 0
in Ω ⊂ Rn ,
where A is a differential operator defined by
Au = div a(x, ∇u).
Here a : Ω × Rn → Rn is a mapping such that x → a(x, z) is measurable for all z ∈ Rn and
z → a(x, z) is continuous for almost every x ∈ Ω. Furthermore, we assume that there exists
K ≥ 1 such that for almost every x ∈ Ω we have
1
2
2
(5.2)
|a(x, z)| + |z| ≤ K +
ha(x, z), zi,
K
where x, z are arbitrary vectors in Rn .
Let us prove the following result (originally proved in [9], in the linear case, by using a
duality technique).
Corollary 5.2. Let 0 < <
1
2
1,2−2ε
and u ∈ Wloc
(Ω) be a very weak solution of
div a(x, ∇u) = 0.
1,2+2ε
Then there exists ε̄ = ε̄(K, n) such that u ∈ Wloc
(Ω) for all 0 < ε < ε̄ and
1
1
Z
2+2ε
Z
2−2ε
2+2ε
2−2ε
≤c
|∇u|
dx
,
|∇u|
dx
Q
2Q
where c is a positive constant depending on K, n.
J. Inequal. Pure and Appl. Math., 1(2) Art. 14, 2000
http://jipam.vu.edu.au/
8
A LBERTO F IORENZA AND F LAVIA G IANNETTI
Proof. Setting
E = ∇u,
B = a(x, ∇u)
we have div B = 0, curl E = 0 and, by (5.2),
2
2
2
2
|E| + |B| = |∇u| + |a(x, ∇u)| ≤
1
K+
K
ha(x, ∇u), ∇ui
so that E, B are div-curl fields of bounded distortion. Applying Proposition 5.1 we get the
assertion.
Another interesting case to which Proposition 5.1 applies is when one considers a homeomorphism
f = (f 1 , f 2 ) : Ω ⊂ R2 → R2 ,
f i ∈ W 1,2−ε (Ω)
f K-quasiregular,
i = 1, 2,
K ≥ 1,
i.e.
2
|Df (x)| ≤
1
K+
K
J(x, f ),
where |Df (x)| denotes the norm of the distributional differential Df (x) and J(x, f ) is the
Jacobian determinant J(x, f ) = det Df (x).
Then, writing the Jacobian J(x, f ) as hB, Ei, where E = ∇f 1 = (fx1 , fy1 ) and B =
(fy2 , −fx2 ) we have div B = 0, curl E = 0 and that (5.1) holds. It follows that for ε sufficiently small f ∈ W 1,2+ε (Ω, R2 ), giving back in this way the celebrated theorem by Bojarski
[1]. Significant results about the Jacobian determinant are in [7].
6. R EGULARITY R ESULTS FOR N ONHOMOGENEOUS E QUATIONS
In this section we consider Φ = (E, B) ∈ L2−2 (Ω, Rn ) × L2−2ε (Ω, Rn ) such that
div B = 0, curl E = 0,
1
2
2
|B(x)| + |E(x)| ≤ K +
hB(x), E(x)i + |F |2 ,
K
(6.1)
(6.2)
where F is a function in Lr (Ω, Rn ), r > 2 − 2ε, for ε sufficiently small.
Theorem 6.1. Let 0 ≤ ε < 12 and E, B vector fields as in (6.1),(6.2). Then there exist ε̄ =
ε̄(K, n) and η̄ = η̄(K, r, ε, n) such that if 0 ≤ ε < ε̄, then Φ = (E, B) ∈ L2−2+η
(Ω, Rn ) ×
loc
2−2ε+η
Lloc
(Ω, Rn ) for all 0 ≤ η < η̄ and
(6.3)
(Z
)
1
1
1
Z
2−2ε+η
2−2ε
Z
2−2ε+η
2−2ε+η
≤c
|Φ|2−2ε+η dx
+
|Φ|2−2ε dx
,
(|F |2 ) 2 dx
Q
2Q
2Q
where c is a positive constant depending on K, r, ε, n.
Proof. Let us fix Q a cube such that 2Q ⊂ Ω and set
Q+ = {x ∈ Q | hB, Ei ≥ 0 a.e.}
Q− = {x ∈ Q | hB, Ei ≤ 0 a.e.}
J. Inequal. Pure and Appl. Math., 1(2) Art. 14, 2000
http://jipam.vu.edu.au/
R EGULARITY R ESULTS F OR V ECTOR F IELDS
9
Let us observe that by (6.2), replacing |F | with f , we have
Z
Z
−hB, Ei
dx ≤
(|B||E|)1−ε dx
ε |E|ε
|B|
−
−
Q
Q
Z
≤
(|B|2 + |E|2 )1−ε dx
Q−
1−ε
1
2
K+
≤
hB, Ei + f
dx
K
Q−
Z
Z
2−2ε
≤
f
dx ≤
f 2−2ε dx
Z
Q−
Q
and therefore
Z
Q
hB, Ei
dx =
|B|ε |E|ε
Z
Q+
Z
≥
Q+
Z
hB, Ei
hB, Ei
dx +
dx
ε
ε
ε
ε
|B| |E|
Q− |B| |E|
Z
hB, Ei
dx −
f 2−2ε dx
(|B|2 + |E|2 + f 2 )ε
Q
Applying Theorem 2.1 with p = q = 2 and r = s =
Z
Q
2n
,
n+1
for ε sufficiently small, we get
Z
hB, Ei
dx ≤ cn ε (|B|2 + |E|2 + f 2 )1−ε dx
(|B|2 + |E|2 + f 2 )ε
2Q
n+1
Z
Z
n
n
(1−ε)
2
2
2
n+1
(|B| + |E| + f )
dx
+
+ cn
f 2−2ε dx.
2Q
2Q
By (6.2)
hB, Ei ≥ cK (|B|2 + |E|2 − f 2 ) = cK (|B|2 + |E|2 + f 2 ) − 2cK f 2
and therefore
Z
(|B|2 + |E|2 + f 2 )1−ε dx
Q
Z
n+1
Z
n
n
(1−ε)
n+1 dx
≤ cn,K ε (|B|2 + |E|2 + f 2 )1−ε dx + cn,K
(|B|2 + |E|2 + f 2 )
2Q
Z
+ cK
2Q
2Q
2
f
dx +
(|B|2 + |E|2 + f 2 )ε
Z
f 2−2ε dx
2Q
Z
n+1
Z
n
n
2
2
2 1−ε
2
2
2 (1−ε) n+1
≤ cn,K ε (|B| + |E| + f ) dx + cn,K
(|B| + |E| + f )
dx
2Q
2Q
Z
+ (cK + 1)
f 2−2ε dx.
2Q
Setting g 2 = |B|2 + |E|2 + f 2 , the last inequality implies
Z
g
Q
2−2ε
Z
dx ≤ cn,K ε
g
2−2ε
Z
dx + cn,K
2Q
J. Inequal. Pure and Appl. Math., 1(2) Art. 14, 2000
g
2Q
n
(2−2ε) n+1
n+1
n
dx
Z
+ (cK + 1)
f 2−2ε dx.
2Q
http://jipam.vu.edu.au/
10
A LBERTO F IORENZA AND F LAVIA G IANNETTI
By Theorem 3.1 there exist ε̄ = ε̄(K, n) and η̄ = η̄(K, r, ε, n) such that if 0 ≤ ε < ε̄, then
2−2ε+η
g ∈ Lloc
(Ω) for all 0 ≤ η < η̄ and
)
(Z
1
1
1
2−2ε+η
2−2ε
Z
2−2ε+η
Z
2−2ε+η
(f 2 ) 2 dx
g 2−2ε dx
+
g 2−2ε+η dx
≤c
2Q
2Q
Q
and then the assertion.
Let us consider now the following equation on a bounded open set Ω ⊂ Rn
div a(x, ∇u) = div F,
(6.4)
where F ∈ Lr (Ω), r > 2 − 2ε, for ε sufficiently small and a : Ω × Rn → Rn is a mapping
satisfying the assumptions of Section 5.
1,2−2ε
Corollary 6.2. Let 0 ≤ ε < 21 and u ∈ Wloc
(Ω) be a very weak solution of the equation
(6.4). Then there exist ε̄ = ε̄(K, n) and η̄ = η̄(K, r, ε, n) such that if 0 ≤ ε < ¯, then
1,2−2ε+η
u ∈ Wloc
(Ω) for all 0 ≤ η < η̄ and
)
(Z
1
1
1
2−2ε+η
2−2ε
Z
2−2ε+η
Z
|F |2−2ε+η dx
|∇u|2−2ε dx
+
|∇u|2−2ε+η dx
≤c
2Q
2Q
Q
for all cubes Q such that 2Q ⊂ Ω and where c is a positive constant depending on K, r, ε, n.
Proof. Setting
E = ∇u,
B = a(x, ∇u) − F,
we have
|E|2 + |B|2 ≤ |∇u|2 + (|a(x, ∇u)| + |F |)2
≤ 2(|a(x, ∇u)|2 + |∇u|2 ) + 2|F |2
1
≤2 K+
ha(x, ∇u), ∇ui + 2|F |2
K
1
1
ha(x, ∇u) − F, ∇ui + 2 K +
hF, ∇ui + 2|F |2 .
=2 K+
K
K
Since by Young’s inequality
1
1
|∇u|2 − hF, ∇ui
hF, ∇ui = 2hF, ∇ui − hF, ∇ui ≤ 2 · 2 K +
|F |2 + 2
K
2 K + K1
1
1
2
2
≤4 K+
|F |2 +
1 (|a(x, ∇u)| + |∇u| ) − hF, ∇ui
K
K+K
1
≤4 K+
|F |2 + ha(x, ∇u), ∇ui − hF, ∇ui
K
1
=4 K+
|F |2 + ha(x, ∇u) − F, ∇ui
K
we get
" #
2
1
1
|E|2 + |B|2 ≤ 4 K +
hB, Ei + 8 K +
+ 2 |F |2 ,
K
K
i.e. E, B are vector fields satisfying (6.1) and (6.2). From Theorem 6.1 we get the higher
integrability for |E|2 + |B|2 and then for |∇u|; the estimate follows directly from (6.3).
J. Inequal. Pure and Appl. Math., 1(2) Art. 14, 2000
http://jipam.vu.edu.au/
R EGULARITY R ESULTS F OR V ECTOR F IELDS
11
1,2
Remark 6.3. Let u ∈ Wloc
(Ω) be a solution of the equation (6.4). Corollary 6.2 asserts that the
function g = |∇u| verifies inequality (4.1) and, surprisingly, satisfies also inequality (4.2) with
λ < 1 sufficiently small.
Remark 6.4. We note that, arguing as in the end of Section 5, our result of higher integrability
applies also to (K, K 0 )-quasiregular mappings (see [4]), i.e. functions f verifying
f = (f 1 , f 2 ) : Ω ⊂ R2 → R2 ,
f i ∈ W 1,2−ε (Ω)
i = 1, 2,
1
J(x, f ) + K 0 .
|Df (x)|2 ≤ K +
K
R EFERENCES
[1] B. BOJARSKI, Homeomorphic solutions of Beltrami system, Dokl. Akad. Nauk. SSSR, 102 (1955),
661–664.
[2] E. GIUSTI, Metodi Diretti nel Calcolo delle Variazioni, Unione Matematica Italiana (1994).
[3] M. GIAQUINTA AND G. MODICA, Regularity results for some classes of higher order non linear
elliptic systems, J. Reine Angew. Math., 311/312 (1979), 145–169.
[4] D. GILBARG AND N.S. TRUDINGER, Elliptic Partial Differential Equations of Second Order,
Springer (1983).
[5] T. IWANIEC, Integrability Theory of the Jacobians, Lipschitz Lectures (1995).
[6] T. IWANIEC, The Gehring Lemma, Quasiconformal Mappings and Analysis, A Collection of Papers Honoring F.W. Gehring P.L. Duren, J.M. Heinonen, B.G. Osgood, B.P. Palka (1998) SpringerVerlag, 181–204.
[7] T. IWANIEC AND C. SBORDONE, On the integrability of the Jacobian under minimal hypothesis,
Arch. Rat. Mech. Anal., 119 (1992), 129–143.
[8] T. IWANIEC AND C. SBORDONE, Weak minima of variational integrals, J. Reine Angew. Math.,
454 (1994), 143–161.
[9] N. MEYERS AND A. ELCRAT, Some results on regularity for solutions of nonlinear elliptic systems and quasiregular functions, Duke Math. J., 42(1) (1975), 121–136.
[10] C. SBORDONE, New estimates for div-curl products and very weak solutions of PDEs, Annali
Scuola Normale Superiore di Pisa (IV), 25(3-4) (1997), 739–756.
J. Inequal. Pure and Appl. Math., 1(2) Art. 14, 2000
http://jipam.vu.edu.au/