A Generalization of Weak
WT
2-Class
of Differential Forms and Its Applications
Gao Hongya
College of Mathematics and Information Science, Hebei University, Baoding, 071002, China. email: [email protected]
Wang Tian
College of Mathematics and Information Science, Hebei University, Baoding, 071002, China. email: [email protected]
Di Qinghua
College of Mathematics and Information Science, Hebei University, Baoding, 071002, China. email: [email protected]
Abstract
A generalized class of weakWT2-class of differential forms is defined,
a weak reverse H¨older inequality is obtained for this class, and some ap-plications to the regularity theory of weakly (K1, K2)-quasiregular map-pings, very weak solutions of nonhomogeneous A-harmonic equations, and generalized solutions of Beltrami system with three characteristic matrices are given.
Mathematics Subject Classification: 30C65, 35J60.
Keywords: generalized weakWT2-class of differential forms, weak reverse H¨older inequality, nonhomogeneousA-harmonic equation, (K1, K2)-quasiregular mapping, Beltrami system.
1
Introduction
We first introduce some symbols and notations used in this paper. Let Ω be a connected open subset of Rn, n ≥ 2. We use e1, e2,· · ·, en to denote the
standard unit basis of Rn. Let Vℓ
=Vℓ
( Rn) be the linear space of ℓ-covectors, spanned by the exterior products eI =ei1 ∧ei2 ∧ · · · ∧eiℓ corresponding to all
ordered ℓ-tuples I = (i1, i2,· · ·, iℓ),1 ≤i1 < i2 < · · ·< iℓ ≤ n, ℓ = 0,1,· · ·, n.
A differentialℓ-formωon Ω is a Schwartz distribution on Ω with values inVℓ
=
Vℓ
(Rn). We write ω ∈ D′
Ω,Vℓ
. For α=P
αIe
I ∈Vℓ and β =PβIeI ∈Vℓ,
the inner product in Vℓ
is given by hα, βi =P
αIβI with summation over all
ℓ-tuples I = (i1, i2,· · ·, iℓ). The Hodge star operator ∗:Vℓ →Vn−ℓ is defined
by the rule ∗1 = e1∧e2∧ · · · ∧en and
α∧ ∗β =β∧ ∗α=hα, βi ∗1 for all α, β ∈ Vℓ
. The norm of α∈Vℓ
is given by the formula |α|2 =hα, αi=
∗(α∧ ∗α)∈V0
= R. The Hodge star is an isometric isomorphism onVℓ
. It is obvious that ∗∗ = (−1)ℓ(n−ℓ) :Vℓ
→ Vℓ
. We set ∗−1ω = (−1)ℓ(n−ℓ)∗ω for ω a differential form of degree ℓ. The operator ∗−1 is an inverse to ∗ in the sense that ∗−1(∗ω) =∗(∗−1ω) =ω forω ∈Vℓ
.
Let 1 ≤p < ∞. We denote the Lp-norm of a measurable function f over
E by
kfkp =kfkp,E =
Z
E|f(x)|
pdx1/p.
We write LpΩ,Vℓ
for theℓ-formsω(x) =P
IωI(x)dxI =Pωi1i2···iℓ(x)dxi1 ∧
dxi2∧ · · · ∧dxiℓ withωI ∈L
p(Ω,R) for all orderedℓ-tuplesI. Thus Lp
Ω,Vℓ
is a Banach space with norm
kωkp,Ω =
Z
Ω|ω(x)|
pdx1/p =Z
Ω
X
|ωI(x)|2
p/2
dx
1/p
.
Similarly,W1,p
Ω,Vℓ
are those differentialℓ-forms on Ω whose coefficients are inW1,p(Ω,R). The notationsW1,p
loc (Ω,R) andW
1,p loc
Ω,Vℓ
are self-explanatory. The exterior derivative is denoted by d : D′
Ω,Vℓ
→ D′
Ω,Vℓ+1
for
ℓ = 0,1,· · ·, n. Its formal adjoint operator d∗ : D′Ω,Vℓ+1
→ D′Ω,Vℓ
is given by d∗ = (−1)nℓ+1∗d∗ on D′Ω,Vℓ+1
, ℓ = 0,1,· · ·, n. The well-known Poincar´e Lemma states thatd◦d = 0. It is easy to see thatd∗◦d∗ = 0 as well.
A differential ℓ-formω ∈ D′
Ω,Vℓ
is called simple if there are differential forms α1,· · ·, αℓ of degree 1 such that
ω =α1 ∧ · · · ∧αℓ.
A useful property is: if α∈Vk
and β∈Vℓ
, then
if at least one of the differential forms α, β is simple. A differential ℓ-formu∈ D′
Ω,Vℓ
is called a closed form if du = 0 in Ω. It is called exact if there exists a differential form α ∈ D′Ω,Vℓ−1
such that
u = dα. Poincar´e Lemma implies that exact forms are closed. Similarly, a differential (ℓ+ 1)-form v ∈ D′Ω,Vℓ+1
is called a coclosed form if d∗v = 0. It is called coexact if there exists a differential form β ∈ D′
Ω,Vℓ
such that
v =d∗β. Balls with radius R are denoted by B
R and BσR is the ball with the
same center asBRand diam(BσR) =σdiam(BR). Then-dimensional Lebesgue
measure of a set E ⊂Rn is denoted by |E|.
The following result can be found in [2]. Let Q ⊂ Rn be a cube or a ball. To each y ∈ Q there corresponds a linear operator Ky : C∞
Q,Vℓ
→
C∞Q,Vℓ−1
defined by
(Kyω)(x;ξ1,· · ·, ξℓ−1) =
Z 1
0 t
ℓ−1ω(tx+y−ty;x−y, ξ
1,· · ·, ξℓ−1)dt and the decomposition
ω =d(Ky) +Ky(dω).
Another linear operator TQ : C∞
Q,Vℓ
→C∞
Q,Vℓ−1
is defined by aver-aging Ky over all pointsy in Q with
TQω =
Z
Qϕ(y)Kyωdy,
where ϕ ∈ C∞
0 (Q) is normalized by
R
Qϕ(y)dy = 1. The ℓ-form ωQ is defined
by
ωQ =|Q|−1
Z
Qω(y)dy
ifℓ= 0; andωQ=d(TQω) ifℓ= 1,· · ·, nfor allω∈Lp
Q,Vℓ
and 1≤p <∞. Franke et al introduced in [3] four classes of differential forms on Rie-mannian manifolds and showed that some differential expressions connected in a natural way to quasiregular mappings are members in these classes. In [4], some counterparts of theorems of Phragm´en-Lindel¨of and of Ahlfors were proved for differential forms of WT-classes. Gao and Wang [1] gave a defi-nition of weak WT2-class of differential forms, and obtained its weak reverse H¨older inequality, an alternative proof for the higher integrability result of weakly A-harmonic tensors due to B.Stroffolini is provided.
Definition 1.1[2]. A differential form α ∈ LplocΩ,Vℓ
is called weakly closed, if for each differential formβ ∈Wloc1,q(Ω,Vℓ+1
), 1
p+
1
q = 1, 1≤p, q≤ ∞
with
we have Z
Ωhα, d
∗βidx= 0.
Remark 1.1 A closed differential form α∈LplocΩ,Vℓ
is weakly closed. Definition 1.2[3]. A weakly closed differential form ω ∈ LplocΩ,Vℓ
,0 ≤
ℓ ≤n, p >1is said to be of the class WT2 onΩ if there exists a weakly closed differential form θ ∈ LqlocΩ,Vn−ℓ
, 1
p +
1
q = 1, such that almost everywhere
on Ω the conditions
ν1|ω|p≤ hω,∗θi (1.1) and
|θ| ≤ν2|ω|p−1 (1.2) are satisfied, with constants ν1, ν2 >0.
Now let
ω∈Lrloc
Ω,^ℓ
, 0≤ℓ≤n, max{1, p−1} ≤r < p, p > 1 (1.3)
be a weakly closed differential form on Ω.
Definition 1.3[1]. A weakly closed differential form ω (1.3) is said to be of the class weak WT2 on Ω, if there exists a weakly closed differential form
θ ∈L
r p−1
loc
Ω,^n−ℓ
, (1.4)
such that the conditions (1.1) and (1.2) hold almost everywhere on Ω, with constants ν1, ν2 >0.
Remark 1.2 The word weak in Definition 1.3 means that the integrable exponent r of ω can be smaller than the natural onep.
We now give a generalization of weak WT2 class of differential forms. Definition 1.4. An exact differential formω (1.3) is said to be of the class weak WT2 on Ω, if there exists a weakly closed differential form θ (1.4) such that the conditions
|ω|p ≤γ
1hω,∗θi+γ2|ω|p−δ+γ3 (1.5) and
|θ| ≤γ4|ω|p−1+γ5 (1.6) hold almost everywhere on Ω, with constants γi >0, i= 1,· · ·,5 and 0< δ <
p−1.
At the end of this section we introduce three lemmas which will be used in the sequel.
Lemma 1.1. Suppose that ω ∈ D′(D,Vℓ
) and dω ∈ Lp(D,Vℓ+1
), ℓ = 0,1,· · ·, n and1< p < n. Then ω−ωD is inLnp/(n−p)(D,Vℓ)and we have the
following uniform estimate
Z
Ω|ω−ωD|
np/(n−p)dx(n−p)/np ≤C(p, n)Z
D|dω|
pdx1/p
for D a cube or a ball in Rn.
The following lemma can be found in [2,6,7]. Lemma 1.2. Let ω ∈ D′(D,Vℓ
) be such that dω ∈ Lp(D,Vℓ+1
), 1 < p <
∞. Then ω−ωD is in W1,p(D,Vℓ) and
kω−ωDkW1,p(D) ≤C(n, p)kdωkp,D
for D a cube or a ball in Rn.
The following lemma comes from [8].
Lemma 1.3. Let BR ⊂ B2R ⊂⊂ Ω be concentric balls centered at x0. Suppose g(x) ∈ Lr(B
2R), 1 < r < ∞, f(x) ∈ Lt(B2R), t > r. If for every
x∈B2R, we have the estimate
−
Z
BR
|g(x)|rdx≤θ−Z B2R
|g(x)|rdx+CZ− B2R
|g(x)|sdx
r s
+−
Z
B2R
|f(x)|rdx,
where 1≤ s < r, 0≤ θ < 1 and −R
BRg(x)dx =
1 |BR|
R
BRg(x)dx stands for the
integral mean. Then there exists an exponent r′ = r′(θ, r, n, C) > r such that
g(x)∈Lr′
loc(Ω), and
−
Z
BR
|g(x)|r′
dx
1 r′
≤C1
(
−
Z
B2R
|g(x)|rdx
1
r
+
−
Z
B2R
|f(x)|r′
dx
1 r′
)
holds for some C1 =C1(n, C, r, θ, R0) independent of the cubeBR, whereR0 = dist(x0, ∂Ω).
In the sequel, we denote C(∗,· · ·,∗), Cj(∗,· · ·,∗) constants that depend
only on the variables ∗,· · ·,∗, whose value may change even on the same line.
2
Weak Reverse H¨
older Inequality for Weak
WT
2Class of Differential Forms
Theorem 2.1. Ifω ∈Lr loc
Ω,Vℓ
, max{1, p−1}< r < p, is of the class weak WT2, then there exists positive constants ε0 = ε0(n, p, γ1, γ2, γ3, γ4, γ5) and
Cj = Cj(n, p, γ1, γ2, γ3, γ4, γ5), j = 1,2, such that the following weak reserve H¨older inequality
−
Z
BR
|ω|rdx≤θ−Z B2R
|ω|rdx+C
1
−
Z
B2R
|ω|sdx
r s
holds with s < r and 0< θ <1, provided that|p−r|< ε0.
Proof Let BR ⊂ B2R ⊂⊂Ω be concentric balls and η(x) ∈C0∞(B2R) be
a cutoff function with 0 ≤ η ≤ 1, η ≡ 1 on BR and |∇η| ≤ CR(n). Since ω
is exact, then there exists a differential form u ∈ Wloc1,rΩ,Vℓ−1
, such that
ω =du. Consider the Hodge decomposition (see [6,7])
|d(ηu)|r−pd(ηu) =dα+h, (2.2)
where dα, h∈Lr/(r−p+1)
B2R,Vℓ
. The following estimate holds
khkr/(r−p+1)≤C(n)|p−r|kd(ηu)krr−p+1. (2.3)
Let
E =|d(ηu)|r−pd(ηu)− |ηdu|r−pηdu. (2.4) By an elementary inequality (see [9])
|X|
−εX− |Y|−εY
≤
2ε(1 +ε)
1−ε |X−Y|
1−ε, 0≤ε <1, X, Y ∈Rn,
we obtain
|E| ≤ 2
p−r(p−r+ 1)
r−p+ 1 |udη|
r−p+1. (2.5) By the weak closedness of θ, one has
Z
B2R
hdα,∗θidx=
Z
B2R
h∗dα,∗ ∗θidx
= (−1)ℓ(n−ℓ)
Z
B2R
h∗dα, θidx= (−1)ℓ−1
Z
B2R
hd∗(∗α), θidx= 0. (2.6)
for all α∈W1,r
B2R,Vℓ
. Combining (2.2) with (2.4) and (2.6) yields
Z
B2R
h|ηdu|r−pηdu,∗θidx=−
Z
B2R
hE,∗θidx+
Z
B2R
hh,∗θidx. (2.7) Now let B1 = {x ∈ B2R : |du| > 1} and B2 = {x ∈ B2R : |du| ≤ 1}. It is
obvious that B1TB2 =φ and B2R=B1SB2. (1.5) and (2.7) imply
Z
BR
|du|rdx≤
Z
B2R
ηr−p+1|du|rdx
=
Z
B1
ηr−p+1|du|rdx+
Z
B2
ηr−p+1|du|rdx
≤
Z
B1
ηr−p+1|du|r−p|du|pdx+|B
2R|
≤ γ1
Z
B1
ηr−p+1|du|r−phdu,∗θidx+γ2
Z
B1
|du|r−δdx+γ3
Z
B1
|du|r−pdx+|B2R|
≤ γ1
Z
B1
h|ηdu|r−pηdu,∗θidx+γ2
Z
B1
|du|r−δdx+ (γ3+ 1)|B2R|
≤ −γ1
Z
B1
hE,∗θidx+γ1
Z
B1
hh,∗θidx+γ2
Z
B1
|du|r−δdx+ (γ
3+ 1)|B2R|
:= γ1I1+γ1I2+γ2I3+ (γ3+ 1)|B2R|,
where we have used the facts |du| ≤1 on B2 and |du|>1 onB1, which imply
Z
B2
ηr−p+1|du|rdx≤ |B
2R| and
Z
B1
|du|r−pdx≤ |B
2R|,
respectively. We now estimate |Ij|,j = 1,2,3. With (1.6), (2.5) and H¨older’s
inequality, we obtain
|I1| =
− Z B1
hE,∗θidx
≤
Z
B2R
|E||θ|dx
≤ 2
p−r(p−r+ 1)
r−p+ 1
Z
B2R h
γ4|du|p−1+γ5
i
|udη|r−p+1dx
≤ C(n)
Rr−p+1 ·
2p−r(p−r+ 1)
r−p+ 1
γ4
Z
B2R
|du|p−1|u|r−p+1dx+γ5
Z
B2R
|u|r−p+1dx
.
(2.9) Notice that ω=du is not affected when an exact form is added tou, thus we may assume thatuB2R is equal to zero. This justifies the application of Lemma
1.1 to the first integral in the right-hand side of (2.9). Hence
C(n)γ4
Rr−p+1 ·
2p−r(p−r+ 1)
r−p+ 1
Z
B2R
|du|p−1|u|r−p+1dx
≤ C(n)γ4
Rr−p+1 ·
2p−r(p−r+ 1)
r−p+ 1
Z
B2R
|du|rdx
p−1
r Z
B2R
|u|rdx
r−p+1
r
≤ C(n)γ4
Rr−p+1 ·
2p−r(p−r+ 1)
r−p+ 1
Z
B2R
|du|rdx
p−r1 Z
B2R
|du|nnr+rdx
(n+r)(nrr−p+1)
≤ C(n, γ4)σ
Z
B2R
|du|rdx+C(n, γ4, σ)
Rr
Z
B2R
|du|nnr+rdx n+r
n
,
(2.10) where σ >0 will be chosen later.
Take 1< p′ <∞ such that n(r−p+1)p′
n+(r−p+1)p′ < r, then using H¨older’s inequality and Lemma 1.1 again, we arrive at
Z
B2R
|u|r−p+1dx≤
Z
B2R
|u|(r−p+1)p′dx
1 p′ Z
B2R
dx
p′ −1 p′
≤ C(n, r)|B2R|
p′−1
p′
Z
B2R
|du|
n(r−p+1)p′
n+(r−p+1)p′dx
!n+(r−p+1)p
′
np′ (2.11)
With (1.6), H¨older’s inequality and (2.3), we obtain
|I2| =
Z B1
hh,∗θidx
≤
Z
B2R
|h||θ|dx≤
Z
B2R h
γ4|du|p−1+γ5
i
|h|dx
≤ γ4|du|
p−1+γ 5
r
p−1
khk r r−p+1
≤C(n)|p−r| γ4|du|
p−1+γ 5
r
p−1
kd(ηu)kr−p+1
With Lemma 1.2, we obtain
||d(ηu)||rr−p+1=||ηdu+udη||rr−p+1 ≤(||ηdu||r+||udη||r)r−p+1
≤ C(n)
R ||u||r+||du||r
!r−p+1
≤ C(n)
R R||du||r+||du||r
!r−p+1
=C(n, p)||du||r−p+1
r .
Therefore
|I2| ≤ C(n, p)|p−r|
γ4|du|
p−1+γ 5
r
p−1
kdukrr−p+1 ≤ C(n, p, σ)|p−r|
γ4|du|
p−1+γ 5
r p−1
r p−1
+σkdukrr ≤ C(n, p, σ)|p−r|γ4
Z
B2R
|du|rdx+C(n, p, σ)|p−r|γ5|B2R|.
(2.12)
where we have used Young’s inequality.
We can assume that p−δ < r < p since Theorem 2.1 holds with |p−r|
sufficiently small. Using H¨older’s inequality and Young’s inequality, |I3| can be estimated as
|I3| =
Z B1
|du|r−δdx
≤
Z
B2R
|du|rdx
p−δ
r Z
B2R
dx
r−p+δ r
≤ ε
Z
B2R
|du|rdx+C(ε)|B
2R|.
(2.13)
Combining (2.8)-(2.13) and dividing both sides by |BR|=ωnRn, we arrive
at
−
Z
BR
|du|rdx ≤ [C(n, γ1, γ4)σ+C(n, p, σ)|p−r|γ4+σ+ε]−
Z
B2R
|du|rdx
+C(n, γ1, γ4, σ)
−
Z
B2R
|du|nnr+rdx n+nr
+C(n, γ1, γ5)
2p−r(p−r+ 1)
r−p+ 1 −
Z
B2R
|du| n(r−p+1)p ′
n+(r−p+1)p′
!n+(r−p+1)p
′
np′
+C(n, p, σ)|p−r|γ5+γ3+ 1 +C(ε).
Take σandε sufficiently small, and thenrsufficiently close top, such thatθ=
C(n, γ1, γ4)σ+C(n, p, σ)|p−r|γ4+σ+ε <1. Lets= max
n
nr n+r,
n(r−p+1)p′
n+(r−p+1)p′
o
< r, we arrive at
−
Z
BR
|du|rdx≤θ−
Z
B2R
|du|rdx+C1
−
Z
B2R
|du|sdx
rs
+C2,
where we have used the fact that t 7→ −R
B2R|du|
tdx1t
3
Some Applications
In this section, we give some applications of the reverse H¨older inequality for weakWT2 classes of differential forms to regularity theory of weakly (K1, K 2)-quasiregular mappings, very weak solutions of nonhomogeneous A-harmonic equations, and generalized solutions of Beltrami system with three character-istic matrices.
3.1
Weakly
(K
1, K
2)
-Quasiregular Mappings
We recall the definition for weakly (K1, K2)-quasiregular mappings, see [7]. Let f = (f1, f2,· · ·, fn) ∈ W1,r
loc(Ω,Rn), 1 ≤ r < ∞. Then f is said to be
weakly (K1, K2)-quasiregular, K1 >0,K2 ≥0, if
|Df(x)|n ≤K1nn/2Jf(x) +K2 (3.1) is satisfied almost everywhere in Ω, hereDf(x) denotes the formal derivative of
f atx, i.e., then×nmatrix∂x∂fi
j
1≤i,j≤nof partial derivatives of the coordinate
functions fi of f. Further, |Df(x)| is the Hilbert-Schmidt norm of Df(x):
|Df(x)|= (TrDtf(x)Df(x))1/2 =
n
X
i,j=1
∂fi
∂xj
!2
1/2
.
Jf(x) = detDf(x) is the Jacobian determinant of f atx.
Quasiregular mappings were first studied by Reshetnyak in 1966-1969. He proved that spatial quasiregular mappings (he used the phrase mappings with bounded distortion for quasiregular mappings) share the fundamental topolog-ical properties of complex-analytic functions: non-constant quasiregular map-pings are discrete, open and sense-preserving. See [10]. The word quasiregular was introduced in this meaning in 1969 by Martio et al. [11], which found an-other approach, i.e., modulus of a family of curves (the extremal length) and capacities of condensers, to quasiregular mappings [12,13]. Their work showed the power of the method of the modulus of a family of curves, made these mappings more widely known, and led to a series of important results. Some notable developments in the theory of spatial weakly quasiregular mappings, including the regularity theory, were obtained by Iwaniec and Martin [14,15], by the technique developed by Donalson and Sullivan [16]. For other develop-ments related to quasiregular mappings and (K1, K2)-quasiregular mappings, see [6,7,17-21].
Theorem 3.1. Letf = (f1, f2,· · ·, fn)∈W1,r
loc(Ω,Rn)be a weakly(K1, K2) -quasiregular mapping. Then for any ℓ-tuple I = (i1,· · ·, iℓ), 1 ≤ i1 < · · · <
iℓ ≤n, the differential formω =dfI =dfi1 ∧ · · ·dfiℓ is of the class weak WT2 with p= n
ℓ.
Proof It is easy to see that
ω =dfi1 ∧ · · · ∧dfiℓ =d(fi1dfi2 ∧ · · · ∧dfiℓ)
is exact. Let
θ= (−1)ℓ(n−ℓ)σ(I, J)dfJ = (−1)ℓ(n−ℓ)σ(I, J)dfj1∧ · · · ∧dfjn−ℓ,
where the complementary (n−ℓ)-tupleJ = (j1,· · ·, jn−ℓ), 1≤j1 <· · ·< jℓ ≤
n, is obtained from the index (1,2,· · ·, n) by simply deleting the elements of
I, andσ(I, J) =±1 is defined by
σ(I, J)dfI ∧dfJ =df1∧ · · · ∧dfn=Jf(x)dx.
It is obvious that θ is closed, thus it is weakly closed.
Because the differential form ω is simple we obtain by the inequality be-tween the geometric and arithmetic means
|ω|1/ℓ ≤
ℓ
Y
m=1
|dfim| !1/ℓ
≤ 1
ℓ
ℓ
X
m=1
|dfim| ≤ 1
ℓ
k
X
m=1
|dfim|2 !1/2
. (3.2)
Similarly,
|θ|1/(n−ℓ) ≤ 1
n−ℓ
n−ℓ
X
m=1
|dfim|2 !1/2
. (3.3)
Combining (3.2) and (3.3), and using (3.1) we have
ℓ|ω|2/ℓ+ (n−ℓ)|θ|2/(n−ℓ)n/2 =|Df(x)|n≤K1nn/2Jf(x) +K2. (3.4) It is obvious that
hω,∗θidx=ω∧ ∗ ∗θ =σ(I, J)dfi1∧ · · ·dfiℓ∧dfj1∧ · · ·dfjn−ℓ =J
f(x)dx. (3.5)
(3.4) and (3.5) imply
ℓn/2|ω|n/ℓ≤
ℓ|ω|2/ℓ+ (n−ℓ)|θ|2/(n−ℓ)n/2
≤K1nn/2hω,∗θi+K2. Thus
|ω|n/ℓ ≤K1
n
ℓ
n/2
hω,∗θi+ K2
ℓn/2 ≤K1
n
ℓ
n/2
hω,∗θi+ K2
Since bothω and θ are simple, then by (3.4), (3.5) and Young’s inequality,
(n−ℓ)n/2|θ|n/(n−ℓ)≤ℓ|ω|2/ℓ+ (n−ℓ)|θ|2/(n−ℓ)n/2 ≤K1nn/2|ω||θ|+K2 ≤K1nn/2
C(ε)|ω|n/ℓ+ε|θ|n/(n−ℓ)
+K2
Take εsmall enough such thatK1nn/2ε= (n−ℓ)n/2/2, and divided (n−ℓ)n/2/2 in both sides of the above inequality yields
|θ| ≤γ4|ω|(n−ℓ)/ℓ+γ5. This ends the proof of Theorem 3.1.
Corollary 3.1. Let f = (f1, f2,· · ·, fn) ∈ W1,r
loc(Ω,Rn), max{1, p −
1} ≤ r < p, be a weakly (K1, K2)-quasiregular mapping, there exists p1 =
p1(n, K1, K2) and p2 = p2(n, K1, K2), p1 < n < p2, such that every weakly (K1, K2)-quasiregular mappingf ∈Wloc1,p1(Ω,Rn)actually belongs toW
1,p2
loc (Ω,Rn),
that is, f is a (K1, K2)-quasiregular mapping in the usual sense.
Proof. This corollary is a direct consequence of Theorem 2.1, Theorem 3.1 and Lemma 1.3.
3.2
Weakly
A
-Harmonic Tensors
Let us now consider the nonhomogeneous A-harmonic equation
d∗A(x, du(x)) =d∗B(x, du), (3.6) where A,B: Ω×Vℓ
(Rn)→Vℓ
(Rn) satisfy the conditions (i)|A(x, ξ)| ≤a|ξ|p−1,
(ii) hA(x, ξ), ξi ≥ |ξ|p,
(iii) |B(x, ξ)| ≤a|ξ|p−δ
for almost every x∈ Ω and allξ ∈Vℓ
(Rn). Here p= nℓ, a ≥1, 0< δ < p−1 are constants, and 1< p <∞ is a fixed exponent associated with (3.6).
Definition 3.2. A very weak solution to (3.6) is an element u of the Sobolev space Wloc1,r(Ω,Vℓ−1
), max{1, p−1} ≤r < p, such that
Z
ΩhA(x, du), dϕidx=
Z
ΩhB(x, du), dϕidx for all ϕ∈W1,r−rp+1
Ω,Vℓ−1
with compact support.
Theorem 3.2. If u ∈ Wloc1,r(Ω,Vℓ−1
), max{1, p−1} ≤ r < p, be a very weak solution to (3.6) with conditions (i), (ii) and (iii), then ω =du is of the class weak WT2.
Proof. It is obvious that the differential form ω = du ∈ Lr loc
Ω,Vℓ
is exact. Letθ =∗−1[A(x, du)−B(x, du)]∈Lp−r1
loc
Ω,Vn−ℓ
. The weak closedness of θ follows from
(−1)n+nℓ+1
Z
Ωhθ, d
∗ψidx=Z
Ωhθ,∗d∗ψidx =
Z
Ωh∗θ, d∗ψidx=
Z
ΩhA(x, du)− B(x, du), d∗ϕidx= 0 for all ψ =∗−1ϕ ∈W1, r
r−p+1(Ω,Vn−ℓ+1
) with suppϕ∩∂Ω =∅. By (i), (iii) and Young’s inequality,
|θ| =|A(x, du)− B(x, du)| ≤ |A(x, du)|+|B(x, du)| ≤a|du|p−1+a|du|p−1−δ ≤a
|du|p−1+ε|du|p−1+C(ε)
=a(1 +ε)|du|p−1+aC(ε).
(3.7)
Further, by (i), (ii) and (iii) we get
hω,∗θi=hdu,A(x, du)− B(x, du)i
= hdu,A(x, du)i − hdu,B(x, du)i ≥ |du|p−a|du|p−δ.
This implies
|du|p ≤ hdu,∗θi+a|du|p−δ,
completing the proof of Theorem 3.2. Corollary 3.2. If u ∈ Wloc1,r(Ω,Vℓ−1
), max{1, p−1} ≤ r < p, be a very weak solution to (3.6) with conditions (i), (ii) and (iii), then there exists p1 =
p1(n, a) and p2 = p2(n, K1, K2), p1 < p < p2, such that every very weak solution u ∈ W1,p1
loc (Ω,
Vℓ−1
) to (3.6) actually belongs to W1,p2
loc (Ω,
Vℓ−1
), that is, u is a weak solution to (3.6) in the usual sense.
Proof. This corollary is a direct consequence of Theorem 2.1, Theorem 3.2 and Lemma 1.3.
3.3
Generalized Solutions to Beltrami System with Three
Characteristic matrices
We now consider the Beltrami system with three characteristic matrices
(I) α1|ξ|2≤ hH(x)ξ, ξi ≤ β1|ξ|2, (II) α2|η|2 ≤ hG(x)η, ηi ≤β2|η|2, (III)α3|ζ|2 ≤ hH(x)ζ, ζi ≤ β3|ζ|2
for all ξ, η, ζ ∈Rn and some positive constants αi, βi, i= 1,2,3.
The Beltrami system with one, two or three characteristic matrices have been studied by many authors, see [6,7,22,23]. In this subsection, we will give a regularity result for generalized solutions to (3.8).
Theorem 3.3. Let f = (f1, f2,· · ·, fn) ∈ W1,r
loc(Ω,Rn) be a generalized
solution of the Beltrami system (3.8), then ω = dfI =dfi1 ∧dfi2 ∧ · · · ∧dfiℓ,
1≤ℓ≤ n is of the class weak WT2.
Proof. We prove Theorem 3.3 by showing that any generalized solution
f = (f1, f2,· · ·, fn)∈W1,r loc (Ω,R
n
) of (3.8) be a weakly (K1, K2)-quasiregular mapping, thus Theorem 3.3 follows from Theorem 3.1. In fact, (I), (II), (III) and (3.8) imply
α1|Df|2≤ hH(x)Df(x), Df(x)i=hDtf(x)H(x)Df(x), Idi = Jf2/n(x)hG(x), Idi+hF(x), Idi ≤β2Jf2/n(x) +β3.
Therefore
|Df(x)|n≤2(n−2)/2
β2
α1
!n/2
Jf(x) +
β3
α1
!n/2
.
Corollary 3.3. Let f = (f1, f2,· · ·, fn) ∈ W1,r
loc(Ω,Rn) be a generalized
solution of the Beltrami system (3.8), then there exists p1, p2, p1 < n < p2, such that every generalized solution f ∈W1,p1
loc (Ω,Rn)to (3.8) actually belongs
to W1,p2
loc (Ω,Rn).
Proof. This corollary is a direct consequence of Theorem 2.1, Theorem 3.3 and Lemma 1.3.
ACKNOWLEDGEMENT. The first author was supported by NSF of Hebei Province (A2015201149).
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