Journal of Inequalities in Pure and
Applied Mathematics
REGULARITY RESULTS FOR VECTOR FIELDS OF BOUNDED
DISTORTION AND APPLICATIONS
volume 1, issue 2, article 14,
2000.
ALBERTO FIORENZA AND FLAVIA GIANNETTI
Dipartimento di Costruzioni e Metodi Matematici in Architettura
via Monteoliveto 3
-80134 Napoli, ITALY
EMail: fiorenza@unina.it
URL: http://cds.unina.it/~fiorenza/
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”
via Cintia
80126 Napoli, ITALY
EMail: giannett@matna2.dma.unina.it
Received 17 January, 2000;
accepted 4 April, 2000.
Communicated by: S. Saitoh
Abstract
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School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
022-99
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Abstract
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
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
div a(x, ∇u) = div F.
If |a(x, z)|2 +|z|2 ≤ (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.
2000 Mathematics Subject Classification: 35J60, 26D15
Key words: Reverse Inequalities, Finite Distortion Vector Fields, Div-Curl Vector
Fields, Elliptic Partial Differential Equations
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.
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2
Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Reverse Hölder Inequalities and Higher Integrability . . . . . . . 8
4
A Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5
Higher Integrability Results and Applications . . . . . . . . . . . . . 15
6
Regularity Results for Nonhomogeneous Equations . . . . . . . . 19
References
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
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1.
Introduction
1,2+ε
The usual way to establish the Wloc
(Ω), ε > 0, regularity of solutions u ∈
1,2
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 |F |2 dx ,
|∇u|2 dx ≤ c
2R dx +
2Q
2Q
Q
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
≤
,
|∇u| n+2 dx
R dx
Q
Q
to obtain the nonhomogeneous reverse Hölder inequality
)
(Z
n+2
Z
Z
n
n
|F |2 dx .
(|∇u|2 ) n+2 dx
+
|∇u|2 dx ≤ c
(1.1)
Q
2Q
2Q
The higher integrability result then arises by using the well-known GiaquintaModica 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
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
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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
(|∇u| )
dx
+c3 (|F |2 )(1−ε) dx.
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).
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
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2.
Preliminary Results
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
∂xi
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
1q
p1 Z
Z
hB,
Ei
(1−ε)q
(1−ε)p
|B|
dx
|E|
dx
|B|ε |E|ε dx ≤ cn ε
2Q
2Q
Q
r1
1sZ
Z
(1−ε)r
(1−ε)s
|B|
dx ,
|E|
dx
+ cn
2Q
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
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whenever 0 ≤ 2ε ≤ min
p−1 q−1 r−1 s−1
,
,
,
p
q
r
s
and div B = 0,
curl E = 0.
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 nonnegative functions and suppose that for every cube Q such that 2Q ⊂ Q0 the
following estimate holds
α Z
Z
Z
Z
α
α
g α dx
f dx + θ
gdx +
g dx ≤ b
(2.3)
2Q
2Q
2Q
Q
with b > 1. There exist constants θ0 = θ0 (α, n), σ0 = σ0 (b, θ, α, r, n) such that
α+σ
if θ < θ0 , then g ∈ Lloc
(Q0 ) for all 0 < σ < σ0 and
(Z
1 )
1 Z
1
Z
g
(2.4)
Q
α+σ
α+σ
dx
α
≤c
g dx
2Q
α
f
+
2Q
where c is a positive constant depending on b, θ, α, r, n.
α+σ
Regularity results for vector
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Alberto Fiorenza and
Flavia Giannetti
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α+σ
dx
,
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3.
Reverse Hölder Inequalities and Higher Integrability
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).
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
(3.1)
Q
2Q
g 2(1−ε) dx + c2
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g 2(1−ε) dx
Z
≤ c1 ε
Regularity results for vector
fields of bounded distortion and
applications
(Z
g
2Q
n
2(1−ε) n+1
Contents
n+1
n
dx
Z
+
f 2(1−ε) dx
)
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,
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2(1−ε)+η
then g ∈ Lloc
Z
g
2(1−ε)+η
(Q0 ), ∀ 0 ≤ η < η̄ and
1
2(1−ε)+η
dx
Q
(Z
g 2(1−ε) dx
≤c
1
2(1−ε)
Z
f 2(1−ε)+η dx
+
2Q
)
1
2(1−ε)+η
,
2Q
where c is a positive constant depending on c2 , r, ε, n.
n
2(1−ε) n+1
n
2(1−ε) n+1
, fε = f
verify the inequalProof. Since the functions gε = g
ity
(Z
)
n+1
Z
Z
Z n+1
n
n+1
n+1
n
n
gε n dx
fε dx
+c1 ε
gε dx
+
gε dx ≤ c2
(3.2)
2Q
2Q
2Q
Q
we can apply Proposition 2.2 with α = n+1
and b = c2 . We get θ0 = θ0 (n) and
n
σ0 = 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
g
n
2(1−ε) n+1
i n+1
+σ
n
∈ L1loc (Q0 )
Regularity results for vector
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∀ 0 < σ < σ0
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and
(3.3)
Z
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n+1
+σ
n
gε
Q
n+11
dx
n +σ
(Z
≤c
n+1
n
gε
2Q
n
n+1
dx
Z
n+1
+σ
n
fε
+
2Q
)
n
n+1
+σ
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dx
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with c depending on c2 , r, ε, n.
Set
θ0
,
0 < ε̄ <
2c1
0 < η̄ < (1 − ε̄)
2nσ0
.
n+1
If 0 ≤ ε < ε̄ and 0 ≤ η < η̄, we have
ε < ε̄ < 1 − η̄
n+1
n+1
<1−η
2nσ0
2nσ0
Regularity results for vector
fields of bounded distortion and
applications
or, equivalently,
n+1
n
2(1 − ε) + η < 2(1 − ε)
+ σ0 .
n+1
n
Alberto Fiorenza and
Flavia Giannetti
Therefore we get
2(1−ε)+η
g ∈ Lloc
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(Q0 )
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and inequality (3.3) becomes
Z
g 2(1−ε)+η dx
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1
2(1−ε)+η
Q
(Z
≤c
g
2Q
2(1−ε)
1
2(1−ε)
dx
Z
f
+
2(1−ε)+η
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1
2(1−ε)+η
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Let us observe that upon closer inspection of the proof of Theorem 3.1, one
can note that the gain of integrability given by σ0 = σ0 (c2 , r, ε, n) is actually
r
, n. Nevertheless, if f ≡ 0 a.e. in Q0 , the number
dependent only on c2 , 2(1−ε)
σ0 , and therefore also η̄ and c, do not depend on ε. This remark is crucial to
prove the following.
Corollary 3.2. Let 0 ≤ ε <
Z
g
2(1−ε)
≤
dx
1
2
and g ∈ L2(1−ε) (Q0 ), Q0 ⊂ Rn , be such that
Z
c1 ε
g
2(1−ε)
Z
n+1
n
dx
2Q
2Q
Q
g
dx + c2
n
2(1−ε) n+1
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)
Q
2Q
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.
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
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4.
A Counterexample
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
Z
α
α1
g dx
(4.1)
Z
≤a
Q
Z
gdx + b
2Q
α
α1
∀Q, 2Q ⊂ Q0 .
f dx
2Q
In this case it is known, [6], that if λ is sufficiently close to 1, g ∈ Lλα
loc (Q0 ) and
Z
λα
g dx
(4.2)
Q
1
λα
Z
≤ aλ
λ
g dx
2Q
λ1
Z
+ bλ
f
λα
1
λα
dx
,
Regularity results for vector
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applications
Alberto Fiorenza and
Flavia Giannetti
2Q
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 (4.2) holds for T
any g ∈ Lα (Q0 ), f ∈ LRα (Q0 ) verifying (4.1). If we
consider a function f ∈ 1≤p<α Lp (Q0 ) such 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.
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Let g ∈ C(Q0 ), g > 0. Then there exists δ > 0 such that
Z
α1
Z
α
g dx
≤2
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
Regularity results for vector
fields of bounded distortion and
applications
sup g ≤ 2 inf g.
2Q
2Q
Let us divide Q0 into a finite number of disjoint cubes
Q0 =
N
[
Qj
j=1
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such that
2Q ⊂ Q0 , diam 2Q ≥ δ
Alberto Fiorenza and
Flavia Giannetti
⇒
∃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
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then there exist
k min{f, k}
fk = max
j=1,...,N
Z
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1
! 2α
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α
(min{f, k}) dx
Qj
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such that
Z α α1
Z
α1
1
≥ max g ≥
fk
g α dx
k
Q0
2Q
2Q
Therefore
Z
α1
Z
α
≤2
g dx
2Q
Z
gdx +
2Q
2Q
∀Q, 2Q ⊂ Q0 , diam 2Q ≥ δ.
1
fk
k
α α1
∀Q, 2Q ⊂ Q0 ,
Regularity results for vector
fields of bounded distortion and
applications
i.e. g, k1 fk satisfy inequality (4.1).
Let us suppose to the contrary that for some λ < 1
Z
λα
1
λα
Z
≤ aλ
g dx
λ
g dx
λ1
Z
+bλ
2Q
2Q
Q
Letting k tend to infinity, we have the inequality
1
λ1
λα
Z
Z
λ
λα
g dx
g dx
≤ aλ
Q
1
fk
k
1
λα ! λα
∀Q, 2Q ⊂ Q0 .
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∀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
λ1
λα
Z
Z
λ
λα
g0 dx
∀Q, 2Q ⊂ Q0,
g0 dx
≤ aλ
Q
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Flavia Giannetti
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which is absurd, since this inequality implies a higher integrability for g0 .
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5.
Higher Integrability Results and Applications
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
hB(x), E(x)i
a.e. in Ω,
(5.1)
|B(x)| + |E(x)| ≤ K +
K
n
where K ≥ 1. Then there exists ε̄ = ε̄(K, n) such that Φ ∈ L2+ε
loc (Ω, R ) ×
2+ε
n
Lloc (Ω, R ) for all 0 < ε < ε̄ and
Z
|Φ|2+ε dx
1
2+ε
Q
Z
≤c
|Φ|2−ε dx
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
1
2−ε
∀Q, 2Q ⊂ Ω,
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2Q
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
2n
p = q = 2 and r = s = n+1
, from inequality (5.1) we get
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Z
(|B|2 + |E|2 )1−ε dx
Close
Q
n+1
Z
Z
n
n
2
2 (1−ε) n+1
2
2 1−ε
(|B| + |E| )
dx
≤ cn,K ε (|B| + |E| ) dx + cn,K
2Q
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for ε sufficiently small. Substituting g 2 for |B|2 + |E|2 in the last inequality
gives
Z
Z
2−2ε
g
dx ≤ cn,K ε
Q
g 2−2ε dx + cn,K
Z
g
2Q
n
(2−2ε) n+1
n+1
n
dx
.
2Q
By Corollary 3.2 there exists ε̄ = ε̄(K, n) such that if 0 ≤ ε < ε̄, then g ∈
L2+2ε
loc (Ω) and
Z
g 2+2ε dx
1
2+2ε
Z
≤c
g 2−2ε dx
1
2−2ε
Regularity results for vector
fields of bounded distortion and
applications
2Q
Q
Alberto Fiorenza and
Flavia Giannetti
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).
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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
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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
Z
|∇u|2+2ε dx
1
2+2ε
Z
≤c
|∇u|2−2ε dx
1
2−2ε
,
2Q
Q
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
where c is a positive constant depending on K, n.
Title Page
Proof. Setting
E = ∇u,
B = a(x, ∇u)
we have div B = 0, curl E = 0 and, by (5.2),
1
2
2
2
2
|E| + |B| = |∇u| + |a(x, ∇u)| ≤ K +
ha(x, ∇u), ∇ui
K
so that E, B are div-curl fields of bounded distortion. Applying Proposition 5.1
we get the assertion.
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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].
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
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6.
Regularity Results for Nonhomogeneous Equations
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)
Regularity results for vector
fields of bounded distortion and
applications
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 ) × L2−2ε+η
(Ω, Rn ) for all 0 ≤ η < η̄ and
loc
loc
Z
2−2ε+η
|Φ|
(6.3)
dx
JJ
J
Q
≤c
|Φ|2−2ε dx
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Contents
1
2−2ε+η
(Z
Alberto Fiorenza and
Flavia Giannetti
1
2−2ε
2Q
Z
+
(|F |2 )
2−2ε+η
2
2Q
where c is a positive constant depending on K, r, ε, n.
)
1
2−2ε+η
dx
,
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Proof. Let us fix Q a cube such that 2Q ⊂ Ω and set
Q+ = {x ∈ Q |
Page 19 of 27
hB, Ei ≥ 0 a.e.}
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Q− = {x ∈ Q | hB, Ei ≤ 0 a.e.}
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
ZQ
(|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−
and therefore
Z
Q
hB, Ei
dx =
|B|ε |E|ε
Z
Q+
Z
≥
Q+
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
2n
Applying Theorem 2.1 with p = q = 2 and r = s = n+1
, for ε sufficiently
small, we get
Z
Z
hB, Ei
dx ≤ cn ε (|B|2 + |E|2 + f 2 )1−ε dx
2
2
2
ε
2Q
Q (|B| + |E| + f )
n+1
Z
Z
n
n
2
2
2 (1−ε) n+1
f 2−2ε dx.
(|B| + |E| + f )
dx
+
+ cn
2Q
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
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2Q
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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
≤ cn,K ε (|B|2 + |E|2 + f 2 )1−ε dx
2Q
Z
2
2
n+1
n
n
2 (1−ε) n+1
(|B| + |E| + f )
+ cn,K
2Q
Z
dx
Z
f2
f 2−2ε dx
dx +
2
2
2
ε
(|B| + |E| + f )
2Q
+ cK
Z 2Q
≤ cn,K ε (|B|2 + |E|2 + f 2 )1−ε dx
2
2
n
2 (1−ε) n+1
(|B| + |E| + f )
Z
f 2−2ε dx.
+ (cK + 1)
+ cn,K
2Q
2Q
Alberto Fiorenza and
Flavia Giannetti
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Contents
2Q
Z
Regularity results for vector
fields of bounded distortion and
applications
n+1
n
dx
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Setting g 2 = |B|2 + |E|2 + f 2 , the last inequality implies
Z
g
2−2ε
Z
dx ≤ cn,K ε
Q
g
2−2ε
Z
dx + cn,K
2Q
g
n
(2−2ε) n+1
n+1
n
dx
2Q
Z
+ (cK + 1)
f 2−2ε dx.
2Q
By Theorem 3.1 there exist ε̄ = ε̄(K, n) and η̄ = η̄(K, r, ε, n) such that if
0 ≤ ε < ε̄, then g ∈ L2−2ε+η
(Ω) for all 0 ≤ η < η̄ and
loc
)
(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
Q
2Q
2Q
and then the assertion.
Alberto Fiorenza and
Flavia Giannetti
Title Page
Let us consider now the following equation on a bounded open set Ω ⊂ Rn
(6.4)
Regularity results for vector
fields of bounded distortion and
applications
div a(x, ∇u) = div F,
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 < 12 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
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1,2−2ε+η
if 0 ≤ ε < ¯, then u ∈ Wloc
(Ω) for all 0 ≤ η < η̄ and
Z
2−2ε+η
|∇u|
Q
1
2−2ε+η
dx
(Z
≤c
|∇u|2−2ε dx
1
2−2ε
Z
+
2Q
|F |2−2ε+η dx
)
1
2−2ε+η
2Q
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,
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
we have
Title Page
2
2
2
2
|E| + |B| ≤ |∇u| + (|a(x, ∇u)| + |F |)
≤ 2(|a(x, ∇u)|2 + |∇u|2 ) + 2|F |2
1
≤2 K+
ha(x, ∇u), ∇ui + 2|F |2
K
1
1
=2 K+
ha(x, ∇u) − F, ∇ui + 2 K +
hF, ∇ui + 2|F |2 .
K
K
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Since by Young’s inequality
hF, ∇ui = 2hF, ∇ui − hF, ∇ui
1
1
|∇u|2 − hF, ∇ui
≤2·2 K +
|F |2 + 2
K
2 K + K1
1
1
≤4 K+
|F |2 +
(|a(x, ∇u)|2 + |∇u|2 ) − hF, ∇ui
K
K + K1
1
≤4 K+
|F |2 + ha(x, ∇u), ∇ui − hF, ∇ui
K
1
=4 K+
|F |2 + ha(x, ∇u) − F, ∇ui
K
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
we get
1
|E|2 + |B|2 ≤ 4 K +
K
" #
2
1
+ 2 |F |2 ,
hB, Ei + 8 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).
1,2
Wloc
(Ω)
Remark 6.1. Let u ∈
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 6.2. 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.
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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
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
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References
[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) Springer-Verlag, 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.
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
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[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.
Regularity results for vector
fields of bounded distortion and
applications
Alberto Fiorenza and
Flavia Giannetti
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