Global integrability of the Jacobian of a composite mapping

GLOBAL INTEGRABILITY OF THE JACOBIAN OF A

COMPOSITE MAPPING

SHUSEN DING AND BING LIU

Received 18 September 2005; Accepted 24 October 2005

We first obtain an improved version of the H¨older inequality with Orlicz norms. Then, as an application of the new version of the H¨older inequality, we study the integrability of the Jacobian of a composite mapping. Finally, we prove a norm comparison theorem.

Copyright © 2006 S. Ding and B. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Carl Gustav Jacob Jacobi (1804–1851), one of the nineteenth century Germany’s most accomplished scientists, developed the theory of determinants and transformations into a powerful tool for evaluating multiple integrals and solving differential equations. Since then, the Jacobian (determinant) has played a critical role in multidimensional analysis and related fields, including nonlinear elasticity, weakly differentiable mappings, con-tinuum mechanics, nonlinear PDEs, and calculus of variations. The integrability of Ja-cobians has become a rather important topic in the study of JaJa-cobians because one of the major applications of Jacobians is to evaluate multiple integrals. Higher integrabil-ity properties of the Jacobian first showed up in [2], where Gehring invented reverse H¨older inequalities and used these inequalities to establish theL1+ε_{-integrability of the} Jacobian of a quasiconformal mapping,ε >0. Recently, the integrability of Jacobians of orientation-preserving mappings of Sobolev classW_loc1,n(Ω,Rn_{) has attracted the} atten-tion of mathematicians, see [1,3–7], for instance. The purpose of this paper is to study theLp_(logL)α_{(Ω)-integrability of the Jacobian of a composite mapping.}

Let 0< p <∞andα≥0 be real numbers and letEbe any subset ofRn_{. We define the} functional on a measurable function f overEby

[f]Lp_(logL)α(E)=

E|f| p_logα

e+ |f|

fp

dx

1/ p

, (1.1)

wherefp=(

E|f(x)|pdx)1/ p. In this paper, we always assume thatΩis a bounded open subset ofRn_{,n≥2. We writeL}p_(logL)α_{(Ω) for the space of all measurable functions}

f onΩsuch that [f]Lp_(logL)α_(Ω)<∞. As usual, we simply writeLp(Ω)=L1(logL)0(Ω) and

LlogL(Ω)=L1_(logL)1_{(Ω), respectively.}

A continuously increasing functionϕ: [0,∞]→[0,∞] withϕ(0)=0 andϕ(∞)= ∞is called an Orlicz function. The Orlicz spaceLϕ_{(Ω) consists of all measurable functions f} onΩsuch that

Ωϕ

_|f|

λ

dx <∞ (1.2)

for someλ=λ(f)>0.Lϕ_{(Ω) is equipped with the nonlinear Luxemburg functional}

fϕ=inf

λ >0 :

Ωϕ

_|f|

λ

dx≤1 . (1.3)

A convex Orlicz functionϕis often called a Young function. Ifϕis a Young function, then · ϕ defines a norm inLϕ(Ω), which is called the Luxemburg norm. Forϕ(t)=

tp_logα_{(e+t), 0< p <∞andα≥0, we have}

fLp_logα_L=inf

k:

Ω|f|

p_logα_e+|f|

k

dx≤kp . (1.4)

FromTheorem 4.2 that will be proved later in this paper, we see that the Luxemburg normfϕis equivalent to [f]Lp_(logL)α(Ω) defined in (1.1) for any 0< p <∞andα≥0. Hence, the Orlicz spaceLψ_{(Ω) withψ(t)=t}p_logα_{(e+t) can be denoted byL}p_(logL)α_(Ω) and the corresponding norm can also be written as [f]Lp_(logL)α_(Ω). The following version

of H¨older inequality appears in [3, Proposition 2.2].

Theorem 1.1. Let 1< p, q <∞, α,β >0, 1/ p+ 1/q=1/r, α/ p+β/q=γ/r and f ∈

Lp_(logL)α_(Ω),g∈Lq_(logL)β_{(Ω). Then fg∈L}r_(logL)γ_(Ω)and

f gLr_logγ_L≤Cf_Lp_logα_Lg

Lq_logβ_L. (1.5)

In this paper, we improve the condition 1< p,q <∞into 0< p,q <∞inTheorem 2.1. We enjoy the elementary method used in the proof ofTheorem 2.1. Then, using the im-proved H¨older inequality, we study theLp_(logL)α_{(Ω)-integrability of the Jacobian of the} composition of mappings.

2. Improved H¨older inequality

Theorem2.1. Let m,n,α,β >0, 1/s=1/m+ 1/n,α/m+β/n=γ/s. Assume that f ∈Lm (logL)α_(Ω)andg∈Ln_(logL)β_{(Ω). Then, fg∈L}s_(logL)γ_(Ω)and

Ω|f g| s_logγ

e+ |f g|

f gs

dx

1/s

≤C

Ω|f| m_logα

e+ |f|

fm

dx

1/m

Ω|g| n_logβ

e+ |g|

gn

dx

1/n ,

(2.1)

whereCis a positive constant. Note that (2.1) can be written as

[f g]Ls_(logL)γ_(Ω)≤C[f]_Lm_(logL)α_(Ω)[g]_Ln_(logL)β_(Ω). (2.2)

Proof. Using the elementary inequality log(e+xa_)≤log(e+x)a+1_{fora >0,x >0, we have}

log

e+

|f|s_|g|s1/s

_|f|s_|g|s1/s 1

≤log

e+ |f| s_|g|s

_|f|s_|g|s 1

1/s+1

=

₁

s + 1

log

e+ |f| s_|g|s

_|f|s_|g|s 1 , log e+

|f|

fm

s

≤log

e+ |f|

fm

s+1

≤(s+ 1) log

e+ |f|

fm

, log e+

|g| gn

s

≤log

e+ |g|

gn

s+1

≤(s+ 1) log

e+ |g|

gn

.

(2.3)

From H¨older inequality (1.5) with 1=1/m/s+ 1/n/s (note thatm/s >1,n/s >1 since 1/s=1/m+ 1/n) and (2.3), we have

Ω|f g| s_logγ

e+ |f g|

f gs

dx = Ω

|f|s_|g|s_logγ

e+ f|

s_|g|s_|1/s

_|f|s_|g|s1/s 1

dx

≤C1

Ω

|f|s_|g|s_logγ_e+ |f|s|g|s

_|f|s_|g|s 1

dx

≤C2

Ω

|f|sm/s_logα

e+ |f| s

_|f|s m/s dx s/m Ω

|g|sn/s_logβ

e+ |g|

s

_|g|s n/s

dx

s/n

=C2

Ω|f| m_logα

e+

|f|

fm

s

dx s/m

Ω|g| n_logβ

e+

|g| gn

s

dx s/n

≤C3

Ω|f| m_logα

e+ |f|

fm

dx

s/m

Ω|g| n_logβ

e+ |g|

gn

dx

s/n

.

Hence, we conclude that

Ω|f g| s_logγ

e+ |f g|

f gs

dx

1/s

≤C4

Ω|f| m_logα

e+ |f|

fm

dx

1/m

Ω|g| n_logβ

e+ |g|

gn

dx

1/n

.

(2.5)

The proof ofTheorem 2.1has been completed.

FromTheorem 2.1, we have the following general result immediately.

Corrollary2.2. Let pi>0,αi>0fori=1, 2,. . .,k,1/ p1+ 1/ p2+···+1/ pk=1/ p, and

α1/ p1+α2/ p2+···+αk/ pk=α/ p. Assume that fi∈Lpi(logL)αi(Ω)fori=1, 2,. . .,k. Then

f1f2···fk∈Lp(logL)α(Ω)and

f1f2···fk

Lp_(logL)α_(Ω)≤C

f1

Lp1(logL)α1(Ω)

f2

Lp2(logL)α2(Ω)···

fk

Lpk(logL)αk(Ω), (2.6)

whereC is a positive constant and the norms[f1f2···fk]Lp_(logL)α_(Ω) and[fi]Lpi(logL)αi(Ω),

i=1, 2,. . .,k, are defined in (1.1).

3. Integrability of Jacobians of composite mappings

In this section, we explore applications of the new version of the H¨older inequality estab-lished in the last section. Specifically, we study the integrability of the Jacobian of the com-position of mappings f :Ω→Rn_{, f =(f}1_(u

1,u2,. . .,un),f2(u1,u2,. . .,un),. . .,fn(u1,u2,

. . .,un)) of Sobolev classW_loc1,p(Ω,Rn_{), whereu}

i=ui(x1,x2,. . .,xn),i=1, 2,. . .,n, are func-tions ofx=(x1,x2,. . .,xn)∈Ωwith continuous partial derivatives∂ui/∂xj,j=1, 2,. . .,n. Assume that the distributional differentialD f(u)=[∂ fi_{/∂uj] andDu(x)=[∂u}

i/∂xj] are locally integrable functions with values in the space GL(n) of alln×n-matrices. As usual, we write

J(x,f)=detD f(u(x))=∂

f1_···fn

∂x1···xn, (3.1)

J(u,f)=detD f(u)=∂

f1_···fn

∂u1···un

, (3.2)

J(x,u)=detDu(x)=∂

u1···un

∂x1···xn

, (3.3)

respectively. UsingTheorem 2.1, we have the following integrability theorem for the Ja-cobian of the composition of mappings.

Theorem 3.1. Let s,t,β,γ >0, with 1/ p=1/s+ 1/t and β/s+γ/t=α/ p. Assume that

J(u(x),f)∈Ls_(logL)β_{(Ω)andJ(x,u)∈L}t_(logL)γ_{(Ω), thenJ(x,f)∈L}p_(logL)α_(Ω)and

Ω

_J(x,f)p logα

e+J(x,f)

J(x,f)_p

dx

1/ p

≤C

Ω

_J(u,f)s logβ

e+J(u,f)

J(u,f)_s

dx

1/s

×

Ω

_J(x,u)t logγ

e+J(x,u)

J(x,u)_t

dx

1/t ,

(3.4)

whereCis a positive constant.

Proof. Note that the Jacobian of the composition of f anducan be expressed as

J(x,f)=∂

f1_···fn

∂x1···xn

=∂(f1···fn)

∂u1···un

·∂

u1···un

∂x1···xn

=J(u,f)·J(x,u). (3.5)

ApplyingTheorem 2.1and (3.5) yields

Ω

_J(x,f)p logα

e+J(x,f)

J(x,f)_p

dx

1/ p

=

Ω

_{J(u,f)·J(x,u)}p logα

e+J(u,f)·J(x,u)

J(u,f)·J(x,u)_p

dx

1/ p

≤C

Ω

_J(u,f)s logβ

e+J(u,f)

J(u,f)_s

dx

1/s

×

Ω

_J(x,u)t logγ

e+J(x,u)

J(x,u)_t

dx

1/t

<∞

(3.6)

sinceJ(u(x),f)∈Ls_(logL)β_{(Ω) andJ(x,u)∈L}t_(logL)γ_{(Ω). Thus,J(x,f)∈L}p_(logL)α_(Ω)

from (3.6). The proof ofTheorem 3.1has been completed.

Applying the H¨older inequality withLp_-norms

f gs,E≤ fα,E· gβ,E, (3.7)

where 0< α,β <∞,s−1_=α−1_+β−1_{, and f andg are any measurable functions on a} measurable setE⊂Rn_{, we have the followingL}p_{-integrability theorem for the Jacobian} of a composite mapping.

Theorem3.2. LetJ(x,f),J(u,f), andJ(x,u)be the Jacobians defined in (3.1), (3.2), and (3.3), respectively. IfJ(u(x),f)∈Ls_{(Ω)andJ(x,u)∈L}t_{(Ω),s,t >0, thenJ(x,f)∈L}p_(Ω) and

where Cis a positive constant and the integrability exponent p ofJ(x,f)determined by 1/ p=1/s+ 1/tis the best possible.

The following example shows that the integrability exponent pof J(x,f) cannot be improved anymore.

Example 3.3. We consider the mappings

f(x,y)=f1,f2=

x

x2_+y2σ,

y

x2_+y2σ

, (x,y)∈D=(x,y) : 0< x2+y2≤ρ2,

x=r−kcosθ, y=r−ksinθ, (r,θ)∈Ω= {(r,θ) : 0< r < ρ, 0< θ≤2π}, (3.9)

whereσandρare positive constants. After a simple calculation, we obtain the following Jacobians:

J1=∂(f 1_,f2₎

∂(r,θ) =

k(2σ−1)

r4σ+2k+1 , J2=

∂(f1_,f2₎

∂(x,y) = 1−2σ

r4σ ,

J3=

∂(x,y)

∂(r,θ)=

−k

r2k+1, 0< r < ρ.

(3.10)

It is easy to see thatJ1∈L1/(4σ+2k+1)(Ω) butJ1 ∈Lp(Ω) for anyp >1/(4σ+ 2k+ 1). Sim-ilarly,J2∈L1/4σ(Ω) butJ2 ∈Ls(Ω) for anys >1/4σ andJ3∈L1/(2k+1)(Ω) butJ3 ∈Lt(Ω) for anyt >1/(2k+ 1). Here, the integrability exponent p=1/(4σ+ 2k+ 1) of∂(f1_,f2_)/∂ (r,θ) is determined by

1

p =(4σ+ 2k+ 1)=

1

s +

1

t, (3.11)

where s =1/4σ and t =1/(2k+ 1) are the integrability exponents of Jacobians

∂(f1_,f2_{)/∂(x,y) and∂(x,y)/∂(r,θ), respectively.}

The above example shows that, inTheorem 3.2, the integrability exponentpofJ(x,f) that is determined by 1/ p=1/s+ 1/tis the best possible, wheresis the integrability expo-nent ofJ(u(x),f) andtis the integrability exponent ofJ(x,u).

Example 3.4. LetJ1=∂(f1,f2)/∂(r,θ),J2=∂(f1,f2)/∂(x,y), andJ3=∂(x,y)/∂(r,θ) be the Jacobians obtained inExample 3.3. For anyε >0, there exists a constantC1>0 such that

_J1p log

e+J1

J1_p

Using (3.10) and (3.12), we have

Ω

_J1p log

e+J1

J1_p

drdθ

=2π ρ

0

_J1p log

e+J1

J1_p

dr

=C2

ρ

0

₁

r4σ+2k+1

p log

e+ (2σ−1)k/r 4σ+2k+1

_(2σ−1)k/r4σ+2k+1 p

dr

≤C3

ρ

0r

−(4σ+2k+1)p_r−(4σ+2k+1)ε/4σ+2k+1_dr≤C 4

ρ

0 r

−(4σ+2k+1)p−ε_dr=C 5<∞

(3.13)

for anypsatisfying 0< p≤1/(4σ+ 2k+ 1)−ε/(4σ+ 2k+ 1). Sinceε >0 is arbitrary, we know thatJ1∈LplogL(Ω) for anypwith 0< p <1/(4σ+ 2k+ 1). Similarly, we haveJ2∈

Ls_{logL(Ω) for anyswith 0< s <1/4σandJ}

3∈LtlogL(Ω) for anytwith 0< t <1/(2k+ 1). This example shows that the integrability exponent p of∂(f1_,f2_{)/∂(r,θ) that is} deter-mined by 1/ p=1/s+ 1/tis the best possible whenα=β=γ=1 inTheorem 3.1.

4. The norm comparison theorem

In this section, we discuss the relationship between normsfLp_logα_Land [f]Lp_(logL)α_(Ω),

which will provide a different way to prove Theorems 2.1and 3.1. First, we recall the following more general inequality appearing in [3, Theorem A.1].

Theorem4.1. Suppose thatA,B,C: [0,∞)→[0,∞)are continuous, monotone increasing functions for which there exist positive constantscanddsuch that

(i)B−1(t)C−1(t)≤cA−1(t)for allt >0, (ii)A(t/d)≤1/2A(t)for allt >0.

Suppose thatGis an open subset ofRn_{, for f ∈LB(G)andg∈LC(G). Thenf g∈LA(G)} and

f gA≤cdfBGC. (4.1)

In [6], Iwaniec and Verde prove that the normfLp_logα_L is equivalent to the norm

[f]Lp_(logL)α_(Ω)for 1< p <∞. Similar to the proof of [6, Lemma 8.6], we have the

relation-ship between the normfLp_logα_Land the norm [f]Lp_(logL)α(Ω).

Theorem4.2. For each f ∈Lp_(logL)α_{(Ω),0< p <∞andα≥0,}

fp≤ fLp_logα_L≤[f]Lp_(logL)α(Ω)≤CfLp_logα_L, (4.2)

Proof. LetK= fLp_logα_L. Then, by the definition of the Luxemburg norm, we have

K=

Ω|f|

p_logα_e+|f|

K

dx 1/ p

. (4.3)

It is clear thatK≥ fpand

K≤

Ω|f| p_logα

e+ |f|

fp

dx

1/ p

=[f]Lp_(logL)α_(Ω), (4.4)

that is,

fLp_logα_L≤[f]Lp_(logL)α(Ω). (4.5)

On the other hand, usingK≥ fp and the elementary inequality|a+b|s≤2s(|a|s+

|b|s_{),s≥0, we obtain that}

Ω|f| p_logα

e+ |f|

fp

dx=

Ω|f| p_logα

e+|f|

K ·

K

fp

dx

≤

Ω|f| p

log

e+|f|

K

+ log

K

fp

α

dx

≤2α

Ω|f|

p_logα_e+|f|

K

+ 2α

Ω|f| p_logα

K

fp

dx

=2αKp_{+ 2}α_fp plogα

K

fp

.

(4.6)

Note that the functionh(t)=tp_logα_{(K/t), 0< t≤K, has its maximum value (α/ep)}α_Kp att=K/eα/ p_{. Then}

fpplogα

K

fp

≤

α ep

α

Kp_{. (4.7)}

Combining (4.6) and (4.7) gives

Ω|f| p_logα

e+ |f|

fp

dx≤2α

1 +

_α

ep α

Kp, (4.8)

which is equivalent to

[f]Lp_(logL)α_(Ω)≤Cf_Lp_logα_L, (4.9)

whereC=2α/ p_{(1 + (α/ep)}α₎1/ p_{. The proof ofTheorem 4.2has been completed.}

It is easy to see thatTheorem 4.2indicates that, for any 0< p <∞andα≥0, the Lux-emburg normfLp_logα_Lis equivalent to the norm [f]_Lp_(logL)α_(Ω)defined in (1.1). Hence,

References

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[2] F. W. Gehring,TheLp_{-integrability of the partial derivatives of a quasiconformal mapping, Acta} Mathematica130(1973), 265–277.

[3] J. Hogan, C. Li, A. McIntosh, and K. Zhang,Global higher integrability of Jacobians on bounded domains, Annales de l’Institut Henri Poincar´e. Analyse Non Lin´eaire17(2000), no. 2, 193–217. [4] T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses,

Archive for Rational Mechanics and Analysis119(1992), no. 2, 129–143.

[5] ,Weak minima of variational integrals, Journal f¨ur die reine und angewandte Mathematik 454(1994), 143–161.

[6] T. Iwaniec and A. Verde,On the operatorᏸ(f)=flog|f|, Journal of Functional Analysis169 (1999), no. 2, 391–420.

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und angewandte Mathematik412(1990), 20–34.

Shusen Ding: Department of Mathematics, Seattle University, Seattle, WA 98122, USA

E-mail address:[email protected]

Bing Liu: Department of Mathematical Sciences, Saginaw Valley State University, University Center, MI 48710, USA