SOME REMARKS ON SOBOLEV IMBEDDING THEOREM
IN BORDERLINE CASES
by
Nicola Fusco
(1)
Pierre Louis Lions
Carlo Sbordone
(2)
(1)
Abstract: An imbedding theorem is given for functions whose gradient belongs to a class
slightly larger than Ln (Ω), Ω ⊂ IRn .
1.Introduction
We wish here to come back on the well known imbedding theorem for W 1,n (Ω) functions due to N.Trudinger [T], J.Moser [Mo].
One way to shed some light on the phenomena involved consists in extending this result
to other spaces closely related to W 1,n , like in [ALT].
We
here spaces of functions that are larger than W 1,n , but are contained in
T consider
1,p
. More precisely, we consider functions u whose gradient Du satisfies for
1<p<n W
some σ > 0
Z
(1)
|Du|n log −σ (e + |Du|)dx < ∞ ,
Ω
(1)
Dipartimento di Matematica e Applicazioni, Università di Napoli, Monte S.Angelo,
via Cintia, 80126 Napoli, Italy
(2)
CEREMADE, Place du Marèchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
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a class of functions motivated by recent work on the regularity properties of Jacobians (see
[BFS], [CLMS], [MU], [IS], [Gr]).
In this paper we prove, in particular, that, if u ∈ W01,1 and (1) holds, then
Z
(2)
exp
|u(x)|α λ
Ω
dx < ∞
n
.
for any λ > 0, where α = n−1+σ
This result relies on the simple observation that condition (2) is equivalent to saying that
Z
1/p
1
lim
− |u|αp dx
=0
p→∞ p
Ω
R
R
1
).
(where −Ω stands for |Ω|
Ω
More generally, our results apply to functions such that
Z
sup ε − |Du|n−ε dx < ∞
σ
ε>0
Ω
(see theorem 2).
2. Some preliminary results
Let Ω be an open set in IRn , with finite measure |Ω|. We denote with EXP = EXP (Ω)
the set of functions g : Ω → IR such that there exists λ > 0 for which
Z
|g| dx < ∞ .
− exp
λ
Ω
One way to test if a function f belongs to EXP is given by the following proposition,
whose proof is an easy exercise (see [G] ch VI, ex 17).
Proposition 1: Let g : Ω → IR be a measurable function. Set
Z
1/p
1
E(g) = e lim sup
− |g|p dx
p→∞ p
Ω
.
Then
Z
|g| dx < ∞}
E(g) = inf{λ > 0 : − exp
λ
Ω
2
.
Remark: This proposition says that g ∈ EXP if and only if E(g) < ∞. In particular, if
f ∈ L∞ (Ω) we have E(f ) = 0 and E(g − f ) = E(g) for any g ∈ EXP .
We recall that EXP is a Banach space under the norm (see [RR])
kgkEXP
Z
|g| dx < 2} .
= inf{λ > 0 : − exp
λ
Ω
We also remark that L∞ is not a dense subspace of EXP . Indeed one can easily prove
the following
Proposition 2: E(g) = 0 if and only if there exists a sequence fh ∈ L∞ such that
kfh − gkEXP → 0.
Proof: If fh verifies kfh − gkEXP → 0, then by the previous remark:
E(g) = E(fh − g) ≤ kfh − gkEXP
.
Conversely, assume that E(g) = 0, and let ε > 0. By definition, we have
Z
|g| dx < ∞ ,
− exp
ε
Ω
then there exists hε such that, for h > hε ,
1
|Ω|
Z
exp
|g| {|g|>h}
If we set
(
ε
dx ≤ 1 .
g(x)
if |g(x)| ≤ h
0
otherwise
fh (x) =
then
Z
Z
|f − g| |g| 1
|{|g| ≤ h}|
h
− exp
dx =
exp
dx +
≤2 .
ε
|Ω| {|g|>h}
ε
|Ω|
Ω
Therefore we have
kfh − gkEXP ≤ ε
f or
h > hε
.
3. The main results
The first Sobolev type result we want to deduce is the following
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Theorem 1:If u ∈ W01,1 (Ω) satisfies for some σ > 0 the condition
Z
lim+ ε − |Du|n−ε dx = 0
σ
(3)
ε→0
,
Ω
then, for any c > 0, we have
Z
|u|α dx < ∞
− exp
c
Ω
n
with α = n − 1 + σ , i.e. E(|u|α ) = 0.
Let us recall that the Riesz potential If of a function f ∈ L1 (Ω) is defined as
Z
f (y)|x − y|1−n dy
If (x) =
.
Ω
We will use the following theorem ([GT], Th 7.34)
1 1
1
Theorem: Let 1 ≤ p, q ≤ ∞ and assume that 0 ≤ δ = p − q < n . Then
kIf kLq (Ω) ≤
(5)
1−δ
1
−δ
n
!1−δ
ωn1−1/n |Ω|1/n−δ kf kLp (Ω)
where ωn is the measure of the unit ball in IRn .
We can now pass to the
Proof of theorem 1: Since
|u(x)| ≤
1
|I(|Du|)(x)| ,
nωn
using proposition 1, it will be enough to prove that
lim ε
(6)
1/α
ε→0+
Z 1/ε ε
− I(|Du|)
dx = 0 .
Ω
1
1
In (5) let us take q = 1ε , p = n − ε, and note that, if 0 < ε ≤ n , then certainly δ < n ,
hence we have
Z 1/ε ε
1
ε − I(|Du|)
dx ≤ ε α
1
α
Ω
1−δ
1
−δ
n
!1−δ
ωn1−1/n |Ω|1/n−δ−ε kDukLn−ε (Ω) .
4
1
It easy to check that, if 0 < ε ≤ n , then:
1−δ
1
−δ
n
!1−δ
≤ c(n)ε−
n−1
n
,
so we have:
1
Z Z
n−ε
1/ε ε
−σ
n−1
1
−
n−ε
ε − I(|Du|)
εσ − |Du|n−ε dx
dx ≤ c(n, |Ω|)ε α n ε
Ω
Ω
1
Z
n−ε
n−1
1
σ
≤ c(n, |Ω|)ε α − n − n εσ − |Du|n−ε dx
1
α
Ω
which proves (6), since
1
α
−
n−1
n
−
σ
n
= 0.
Remark 1: If g ∈ L1loc (Ω) satisfies
Z
− |g|n log −σ (e + g)dx < +∞
,
Ω
then (see [BFS] lemma 3)
Z
lim+ ε − |g|n−ε dx = 0 .
σ
From this it follows that if u ∈
ε→0
Ω
W01,1 (Ω)
and |Du|n log −σ (e + |Du|) ∈ L1 (Ω), then
Z
|u|α <∞
− exp
c
Ω
for any c > 0, where α is given by theorem 1. Note that, in any dimension, if σ = 1, then
α = 1.
Remark 2: Theorem 1 is optimal in the sense that the exponent α cannot be improved.
Infact, if
log|x|
u(x) =
|log|log|x|||ϑ
for |x| small, with ϑ > 1/n, then |Du|n log −1 (e + |Du|) is in L1 , while, for any c > 0, δ > 0
Z
exp
|u|1+δ c
=∞ .
We conclude by proving an imbedding theorem in terms of the quantity on the left hand
side of (3), which can be regarded as an extention of Trudinger’s imbedding theorem.
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Theorem 2:Let u ∈ W01,1 (Ω) satisfy for some σ ≥ 0
1
Z
n−ε
(7)
M = sup εσ − |Du|n−ε dx
<∞
0<ε≤1
.
Ω
n
Then, if α = n − 1 + σ , there exist c1 = c1 (n, σ), c2 = c2 (n, σ) such that
Z
α
|u|
(8)
− exp
dx ≤ c2 .
c1 M |Ω|1/n
Ω
1
Proof: Arguing as in the proof of theorem 1, we have for any 0 < ε ≤ n
1
Z
Z
ε
1/ε ε
εα 1
1
α
ε
ε − |u(x)|
dx
≤
− I(|Du|)
nωn Ω
Ω
1
Z
n−ε
n−1
1
1
−
≤ c(n)ε α n |Ω| n εσ − |Du|n−ε dx
Ω
≤ c(n)ε
1
1
σ
α + n −1− n
1
n
|Ω| M
1
n
= c(n)|Ω| M
From this inequality, taking αε = p1 , we get
Z
p 1/p α
1
1
sup
≤ c(n)|Ω| n M
− |u|α dx
n p
p≥ α
Ω
.
The result now follows noting that if
Z
1/p
1
− |u|p dx
λ = sup
p≥p0 p
Ω
,
where p0 ≥ 1, then
Z
|u| − exp
dx ≤ c(p0 ) .
2eλ
Ω
Remark 3: If |Du| ∈ Ln,∞ = weak − Ln , then (7) holds with σ = 1 (see[IS]) and then (8)
holds with α = 1. for this and similar results see [ALT]. Note that Theorem 2 holds true
also in the case σ = 0. In this case the above result reduces to Trudinger’s theorem (see
[T]).
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References
[ALT]
A.Alvino, P.L.Lions, G.Trombetti - On Optimization Problems with Prescribed
Rearrangements, Nonlinear Analysis TMA, 13 (2) (1989), 185-220
[BFS]
H.Brèzis, N.Fusco, C.Sbordone - Integrability for the Jacobian of Orientation Preserving Mappings, to appear in J. Functional Anal.
[CLMS]
R.Coifman, P.L.Lions, Y.Meyer, S.Semmes - Compensated Compactness and
Hardy Spaces,J. Math. Pures Appl. 72, 3 (1993), 247-286
[G]
J.B.Garnett - *Bounded Analitic Functions, Acad. Press (1981)
[GT]
D.Gilbarg, N.S.Trudinger - *Elliptic Partial Differential Equations of Second Order,
Springer (1977)
[Gr]
L.Greco - A Remark on the Equality detDu = DetDu, to appear on J. Integral and
Diff. equations
[IS]
T.Iwaniec, C.Sbordone - On the Integrability of the Jacobian under Minimal Hypotheses, Arch. Rational Mech. Anal. 119 (1992), 129-143
[Mo]
J.Moser - A Sharp Form of an Inequality by N.Trudinger, Indiana U. Math. J., 20 (11)
(1971), 1077-1092
[M]
S.Müller - Higher Integrability of Determinants and Weak Convergence in L1 , J. Reine
Angew. Math. 412 (1990), 20-34
[RR]
[T]
M.M.Rao, Z.D.Ren - *Theory of Orlicz Spaces, M.Dekker (1991)
N.S.Trudinger - On Imbeddings into Orlicz Spaces and some Applications, J. Math.
and Mech. 17 (5) (1967), 473-483
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