Some critical minimization problems for functions of
bounded variations
Thomas Bartsch
Mathematisches Institut
Universität Giessen
Arndtstr.2
35392 Giessen
Germany
e-mail: [email protected]
Michel Willem
Département mathématique
Université Catholique de Louvain
2 chemin du Cyclotron
1348 Louvain-la-Neuve
Belgique
e-mail : [email protected]
Abstract
Using a new elementary method, we prove the existence of minimizers for
various critical problems in BV (Ω) and also in W k,p (Ω), 1 < p < ∞.
2000 Mathematics subject classification: 35J20, 35J25, 35J60, 35J65.
Key words: functions of bounded variation, critical exponents, Gagliardo-Nirenberg
inequality, isoperimetric inequalities.
1
1.
Introduction
After the classical results due to Brezis and Nirenberg ([2]), many papers were devoted
to critical minimization problems on W 1,p (Ω) (1 < p < ∞) or on some subspaces. See
e.g. the list of references in [10].
When p = 1, it is necessary to replace W 1,1 (Ω) by BV (Ω), the space of integrable
functions with bounded variations on Ω. We know only 3 papers devoted to critical
minimization problems on BV (Ω) : [1], [5] and [7]. The critical trace problem in BV (Ω),
treated in [8], is different since it is convex. We exclude this problem.
The existence of optimal functions for the sharp Poincaré inequality
Z
1
u dx
c u −
N/(N −1) ≤ ||Du||Ω
m(Ω) Ω
L
(Ω)
is proved in [5] when Ω is a ball or a sphere and in [1] when Ω is a bounded domain
with C 2 boundary. The proof in [5] uses a specific isoperimetric inequality and in [1]
the concentration-compactness principle in BV (Ω). When Ω ⊂ R2 , the results in [1]
solve a problem of [3].
The minimization problem

Z
Z


|u|dσ
 inf ||Du||Ω + a|u|dx +
∂Ω
Ω



u ∈ BV (Ω), ||u||LN/(N −1) (Ω) = 1,
is treated in [7] using approximation by subcritical problems
and the concentrationZ
|u|dσ replaces the Dirichlet
compactness principle in BV (Ω). The penalization term
∂Ω
boundary condition (see [7] and [11]). See also [6] and [15] for the existence of critical
points.
A general existence theorem for subcritical minimization problems on BV (Ω) is
contained in [11].
In this paper, we solve critical minimization problems on BV (Ω) by using a new
elementary lemma (Lemma 3.2) or a variant (Lemma 4.1). This method is also applicable to critical minimization problems on W 1,p (Ω) (1 < p < ∞) (see Lemma 5.1), is
rather simple and avoids any concentration-compactness type argument.
In section 2 we recall some basic properties of functions of bounded variations (see
[10] and [16]).
2
2.
Functions of bounded variations
Let Ω be an open subset of RN . The variation of u ∈ L1loc (Ω) is defined by
Z
N
||Du||Ω = sup
u div v dx : v ∈ D(Ω, R ), ||v||∞ ≤ 1
Ω
where
||v||∞
N
X
= sup
(vk (x))2
x∈Ω
!1/2
.
k=1
The variation is lower semi-continuous.
L1 (Ω)
loc
un −→
u ⇒ ||Du||Ω ≤ lim ||Dun ||Ω .
n→∞
On
BV (Ω) = {u ∈ L1 (Ω) : ||Du||Ω < ∞}
we define the norm
||u||BV (Ω) = ||Du||Ω + ||u||L1 (Ω)
and the distance of strict convergence
d(f, g) = ||Df ||Ω − ||Dg||Ω + ||f − g||L1 (Ω) .
The sequence (un ) converges weakly to u in BV (Ω) (written un * u) if
||un − u||L1 → 0,
∂k un * ∂k u in [C0 (Ω)]∗ ,
n → ∞,
n → ∞, 1 ≤ k ≤ N.
where [C0 (Ω)]∗ denotes the space of finite measures on Ω.
It is clear that
norm convergence ⇒ strict convergence ⇒ weak convergence.
We now assume that Ω is a bounded domain of RN (N ≥ 2) with Lipschitz boundary.
Let us recall (see [10]) that, for every u ∈ BV (Ω), the trace of u, γ0 (u), belongs to
L1 (∂Ω) and that the extension by 0

 u0 (x) = u(x), x ∈ Ω,

x ∈ RN \{0},
= 0,
belongs to BV (RN ). Moreover,
Z
||Du0 ||RN = ||Du||Ω +
|γ0 (u)|dσ
∂Ω
3
defines an equivalent norm on BV (Ω). The space W 1,1 (Ω) is dense in BV (Ω) with
respect to the strict convergence (not the norm convergence!) and the trace operator
γ0 : BV (Ω) → L1 (∂Ω) is continuous with respect to the strict convergence (not the
weak convergence!).
We will also denote by u the trace of u and the extension of u by 0.
Let us denote by 1∗ the critical exponent N/(N − 1) and by VN the volume of the
unit ball in RN . The following inequality is due to Cherrier [4].
Theorem 2.1. For every ε > 0 there exists cε > 0 such that, for all u ∈ BV (Ω),
(N (VN /2)1/N − ε)||u||L1∗ (Ω) ≤ ||Du||Ω + cε ||u||L1 (Ω) .
Let us recall that, for 1 ≤ p < 1∗ , the imbedding BV (Ω) ⊂ Lp (Ω) is compact and
∗
that the imbedding BV (Ω) ⊂ L1 (Ω) is continuous (but not compact!).
We will also need the sharp Gagliardo-Nirenberg inequality due to Mazýa and Federer and Fleming (see [14]):
∗
Theorem 2.2. For every u ∈ L1 (RN ),
1/N
N VN ||u||L1∗ (RN ) ≤ ||Du||RN .
Moreover equality holds if and only if u is the characteristic function of a ball.
We will use truncation as a basic tool. We define, for h > 0,
Th (s) = min(max(s, −h), h), Rh (s) = s − Th (s).
Proposition 2.3. For every u ∈ BV (Ω),
||Du||Ω = ||DTh u||Ω + ||DRh u||Ω .
Proof. It is clear that
||Du||Ω ≤ ||DTh u||Ω + ||DRh u||Ω .
Let (un ) ⊂ W 1,1 (Ω) be such that un → u strictly in BV (Ω). Then, by lower semicontinuity,
||DTh u||Ω + ||DRh u||Ω ≤
≤
lim ||∇Th un ||L1 (Ω) + lim ||∇Rh un ||L1 (Ω)
n→∞
n→∞
lim ||∇un ||L1 (Ω) = ||Du||Ω .
n→∞
The proof of Proposition 2.3 was communicated to us by J. Van Schaftingen.
4
3.
Critical minimizations problems in BV (Ω)
The following result is due to Degiovanni and Magrone in the case p = 1∗ (see [6]
p. 603). We give the proof for the sake of completeness.
Lemma 3.1. Let Ω be a bounded domain in RN and let 1 ≤ p < ∞ and (un ) ⊂ Lp (Ω)
be such that
a) sup ||un ||p < ∞
b) (un ) converges to u almost everywhere on Ω.
Then
lim (||un ||pp − ||Rh un ||pp ) = (||u||pp − ||Rh u||pp ).
n→∞
Proof. Let us define
f (s) = |s|p − |Rh (s)|p .
For every ε > 0, there exists Cε > 0 such that
|f (s) − f (t)| ≤ ε(|s|p + |t|p ) + Cε .
It follows from Fatou’s lemma that
Z
2ε |u|p dx + Cε m(Ω)
Ω
Z
≤ lim
ε(|un |p + |u|p ) + Cε − |f (un ) − f (u)|dx
n→∞ Ω
Z
Z
Z
p
p
|un | dx + ε |u| dx + Cε m(Ω) − lim
|f (un ) − f (u)|dx.
≤ ε sup
n
n→∞
Ω
Ω
Hence
Z
|f (un ) − f (u)|dx ≤ ε sup
lim
n→∞
Z
n
Ω
Ω
|un |p dx.
Ω
Since ε > 0 is arbitrary, the proof is complete.
In this section, we assume that Ω is a bounded domain of RN (N ≥ 2) with Lipschitz
boundary.
Lemma 3.2. Let a ∈ C(Ω̄) and b ∈ C(∂Ω) be such that ϕ defined on BV (Ω) by
Z
Z
ϕ(u) = ||Du||Ω + a|u|dx +
b|u|dσ
Ω
∂Ω
satisfies
c = inf{ϕ(u)/||u||L1∗ (Ω) : u ∈ BV (Ω)\{0}} > 0.
Let (un ) ⊂ BV (Ω) be such that ||un ||L1∗ (Ω) = 1, ϕ(un ) → c, n → ∞, and un * u in
BV (Ω). Then either ||u||L1∗ (Ω) = 0 or ||u||L1∗ (Ω) = 1.
5
Proof. By going if necessary to a subsequence, we can assume that un → u a.e. on Ω.
We have, using the preceding lemma,
c =
lim [ϕ(Th un ) + ϕ(Rh un )]
n→∞
≥ c lim [||Th un ||1∗ + ||Rh un ||1∗ ]
n→∞
∗
∗
∗
= c ||Th u||1∗ + (1 + ||Rh u||11∗ − ||u||11∗ )1/1 .
When h → ∞, we obtain
∗
∗
∗
∗
1 ≥ (||u||11∗ )1/1 + (1 − ||u||11∗ )1/1 ,
so that ||u||1∗ = 0 or ||u||1∗ = 1.
We consider first the case when b = 0. We assume that a ∈ C(Ω̄) and
Z
(A1) 0 < S0 (a, Ω) = inf
||Du||Ω + a|u|dx /||u||L1∗ (Ω) : u ∈ BV (Ω)\{0} ,
Ω
(A2)
S0 (A, Ω) < N (VN /2)1/N .
Theorem 3.3. Under assumptions (A1), (A2), there exists u ∈ BV (Ω)\{0} such that
u ≥ 0 and
Z
S0 (a, Ω)||u||L1∗ (Ω) = ||Du||Ω + a|u|dx.
Ω
Proof. Let (un ) ⊂ BV (Ω) be such that ||un ||1∗ = 1 and
Z
||Dun ||Ω + a|un |dx → S0 (a, Ω).
Ω
Since (un ) is bounded in BV (Ω), we can assume that un * u in BV (Ω). Let 0 < ε <
N (VN /2)1/N − S0 (a, Ω). It follows from Theorem 2.1 that, for some cε > 0,
Z
S0 (a, Ω) = lim ||Dun ||Ω + a|un |dx
n→∞
Ω
Z
Z
1/N
≥ N (VN /2)
− ε − cε |u|dx + a|u|dx.
Ω
Ω
Hence, u 6= 0. The preceding lemma implies that ||u||1∗ = 1. Since, by lower semicontinuity,
Z
||Du||Ω + a|u|dx ≤ S0 (a, Ω),
Ω
u is a minimizer for S0 (a, Ω). Since ||D|u|||Ω ≤ ||Du||Ω , we can replace u by |u|.
6
The following result gives a concrete sufficient condition for (A2).
Theorem 3.4. Let Ω be a bounded domain with C 2 boundary and let a ∈ C(Ω̄) be such
that, for some y ∈ ∂Ω,
N − 1 VN −1
2
H(y) > a(y),
N + 1 VN
where H(y) denotes the mean curvature of ∂Ω at y. Then (A2) is satisfied.
Proof. We can assume that y = 0. For r > 0 small enough, we have
δ=
N −1
VN
VN −1 H(0) −
A > 0,
N +1
2
where A = sup{a(x) : x ∈ Ω ∩ B(0, r)}. Let us define uε = χΩ∩B(0,ε) . By formula (1) in
[12], we have, for ε → 0,
VN N N − 1 VN −1
ε −
H(0)εN +1 + o(εN +1 ),
2
N +1 2
VN N −1
VN −1
||Duε ||Ω = N
ε
− (N − 1)
H(0)εN + o(εN ),
2
2
Z
VN
a uε dx ≤ A εN + o(εN ).
2
Ω
∗
||uε ||11∗ = m(Ω ∩ B(0, ε)) =
It follows that, for ε → 0,
Z
||Duε ||Ω + a uε dx /||uε ||1∗
Ω
≤
VN
2
1−N
N
VN
N −1
VN
N
−
VN −1 H(0)ε +
Aε + o(ε)
2
N +1
2
= N (VN /2)1/N − δ(VN /2)
1−N
N
ε + o(ε),
so that S0 (a, Ω) < N (VN /2)1/N .
We consider now the case when b = 1. The following result is due to Demengel [7],
but our proof, using Lemma 3.1, is simpler.
Let us recall that, for u ∈ BV (Ω),
Z
|u|dσ.
||Du||RN = ||Du||Ω +
∂Ω
We assume that a ∈ C(Ω̄) and
7
(B1)
Z
0 < S1 (a, Ω) = inf
||Du||RN + a|u|dx /||u||L1∗ (Ω) : u ∈ BV (Ω)\{0} ,
Ω
(B2)
1/N
S1 (a, Ω) < N VN .
Theorem 3.5. Under assumptions (B1), (B2), there exists u ∈ BV (Ω)\{0} such that
u ≥ 0 and
Z
S1 (a, Ω)||u||L1∗ (Ω) = ||Du||RN + a|u|dx.
Ω
Proof. Let (un ) ⊂ BV (Ω) be such that ||un ||1∗ = 1 and
Z
||Dun ||RN + a|un |dx → S1 (a, Ω).
Ω
The sequence (un ) is bounded in BV (RN ). We can assume that un * u in BV (RN ).
It follows from Theorem 2.2 that
Z
S1 (a, Ω) = lim ||Du||RN + a|un |dx
n→∞
Ω
Z
1/N
≥ N VN + a|u|dx.
Ω
By assumption (B2), u 6= 0. Lemma 3.2 implies that ||u||1∗ = 1. Since, by lower
semi-continuity,
Z
||Du||RN + a|u|dx ≤ S1 (a, Ω),
Ω
u is a minimizer for S1 (a, Ω). Since ||D|u|||RN ≤ ||Du||RN , we can replace u by |u|. 4.
Poincaré inequality
Let us recall the general Poincaré inequality in BV (Ω) due to Meyers and Ziemer [13].
Let Ω be a boundedZ domain of RN (N ≥ 2) with Lipschitz boundary and let
f ∈ LN (Ω) be such that
f dx = 1. Then
Ω
Z
S2 (f, Ω) = inf ||Du||Ω /||u||L1∗ (Ω) : u ∈ BV (Ω)\{0},
f u dx = 0 > 0.
Ω
When f ≡ 1/m(Ω), this is the Poincaré inequality.
Z
N
Lemma 4.1. Let f ∈ L (Ω) be such that
f dx = 1 and let (un ) ⊂ BV (Ω) be
Ω
Z
such that ||un ||L1∗ (Ω) = 1, f un dx = 0, ||Dun ||Ω → S2 (f, Ω), n → ∞, and un * u in
Ω
BV (Ω). Then either ||u||L1∗ (Ω) = 0 or ||u||L1∗ (Ω) = 1.
8
Proof. By going if necessary to a subsequence, we can assume that un → u a.e. on Ω.
Let us define, for h > 0 and n ∈ N,
Z
Z
f Th un dx, ch =
f Th u dx.
ch,n =
Ω
Ω
Using Lemma 3.1, we obtain,
S2 (f, Ω) =
lim [||DTh un ||Ω + ||DRh un ||Ω ]
n→∞
≥ S2 (f, Ω) lim [||Th un − ch,n ||1∗ + ||Rh un + ch,n ||1∗ ]
n→∞
≥ S2 (f, Ω) lim [||Th un ||1∗ + ||Rh un ||1∗ − 2||ch,n ||1∗ ]
hn→∞
i
∗
∗ 1/1∗
= S2 (f, Ω) ||Th u||1∗ + 1 + ||Rh u||11∗ − ||u||11∗
− 2||ch ||1∗ .
Z
Since lim ch = lim
h→∞
h→∞
Z
f Th u dx =
Ω
f u dx = 0, we have
Ω
∗ 1/1∗
∗ 1/1∗
1 ≥ ||u||11∗
+ 1 − ||u||11∗
,
so that ||u||1∗ = 0 or ||u||1∗ = 1.
The following theorem was proved by Bouchez and Van Schaftingen in the case
f ≡ 1/m(Ω) (see [1]).
N
Theorem 4.2.
Let Ω be a bounded domain of RN with C 2 boundary and
Z
Z let f ∈ L (Ω)
be such that f dx = 1. Then there exists u ∈ BV (Ω)\{0} such that f u dx = 0 and
Ω
Ω
S2 (f, Ω)||u||L1∗ (Ω) = ||Du||Ω .
Proof. 1) Let us first prove that
(∗)
S2 (f, Ω) < N (VN /2)1/N .
We can assume that 0 ∈ ∂Ω and that H(0), the mean curvature of ∂Ω at 0, is positive.
Let us define, as in [1], for ε > 0 small enough,
uε = χΩ∩B(0,ε) − λε χΩ\B(0,ε) ,
Z
Z
λε =
f dx/
f dx.
Ω∩B(0,ε)
Ω\B(0,ε)
9
Hölder inequality implies that λε = o(εN −1 ). By formula (1) in [12], we have, for ε → 0,
∗
VN N N − 1 VN −1
ε −
H(0)εN +1 + o(εN +1 ),
2
N +1 2
= (1 + λε )||DχΩ∩B(0,ε) ||
VN N −1
VN −1
= N
ε
− (N − 1)
H(0)εN + o(εN ).
2
2
||uε ||11∗ ≥ m(Ω ∩ B(0, ε)) =
||Duε ||Ω
It follows that, for ε → 0,
||Duε ||Ω /||uε ||1∗ ≤
VN
2
1−N
N
VN
N −1
N
−
VN −1 H(0)ε + o(ε) ,
2
N +1
so that (∗) is satisfied.
Z
2) Let (un ) ⊂ BV (Ω) be such that ||un ||1∗ = 1,
f un dx = 0 and
Ω
||Dun ||Ω → S2 (f, Ω),
n → ∞.
We can assume that un * u in BV (Ω). Let 0 < ε < N (VN /2)1/N − S2 (f, Ω). It follows
from Theorem 2.1 that, for some cε > 0,
Z
1/N
S2 (f, Ω) = lim ||Dun ||Ω ≥ N (VN /2)
− ε − cε |u|dx.
n→∞
Ω
Z
Hence u 6= 0. The preceding lemma implies that ||u||1∗ = 1. Since
f u dx = 0 and,
Ω
by lower semi-continuity,
||Du||Ω ≤ S2 (f, Ω),
u is a minimizer for S2 (f, Ω).
5.
Critical minimization problems in W 1,p(Ω)
In this section, we assume that Ω is a smooth bounded domain of RN . We define, for
1 < p < ∞, the critical exponent p∗ = N p/(N − p) and
X0 = W 1,p (Ω),
X1 = W01,p (Ω),
Z
1,p
X2 =
u ∈ W (Ω) :
f u dx = 0
Ω
p∗
Z
where f ∈ L (Ω) and
f dx = 1.
Ω
The following lemma is a variant of Lemma 3.2 and Lemma 4.1 with a similar proof.
10
Lemma 5.1. Let a ∈ C(Ω̄) be such that ϕ defined on Xj (where j = 0, 1 or 2) by
Z
Z
p
|∇u| dx + a|u|p dx
ϕ(u) =
Ω
Ω
satisfies
n
o
cj = inf ϕ(u)/||u||pLp∗ (Ω) : u ∈ Xj \{0} > 0.
Let (un ) ⊂ Xj be such that ||un ||Lp∗ (Ω) = 1, ϕ(un ) → cj , n → ∞, and un * u in Xj .
Then either ||u||Lp∗ (Ω) = 0 or ||u||Lp∗ (Ω) = 1.
The preceding lemma is applicable to many quasilinear critical problems as considered e.g. in [9].
Let us define
Z
p
N
p
N
S(p, R ) = inf
|∇u| dx/||u||Lp∗ (RN ) : u ∈ D(R )\{0} .
RN
The following Theorem is a variant of Theorems 3.3, 3.5 and 4.2.
Theorem 5.2. a) If 0 < c0 < S(p, RN )/2p/N , then c0 is achieved.
b) If 0 < c1 < S(p, RN ), then c1 is achieved.
c) If 0 < c2 < S(p, RN )/2p/N , then c2 is achieved.
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