Sobolev regularity and an enhanced Jensen inequality

0.17 MB·English

by  Mark A. Peletier

PROCEEDINGSOFTHE
AMERICANMATHEMATICALSOCIETY
Volume00,Number0,Pages000–000
S0002-9939(XX)0000-0
SOBOLEV REGULARITY AND AN ENHANCED JENSEN
INEQUALITY
7
0
0
MARKA.PELETIER,ROBERTPLANQUE´,ANDMATTHIASRO¨GER
2

n
a Abstract. Wederiveanewcriterionforareal-valuedfunctionutobeinthe
J Sobolev space W1,2(Rn). This criterion consists of comparing the value of a
functionalRf(u)withthevaluesofthesamefunctionalappliedtoconvolutions
5
of u with a Dirac sequence. The difference of these values converges to zero
1
as the convolutions approach u, and we prove that the rate of convergence

to zero is connected to regularity: u ∈ W1,2 if and only if the convergence
]
A is sufficiently fast. We finally applyour criterium toa minimizationproblem
with constraints, where regularity of minimizers cannot be deduced from the
F
Euler-Lagrangeequation.
.
h
t
a
m
[ 1. Introduction

1 Jensen’s inequality states that if f : R → R is convex and ϕ ∈ L1(Rn) with
v ϕ≥0 and ϕ=1, then
2
1 R
4 f u(x) ϕ(x)dx≥f u(x)ϕ(x)dx ,
1 ZRn (cid:18)ZRn (cid:19)
0 for any u : Rn → R for(cid:0)whic(cid:1)h the integrals make sense. A consequence of this
7
inequality is that
0
/
h f(u) = f(u)∗ϕ ≥ f(u∗ϕ). (1.1)
at ZRn ZRn ZRn
m In this paper we investigate the inequality (1.1) more closely. In particular we
: study the relationship between the regularity of a function u ∈ L2(Rn) and the
v
asymptotic behaviour as ε→0 of
i
X
Tε(u) := f(u)−f(u∗ϕ ) . (1.2)
r f ε
a ZRn
Here ϕ (y) = ε−nϕ(ε−1y) and f is a(cid:2)smooth function(cid:3), but now not necessarily
ε
convex. Since u∗ϕ → u almost everywhere, we find for ‘well-behaved’ f that
ε
Tε(u) → 0 as ε → 0. Our aim is to establish a connection between the rate of
f
convergence of Tε(u) to zero and the regularity of u.
f
ReceivedbytheeditorsFebruary2,2008.
2000 Mathematics Subject Classification. Primary46E35;Secondary 49J45,49J40.
Key words and phrases. Jensen’sinequality,regularityofminimizers,parametricintegrals.
MA and MR were supported by NWO grant 639.032.306. RP wishes to thank the Centrum
voorWiskundeenInformaticafortheirfinancialsupport.
(cid:13)c1997AmericanMathematicalSociety
1
2 PELETIER,PLANQUE´,ANDRO¨GER
Such a connection between the decay rate of Tε(u) and the regularity of u is
f
suggested by the following informal arguments. Taking n = 1, and assuming ϕ to
be even and u to be smooth, we develop u∗ϕ as
ε
ε2
u∗ϕ (x) = u(x−y)ϕ (y)dy ≈ u(x)+ u′′(x) y2ϕ(y)dy, (1.3)
ε ε
R 2 R
Z Z
so that
Tε(u) ≈ −cε2 f′(u)u′′ = cε2 f′′(u)|u′|2, (1.4)
f
R R
Z Z
withc= 1 y2ϕ(y)dy. Thissuggeststhatifu∈W1,2(R)thenTε(u)isoforderε2.
2 R f
′′
Moreover,sincef ≥0inthecasethatf isconvex,(1.4)givesanenhancedversion
R
of (1.1).
Conversely, for functions not in W1,2(Rn) we might not observe decay of Tε(u)
f
with the rate of ε2, as a simple example shows. Take f(u) = u2 and consider a
functionwithajumpsingularitysuchasu(x)=H(x)−H(x−1)6∈W1,2(R),where
H(x) is the Heaviside function. Now choose as regularizationkernels the functions
ϕ (x) = 1 (H(x+ε)−H(x−ε)), after which an explicit calculation shows that
ε 2ε
Tε(u)=2ε/3. Here the decay is only of order ε.
f
The goalofthis paper is to provethe asymptotic development(1.4) in arbitrary
dimension and to show that also the converse statement holds true: if Tε(u) is
f
of order ε2 then u ∈ W1,2(Rn). These results are stated below and proved in
Sections 2 and 3.
The fact that one can deduce regularity from the decay rate is the original
motivation of this work. In Section 4 we illustrate the use of this result with a
minimization problem in which the Euler-Lagrange equation provides no regular-
ity for a minimizer u. Instead we estimate the decay of Tε(u) directly from the
f
minimization property and obtain that u is in W1,2.
OurresultscanbecomparedtothecharacterisationofSobolevspacesintroduced
by Bourgain, Brezis and Mironescu [2, 4]. In fact, the regularity conclusion in
Theorem 1.2 could be derived from [2], with about the same amount of effort as
the self-contained proof that we give here.
1.1. Notation and assumptions. Let ϕ∈L1(Rn) satisfy
ϕ ≥ 0 in Rn, (1.5)
ϕ = 1, (1.6)
ZRn
yϕ(y)dy = 0, (1.7)
ZRn
|y|2ϕ(y)dy < ∞. (1.8)
ZRn
For ε>0 we define the Dirac sequence
1 x
ϕ (x) := ϕ . (1.9)
ε εn ε
(cid:16) (cid:17)
SOBOLEV REGULARITY AND AN ENHANCED JENSEN INEQUALITY 3
Eventually we will restrict ourselves to the case that ϕ is rotationally symmetric,
that is ϕ(x)=ϕ˜(|x|), where ϕ˜:R+ →R+. Then (1.6) is equivalent to
0 0
∞
nω rn−1ϕ˜(r)dr = 1, (1.10)
n ε
Z0
where ω denotes the volume of the unit-ball, i.e. |B (0)|=ω .
n 1 n
For a function u ∈ L1(Rn) and 0 ≤ s ≤ 1 we define the convolution u and
ε
modified convolutions u by
ε,s
u (x) := u∗ϕ (x) = u(x−y)ϕ (y)dy, (1.11)
ε ε ε
ZRn
(cid:0) (cid:1)
u (x) := u∗ϕ (x) = u(x−y)ϕ (y)dy = u(x−sz)ϕ (z)dz.
ε,s εs εs ε
ZRn ZRn
(cid:0) (cid:1) (1.12)
Note that u = u ,u = u .
ε ε,1 ε,0
We also use the notation a⊗b for the tensor product of a,b∈Rn.
1.2. Statement of main results. Our first result proves (1.4).
Theorem 1.1. Let f ∈C2(R) have uniformly bounded second derivative and
′
f(0)=0, f (0) = 0. (1.13)
Let (ϕ ) be a Dirac sequence as in Section 1.1. Then for any u∈W1,2(Rn),
ε ε>0
1 1
′′
lim f(u)−f(u ) dx= f (u(x))∇u(x)·A(ϕ)∇u(x)dx, (1.14)
ε→0ε2 ZRn ε 2ZRn
where (cid:2) (cid:3)
A(ϕ) = y⊗y ϕ(y)dy. (1.15)
ZRn
If ϕ is rotationally symmetric, i.e. ϕ(x)(cid:0)=ϕ˜(|x(cid:1)|), then
1 1
lim f(u)−f(u ) dx= a(ϕ) f′′(u(x))|∇u(x)|2dx, (1.16)
ε→0ε2 ZRn ε 2 ZRn
with (cid:2) (cid:3)
∞
a(ϕ) = ω rn+1ϕ˜(r)dr. (1.17)
n
Z0
The second theorem shows that for uniformly convex f a decay of Tε(u) of
f
order ε2 implies that u∈W1,2(Rn).
Theorem 1.2. Let f ∈ C2(R) have uniformly bounded second derivative, assume
that (1.13) is satisfied, and that there is a positive number c >0 such that
1
f′′ ≥c on R. (1.18)
1
If for u∈L2(Rn),
1
liminf f(u(x))−f(u (x)) dx < ∞, (1.19)
ε→0 ε2 ZRn ε
then u∈W1,2(Rn). In particular(cid:2)(1.14) and (1.16) h(cid:3)old.
Remark 1.3. Weprescribe(1.13)sinceingeneralthedifferencef(u)−f(u )need
ε
not have sufficient decay to be Lebesgue integrable. For functions u ∈ L1(Rn)∩
L2(Rn) Theorems 1.1 and 1.2 hold even without assuming (1.13).
4 PELETIER,PLANQUE´,ANDRO¨GER
2. Proof of Theorem 1.1
We first show that f(u)−f(u ) ∈ L1(Rn). Let η ∈ C0(Rn;[0,1]). Using the
ε c
Fundamental Theorem of Calculus we obtain that
η(x) f(u(x))−f(u (x)) dx=
ε
ZRn
(cid:12)(cid:12) 1 ∂(cid:12)(cid:12)
= η(x) f u (x) ds dx
ε,s
ZRn (cid:12)Z0 ∂s (cid:12)
1 (cid:12) (cid:0) (cid:1) (cid:12)
≤ η(cid:12)(x) f′ u (x) ∇(cid:12)u(x−sy)·yϕ (y) dydxds
ε,s ε
Z0 ZRn ZRn(cid:12) (cid:12)
≤ 1 1 η(x) (cid:12)(cid:12) f(cid:0)′ u (x(cid:1)) 2|y|ϕ (y)dydxds (cid:12)(cid:12)
ε,s ε
2Z0 ZRn ZRn
1 1 (cid:0) (cid:1)
+ η(x) |∇u(x−sy)|2|y|ϕ (y)dydxds, (2.1)
ε
2Z0 ZRn ZRn
where we have used Fubini’s Theorem and Young’s inequality. For the first term
on the right-hand side we deduce by (1.8) and (1.13) that
η(x)f′ uε,s(x) 2|y|ϕε(y)dydx ≤ Cε(ϕ)kf′′k2∞ uε,s(x)2dx
ZRnZRn ZRn
(cid:0) (cid:1) (1.1)
≤ C (ϕ)kf′′k2 u(x)2dx. (2.2)
ε ∞
ZRn
For the second term on the right-hand side of (2.1) we obtain similarly
η(x)|∇u(x−sy)|2|y|ϕ (y)dydx ≤ C (ϕ) |∇u(x)|2dx. (2.3)
ε ε
ZRnZRn Z
By (2.1)-(2.3) we therefore obtain
η(x) f(u(x))−f(u (x)) dx ≤ C (ϕ) 1+kf′′k2 kuk2 < ∞. (2.4)
ε ε ∞ W1,2(Rn)
ZRn
(cid:12) (cid:12) (cid:0) (cid:1)
Letting η ր(cid:12) 1 we deduce tha(cid:12)t f(u)−f(u ) ∈ L1(Rn). Repeating some of the
ε
calculations above, we obtain that
1 ∂
Tε(u) = − f u (x) dsdx
f ZRnZ0 ∂s ε,s
1 (cid:0) (cid:1)
′
= f u (x) ∇u(x−sy)·yϕ (y)dydxds (2.5)
ε,s ε
Z0 ZRnZRn
(cid:0) (cid:1)
and that
s ∂
′ ′ ′
f u (x) −f u (x−sy) = − f u (x−ry) dr
ε,s ε,s ε,s
∂r
Z0
(cid:0) (cid:1) (cid:0) (cid:1) s (cid:0) (cid:1)
′′
= f u (x−ry) ∇u (x−ry)·ydr. (2.6)
ε,s ε,s
Z0
(cid:0) (cid:1)
SOBOLEV REGULARITY AND AN ENHANCED JENSEN INEQUALITY 5
Since for all 0≤s≤1
′
f u (x−sy) ∇u(x−sy)·yϕ (y)dydx =
ε,s ε
ZRnZRn
(cid:0) (cid:1)
′
= f u (z) ∇u(z)·yϕ (y)dzdy
ε,s ε
ZRnZRn
(1.7) (cid:0) (cid:1)
= 0,
we deduce from (2.5), (2.6) that
1
f(u(x))−f(u (x)) dx=
ε2 ZRn ε
(cid:2) 1 1 (cid:3) s
′′
= f u (x−ry) ∇u (x−ry)·ydr
ε2 Z0 ZRnZRnh(cid:16)Z0 (cid:0) ε,s (cid:1) ε,s (cid:17)
∇u(x−sy)·yϕ (y) dydxds
ε
1 1 s i
′′
= f u (x) ∇u (x)·y
ε2 Z0 Z0 ZRnZRnh (cid:0) ε,s (cid:1) ε,s
∇u(x−ry)·yϕ (y) dydxdrds
ε
1 1 s i
′′
= f u (x) ∇u (x)·
εn+2 Z0 Z0 ZRn ε,s ε,s
(cid:0) (cid:1)
y⊗y ·∇u(x−ry)ϕ(ε−1y)dydxdrds
ZRn
1 1 s (cid:0) (cid:1)
′′
= f u (x) ∇u (x)·
εn+2 Z0 Z0 ZRn ε,s ε,s
(cid:0)1 (cid:1)
z⊗z ·∇u(x−z)ϕ(ε−1r−1z)dzdxdrds
rn+2 ZRn
1 s (cid:0) (cid:1)
′′
= f u (x) ∇u (x)· κ ∗∇u (x)dxdrds, (2.7)
ε,s ε,s εr
Z0 Z0 ZRn
(cid:0) (cid:1) (cid:0) (cid:1)
with the modified convolution kernel κ defined by
εr
z
κ (z) := (εr)−nκ , (2.8)
εr
εr
(cid:16) (cid:17)
where κ is given by
κ(y) := y⊗y ϕ(y). (2.9)
As ε→0 we have that for all 0≤r, s≤(cid:0)1, (cid:1)
u → u almost everywhere,
ε,s
∇u → ∇u in L2(Rn), (2.10)
ε,s
κ ∗∇u → κ(y)dy ∇u in L2(Rn). (2.11)
εr
(cid:16)ZRn (cid:17)
Moreover,theintegrandontheright-handsideof (2.7)isdominatedbythefunction
1
2kf′′kL∞(R) |∇uε,s|2+ κεr∗∇u 2 , (2.12)
(cid:16) (cid:12) (cid:12) (cid:17)
(cid:12) (cid:12)
6 PELETIER,PLANQUE´,ANDRO¨GER
which converges strongly in L1(Rn) as ε → 0 by (2.10) and (2.11). We therefore
deduce from (2.7) that
1
lim f(u(x))−f(u (x)) dx=
ε→0ε2 ZRn ε
(cid:2)1 s (cid:3)
′′
= f (u(x))∇u(x)· κ(y)dy ∇u(x)dxdrds
Z0 Z0 ZRn (cid:16)ZRn (cid:17)
1
′′
= f (u(x))∇u(x)· κ(y)dy ∇u(x)dx. (2.13)
2ZRn (cid:16)ZRn (cid:17)
In the rotationally symmetric case we observe that
∞
y⊗y ϕ(y)dy = rn+1ϕ˜(r)dr θ⊗θdθ
ZRn(cid:0) (cid:1) (cid:16)Z0 ∞ (cid:17)ZSn−1
= ω rn+1ϕ˜(r)dr Id. (2.14)
n
(cid:16)Z0 (cid:17)
Here we used that for i,j =1,...,n,
ω if i=j,
n
θ θ dθ =
i j
ZSn−1 (0 if i6=j.
We therefore obtain from (2.7)-(2.14) that in the rotationally symmetric case
∞
1 1
lim f(u)−f(u ) dx = ω rn+1ϕ˜(r)dr f′′(u)|∇u|2dx.
ε→0ε2 ZRn(cid:2) ε (cid:3) n(cid:16)Z0 (cid:17)2ZRn (2.15)
3. Proof of Theorem 1.2
Let
1
Λ := liminf f(u(x))−f(u (x)) dx. (3.1)
ε→0 ε2 ZRn ε
(cid:2) (cid:3)
We first remark that by assumption (1.18) the function r 7→f(r)− c1r2 is convex
2
and we deduce again by (1.1) that
c c
f(u)− 1u2 − f(u )− 1u2 ≥ 0. (3.2)
ZRnh(cid:16) 2 (cid:17) (cid:16) ε 2 ε(cid:17)i
This implies that
1 2 1 2
liminf u2−u2 ≤ liminf f(u)−f(u ) = Λ. (3.3)
ε→0 ε2 ZRn(cid:16) ε(cid:17) c1 ε→0 ε2 ZRn(cid:2) ε (cid:3) c1
SOBOLEV REGULARITY AND AN ENHANCED JENSEN INEQUALITY 7
Next we consider δ > 0 and the regularizations u = u∗ϕ . Set γ := ϕ ∗ϕ .
δ δ ε ε ε
Then we obtain that
u2(x)− u ∗ϕ 2(x) dx=
δ δ ε
ZRn
(cid:2) (cid:0) (cid:1) (cid:3)
= u2− u (x)u (y)γ (x−y)dydx
δ δ δ ε
ZRn ZRnZRn
1 2
= u (x)−u (y) γ (x−y)dydx
δ δ ε
2ZRnZRn
(1.1)1 (cid:0) (cid:1) 2
≤ u(x−z)−u(y−z) ϕ (z)dzγ (x−y)dydx
δ ε
2ZRnZRnZRn
1 (cid:0) 2 (cid:1)
= u(ξ)−u(η) γ (ξ−η)dηdξϕ (z)dz
ε δ
2ZRnZRnZRn
1 (cid:0) 2 (cid:1)
= u(ξ)−u(η) γ (ξ−η)dηdξ
ε
2ZRnZRn
(cid:0) (cid:1)
= u2(ξ)−u2(ξ) dξ. (3.4)
ε
ZRn
(cid:2) (cid:3)
We therefore deduce from Theorem 1.1 and (3.3), (3.4) that
|∇u |2 = C(ϕ)liminf 1 u2(x)− u ∗ϕ 2(x) dx
ZRn δ ε→0 ε2 ZRn δ δ ε
1 (cid:2) (cid:0) (cid:1) (cid:3)2
≤ C(ϕ)liminf u2(x)−u2(x) dx ≤ ΛC(ϕ). (3.5)
ε→0 ε2 ZRn ε c1
(cid:2) (cid:3)
Since this estimate is uniform in δ > 0 it follows that u ∈ W1,2(Rn). By Theo-
rem 1.1 we therefore deduce (1.14) and (1.16).
4. Application: regularity of minimizers in a lipid bilayer model
We illustratethe utility ofTheorems1.1and1.2with anexample. The problem
is to determine the regularity of solutions of the following minimization problem.
Problem 4.1. Let α > 0,h > 0 and a kernel κ ∈ W1,1(R,R+) be given such that
0
κ′ ∈BV(R). Denote by τ the translation operator,
h
(τ u)(x):=u(x−h) for u:R→R.
h
Consider the set K ⊂R,
K := u∈L1(R) u=1, u≥0, u+τ u≤1 a.e. (4.1)
h
R
n (cid:12) Z o
and a strictly convex, increasing(cid:12)(cid:12), and smooth function f with f(0) = f′(0) = 0.
We then define a functional F :K →R by
F(u):= f(u)− αuκ∗u (4.2)
R R
Z Z
and consider the problem of finding a function u∈K that minimizes F in K, i.e.
F(u)=min{F(v) | v ∈K}. (4.3)
8 PELETIER,PLANQUE´,ANDRO¨GER
Remark 4.2. This problem arises in the modelling of lipid bilayers, biological
membranes (see [1] for more details). There, a specific type of molecules is consid-
ered, consisting of two beads connected by a rigid rod. The beads have a certain
volume, but the rod occupies no space.
The beads are assumed to be of sub-continuum size, but the rod length is non-
negligeable at the continuum scale. In the one-dimensional case we also assume
that the rods lie parallel to the single spatial axis. Combining these assumptions
we model the distribution of such molecules over the real line by a variable u that
representsthe volume fraction occupied by leftmost beads. The volume fraction of
rightmost beads is then given by τ u, and in the condition u+τ u ≤ 1 we now
h h
recognize a volume constraint.
The functional F represents a free energy. In the entropy term f(u) the func-
tion f is strictly convex, increasing, and smooth and can be assumed to satisfy
′ R
f(0)=f (0)=0. The destabilizing term − uκ∗u is a highly stylized representa-
tion of the hydrophobic effect, which favours clustering of beads.
R
Without the constraints u ≥ 0 and u + τ u ≤ 1 present in (4.1), we would
h
immediately be able to infer that minimizers (even stationary points) are smooth
usingasimplebootstrapargument: sinceu∈K wehaveu∈Lp(R)andκ∗u∈W1,p
for all 1≤p≤∞, and therefore, using the Euler-Lagrangeequation,
′
f (u)−2ακ∗u=λ,
for some λ∈R, we find u∈W1,p(R). Iterating this procedure we obtain that u∈
Wk,p(R) for all k ∈ N , 1≤ p≤ ∞. However, when including the two constraints,
twoadditionalLagrangemultipliersµandν appearintheEuler-Lagrangeequation,
′
f (u)−2ακ∗u=λ+µ−ν−τ−hν.
Hereµandν aremeasuresonR,andstandardtheoryprovidesnofurtherregularity
thanthis. Inthiscasethelackofregularityintheright-handsideinterfereswiththe
bootstrap process, and this equation therefore does not give rise to any additional
regularity.
The interest of Theorem 1.2 for this case lies in the fact that K is closed under
convolution. In particular, if u is a minimizer of F in K, then u = ϕ ∗u is also
ε ε
admissible, and we can compare F(u ) with F(u). From this comparison and an
ε
application of Theorem 1.2 we deduce the regularity of u:
Corollary 4.3. Let u be a minimizer of (4.3). Then u∈W1,2(R).
Proof. Chooseafunctionϕ∈L1(R)asinSection1.1withDiracsequence(ϕ ) ,
ε ε>0
andsetu :=ϕ ∗u. Firstwe showthatthere existsa constantC ∈R suchthatfor
ε ε
all u∈L2(R)
uκ∗u−u κ∗u ≤ Cε2 u2. (4.4)
ε ε
R R
(cid:12)Z (cid:12) Z
With this aim we obser(cid:12)ve th(cid:0)at (cid:1)(cid:12)
(cid:12) (cid:12)
uκ∗u−u κ∗u = u κ∗u−κ∗u∗γ (4.5)
ε ε ε
R R
Z Z
with (cid:0) (cid:1) (cid:0) (cid:1)
1 x
γ := ϕ∗ϕ, γ (x) := ϕ ∗ϕ (x) = γ ,
ε ε ε
ε ε
(cid:16) (cid:17)
and that γ,γ satisfy the assumptions in S(cid:0)ection(cid:1)1.1.
ε
SOBOLEV REGULARITY AND AN ENHANCED JENSEN INEQUALITY 9
Since κ′ ∈BV(R) we have (κ∗u)′′ =κ′′∗u∈L2(R) and
′′ ′′
kκ ∗ukL2(R) ≤ |κ | kukL2(R).
R
(cid:16)Z (cid:17)
Repeating some arguments of Theorem 1.1 we calculate that
1 d
(κ∗u)(x)−(κ∗u∗γ )(x)= − (κ∗u)(x−sy)γ (y)dyds
ε ε
Z0 dsZR
1
′
= (κ∗u)(x−sy)yγ (y)dyds
ε
Z0 ZR
1 1
= (κ∗u)′′(x−rsy)sy2γ (y)dydsdr
ε
Z0 Z0 ZR
1 1
= ε2 (κ∗u)′′(x−εrsz)sz2γ(z)dzdrds,
Z0 Z0 ZR
and therefore that
u κ∗u−κ∗u∗γ ≤
ε
R
(cid:12)Z (cid:12)
≤(cid:12)(cid:12)ε2 1(cid:0) 1 (κ∗(cid:1)u(cid:12)(cid:12))′′(x−εrsz)2dx 21 u(x)2dx 12sz2γ(z)dzdrds
≤ε2CZ0(γ)Zk0κ′Z′∗RuhkZLR2(R)kukL2(R) i hZR i
≤ε2C(ϕ,κ)kuk2 ,
L2(R)
which by (4.5) implies (4.4).
Since K is closed under convolution with ϕ , u is admissible, and therefore
ε ε
0≤F(u )−F(u)≤ [f(u )−f(u)]+Cε2 u2.
ε ε
R R
Z Z
The last term is bounded by
Cε2kuk kuk ≤Cε2,
L1(R) L∞(R)
which follows from combining both inequality constraints in (4.1). Therefore
[f(u)−f(u )]=O(ε2), and from Theorem 1.2 we conclude that u∈W1,2(R).
ε
(cid:3)
R
As is described in more detail in the thesis [3], Corollory 4.3 paves the way
towards rigorously deriving the Euler-Lagrange equations for Problem (4.3), and
forms an important ingredient of the proof of existence of minimizers.
References
[1] J. G. Blomand M. A.Peletier, A continuum model of lipid bilayers, Euro. Jnl. Appl.Math.
15(2004), 487–508.
[2] J. Bourgain, H. Brezis, and P. Mironescu, Another look at Sobolev spaces, Optimal control
andpartialdifferentialequations(Jos´eLuis(ed.)etal.Menaldi,ed.),Amsterdam: IOSPress;
Tokyo: Ohmsha,2001,pp.439–455.
[3] R.Planqu´e,Constraintsinappliedmathematics: Rods,membranes,andcuckoos,Ph.D.thesis,
Technische UniversiteitDelft,2005.
[4] A.C.Ponce,Anew approach toSobolev spaces andconnections toΓ-convergence,Calc.Var.
PartialDifferentialEquations 19(2004), no.3,229–255.
10 PELETIER,PLANQUE´,ANDRO¨GER
TechnischeUniversiteitEindhoven,DenDolech2,P.O.Box513,5600MBEindhoven
Departmentof Mathematics,Vrije Universiteit, De Boelelaan 1081a,1081HV Ams-
terdam
E-mail address: [email protected]
MaxPlanckInstituteforMathematicsintheSciences,Inselstr. 22,D-04103Leipzig