Full text of DOI:10.1051/m2an:2000105

Mathematical Modelling and Numerical Analysis
Mod´elisation Math´ematique et Analyse Num´erique
ESAIM: M2AN
Vol. 34, No 4, 2000, pp. 811–834
ABOUT STABILITY OF EQUILIBRIUM SHAPES
Marc Dambrine1 and Michel Pierre1
Abstract. We discuss the stability of “critical” or “equilibrium” shapes of a shape-dependent energy
functional. We analyze a problem arising when looking at the positivity of the second derivative in
order to prove that a critical shape is an optimal shape. Indeed, often when positivity -or coercivity-
holds, it does for a weaker norm than the norm for which the functional is twice differentiable and
local optimality cannot be a priori deduced. We solve this problem for a particular but significant
example. We prove “weak-coercivity” of the second derivative uniformly in a “strong” neighborhood
of the equilibrium shape.
R´esum´e. Nous nous int´eressons `a la stabilit´e des formes critiques ou d’“´equilibre” d’une ´energie
d´ependant de la forme. Dans le but de montrer qu’une forme critique est une forme optimale, nous
´etudions la positivit´e de la d´eriv´ee seconde. En effet, quand elle a lieu, la coercivit´e n’est vraie que
dans une norme plus faible que celle pour laquelle l’´energie est diff´erentiable: l’optimalit´e locale ne
peut donc pas en ˆetre d´eduite a priori. Nous r´esolvons cette difficult´e dans un cas particulier mais
n´eanmoins significatif. Nous ´etablissons de la “coercivit´e faible” de la d´eriv´ee seconde uniform´ement
dans un voisinage “fort” de la forme d’´equilibre.
Mathematics Subject Classification. 49Q10, 49K40, 35J25.
Received: August 31, 1999. Revised: March 13, 2000.
1. Introduction
We consider here the question of stability of equilibrium shapes which can be stated as follows. Let Ω → E(Ω)
be a real valued functional defined on a family O of subsets Ω of Rn. Let Ω0 be an equilibrium shape for E(.),
that is a shape at which the first derivative of E(.) on O vanishes (we also say “critical shape”, see below for a
precise definition). By stability, we mean that E(Ω0) is a strict local extremum, say a minimum, that is
E(Ω0) < E(Ω)
(1.1)
for all Ω close enough to Ω0 and in O. If E(.) represents the total energy in some shape equilibrium problem,
this definition coincides with the classical notion of stability.
One of the difficulties is to understand properly the meaning of “being close to Ω0” and therefore to choose
the right topology on O. One of the classical techniques is then to compute the second derivative of E(.) at
Ω0 and to prove that it is strictly positive. However, in many applications, one is led to a situation where
the second derivative of E(.) at Ω0 is coercive (i.e. strictly positive) for a certain norm which turns out to be
Keywords and phrases. Shape optimisation, stability of critical shape, weak coercivity, area-preserving transformations.
1
Antenne de Bretagne de l’ENS Cachan, Institut de Recherche Math´ematique de Rennes, Campus de Ker Lann, 35170 Bruz,
France. e-mail: dambrine@bretagne.ens-cachan.fr; pierre@bretagne.ens-cachan.fr
c
ꢀ EDP Sciences, SMAI 2000

812
M. DAMBRINE AND M. PIERRE
weaker than the norm for which differentiability and Taylor formula hold. Consequently, the existence of a local
minimum does not follow, even for the stronger topology. In order to show what could happen, let us consider
an elementary example of such a situation (not taken from shape optimization). Let L2(0, 1) and H01(0, 1) be
2
1
equipped with their usual norm and recall that || · ||L ≤ || · ||H . Consider the functional defined as
0
2
4
E(u) = kukL2(0,1) − kuk
.
H01(0,1)
One can check that E is twice differentiable in H01 and that
2
E0(0) ≡ 0, ∀h ∈ H01(0, 1), E00(0).(h, h) = 2khkL2(0,1)
.
Therefore, E00(0) is coercive for the weaker norm L2. This yields some “weak stability”: indeed, there is a local
4
minimum in each direction u=0 ∈ H01 since, for t ∈ R, E(tu0) = t2(||u0||L − t2||u0|| ). However, there is no
2
H01
local minimum for E, even for the strong topology, since there is no r > 0 such that
2
4
1
kukH (0,1) < r =⇒ E(u) > E(0) = 0 i.e. kukL2(0,1) > kuk
,
H01(0,1)
0
as one can always construct a sequence in H01(0, 1) such that
1
2
kunk
= r/2, kunkL (0,1) → 0 when n → +∞.
H0 (0,1)
Our goal is to precisely analyze this difficulty in a particular, but significant situation coming from shape
optimization. Here, the second derivative of E(.) will exist in a C2,α-norm around Ω0, but coercivity will only
hold with respect to the H1/2(∂Ω0)-norm. This situation is typical in shapes problems. Here we choose E(.)
to be the energy associated with the classical Dirichlet problem and the measure of the admissible domains is
supposed to be given. This model problem arises in many examples: let us for instance mention the case where
E(.) is the total energy in a problem of equilibrium shapes for liquid metals confined in a electro-magnetic
field (see e.g. [1, 12, 13, 15]) We will restrict ourself to a simple two-dimensional model for which stability was
already investigated in [3,7–9,14,19]. The critical shapes we consider are assumed to be regular. Stability of a
critical shape Ω0 will mean that E(Ω0) is a strict minimum for E(Ω) among the admissible domains Ω in some
C
2,α-neighborhood of Ω0 with the same measure as Ω0.
We prove here for this problem that stability does occur when H1/2-positivity of the second derivative holds
on the tangent subspace of constraints. The main idea is to compute the second derivative not only at the
2,α
equilibrium shape Ω0 but around Ω0 in the C2,α-sense and to prove a uniform H1/2(∂Ω0)-coercivity in a C
-
neighborhood of Ω0. This yields at least the existence of a local minimum in the C2,α-topology. This technique,
while developed here only in a specific case, may actually be used in many other situations like for the “exterior
shaping problem” where the Dirichlet problem is set in the exterior of the shapes, or also when the functional
depends on the perimeter and for more general functional depending on the state and on the gradient of the
state (see [4]). We think that the estimate of the variation of the second derivative is by itself interesting and
might lead to other applications. Note that the above question of choice of topologies was, in particular, raised
in [7].
2. The problem and the results
2
2
Let k be a given real-valued function with compact support in R and belonging to the C0,α(R ) Ho¨lder
space (α ∈ (0, 1)) and let S > meas(support(k)) be a given constant where meas(.) stands for the Lebesgue
measure. We first define the family O of admissible shapes to be the family of open bounded subsets Ω of R2
with C2,α-boundary such that
meas(Ω) = S,
(2.1)
(2.2)
support(k) ⊂ Ω.

ABOUT STABILITY OF EQUILIBRIUM SHAPES
813
We consider the shaping function E from O into R defined by
Z
2
E(Ω) = −
|∇ uΩ| ,
Ω
where |.| denotes here the Euclidian norm and the function uΩ is the solution of the Dirichlet problem
ꢀ
−∆uΩ = k in Ω,
(2.3)
uΩ = 0
on ∂Ω.
Note that E(Ω) is, up to a positive constant, the “energy” associated with the Dirichlet problem (2.3) which is
Z
Z
Z
Z
1
2
1
2
2
2
|∇ uΩ| −
k uΩ = −
|∇ uΩ| = −
k uΩ,
Ω
Ω
Ω
Ω
the above equalities following easily from multiplying (2.3) by uΩ. Note also that (see e.g. Sect. 4), with the
above regularity assumptions
∀Ω ∈ O, uΩ ∈ C2,α(Ω).
We now consider Ω0 a critical point of E under the constraint (2.1) that is to say an open set Ω0 in O where
the derivative of E(Ω) + Λ meas(Ω) with respect to Ω vanishes for some Λ ∈ R. As we will verify later (see
Sect. 5), this means - and we will assume it throughout the paper:
2
|∇uΩ
|
=
Λ on ∂Ω0.
(2.4)
0
The constant Λ is the Lagrange multiplier corresponding to the constraint meas(Ω) = S. We will recall below
2,α
(see beginning of Sect. 3) the definition of the spaces C
and of their norms.
We want to find sufficient conditions for the stability of Ω0 using second derivatives of the augmented
functional J(Ω) = E(Ω) + Λ meas(Ω). Let us first recall some facts about these derivatives.
Let us denote
2,α
(R2, R2); kΘ − IdR k2,α < η}.
(2.5)
2
V(η) := {Θ ∈ C
¯
Note that, for η small enough, any Θ ∈ V(η) is a diffeomorphism. For all Θ ∈ V(η), we set J(Θ) := J(Θ(Ω0)).
¯
Then the second (classical) derivative of J exists and one can show (see e.g. [2, 6, 16–18, 20, 21] or also Sect. 5
here) that, since Ω0 is a critical shape, the second derivative at Θ = Id(= Identity) has a specific structure,
namely
2,α
2
2
00
¯
∀ξ ∈ C (R , R ), J (Id)(ξ, ξ) = B(ξ · n
, ξ · n
)
(2.6)
|∂Ω0
|∂Ω0
where B is a continuous bilinear form on C1,α(∂Ω0, R), n is the unit exterior normal vector field to ∂Ω0 and
·
denotes the restriction of a function to ∂Ω0. As we will see in Section 5, in the situation we consider here,
|∂Ω0
the explicit expression of B is given as follows: set m = ξ · n
, then
|∂Ω0
Z
B(m, m) = 2 Λ
C0(m) m + C m2,
(2.7)
∂Ω0
where C denotes the curvature of ∂Ω0 and C0 denotes the so-called “capacity” or “Steklov-Poincar´e” operator
on ∂Ω0. We refer e.g. to [5] or to Section 5 for a precise definition and properties of this operator but we can
already mention that:
Z
Z
2
C0(m) m =
|∇M|
∂Ω0
Ω0

814
M. DAMBRINE AND M. PIERRE
where M is the harmonic extension of m to Ω0. Consequently, the first part of the integral is always strictly
positive (except if M = 0 which happens only if m is constant on each boundary of the connected components
of Ω0).
¯
A necessary condition for stability is that the second derivative of J be positive on the subspace which is
R
tangent to the constraint, namely the subspace of functions m as above such that
m = 0. Obviously, it is
∂Ω0
the case if, for instance, Ω0 is convex; then B is even coercive in the space H1/2(∂Ω0) if Λ > 0. More general
situations are described in [4,11]. See also more comments on this at the end of this paper.
The question we address here is the converse. For this, one has to assume, as usual, that the second derivative
is strictly positive in some sense. The natural space of coercivity here is H1/2(∂Ω0). The question is then the
following: assume that there exists c > 0 such that
Z
2
∀m ∈ C2,α(∂Ω0, R) with
m = 0, B(m, m) ≥ c ||m||H1/2
.
(2.8)
(∂Ω0)
∂Ω0
Then, is E(Ω0) a strict local minimum with the constraint (2.1), local at least in the C2,α-topology?
We prove here that the answer to this question is positive:
4,α
Theorem 2.1 (Existence of a local strict minimum). Assume ∂Ω0 is of class C
and (2.8) holds for some
ꢁ
ꢂ
c > 0. Then there exists η > 0 such that for all Θ in V(η) with meas Θ(Ω0) = meas(Ω0) and different from
the identity
ꢁ
ꢂ
E Θ(Ω0) > E(Ω0).
The main point in the proof of this result will be contained in Theorem 2.5. In order to state it, we need to
introduce a few notations and to recall some more or less known facts on small regular perturbations of Ω0
and of ∂Ω0. Since we are in dimension two, ∂Ω0 is a union of q disjoint regular Jordan curves. For simplicity,
we will write the proof when q = 1 (i.e. Ω0 is simply connected). The changes needed in the general case are
obvious (see the remark at the end of Sect. 2.2 or also [4]).
Let γ denote a function in Ck,α([0, L], R2), with k ≥ 2, whose image is ∂Ω0, that is

γ([0, L]) = ∂Ω0, γ(0) = γ(L),

γ is one-to-one from [0, L) into ∂Ω0,

∀s ∈ [0, L], ||γ0(s)|| = 1.
Here the parameter s is the length parameter and L is the total length of ∂Ω0. We denote by n the unit exterior
normal derivative to ∂Ω0. The orientation is chosen so that so that n(γ(s)) = R−π/2(γ0(s)) where R
is the
−π/2
rotation of angle −π/2 in R or also n(γ(s)) = (γ20 (s), −γ10 (s)) where γ = (γ1, γ2). We will often write simply
2
n(s) = n(γ(s)).
For τ > 0, we denote by Tτ the tubular neighborhood of ∂Ω0 with radius τ, that is
Tτ = {x ∈ R2; distance(x, ∂Ω0) < τ}.
Lemma 2.2 (Normal representation of small perturbations of ∂Ω0). Assume ∂Ω0 is of class C2,α. Then, there
exists τ1 > 0 such that the mapping
(s, τ) ∈ [0, L) × (−τ1, τ1) → γ(s) + τ n(s) ∈ Tτ
1
is one-to-one. Moreover, there exists η1 > 0 such that for all Θ ∈ V(η1), there exists a unique dΘ
∈
C
1,α([0, L], (−τ1, τ1)) with dΘ(0) = dΘ(L) and such that s → γ(s) + dΘ(s) n(s) is one-to-one from [0, L)
into Θ(∂Ω0). If ∂Ω0 is of class C3,α, then dΘ is of class C
and we have
2,α
kdΘk2,α ≤ C kΘ − Idk2,α
.
(2.9)

ABOUT STABILITY OF EQUILIBRIUM SHAPES
815
Here our goal is to minimize the functional E(.) on open subsets with a prescribed measure. Therefore we want
to go from Ω0 to Θ(Ω0) for any Θ ∈ V(η1) with meas(Θ(Ω0)) = meas(Ω0), by a regular path t ∈ [0, 1] → Ω(t)
where meas(Ω(t)) = meas(Ω0). This can be done through normal deformations obtained from the flow of a
divergence-free vector field as stated in the next proposition.
We first need to extend the vector-field n to the tubular neighborhood Tτ of ∂Ω0. We do it as follows: if
1
x ∈ Tτ , there exists (s, τ) unique in [0, L) × (−τ1, τ1) such that x = γ(s) + τn(s); then we define n(x) as
1
n(γ(s)). Thus for all (s, τ) ∈ [0, L) × (−τ1, τ1),
n(γ(s) + τ n(s)) = n(s).
(2.10)
Proposition 2.3 (Area-preserving normal deformations of Ω0). We assume ∂Ω0 is of class C4,α. There exist
η2 > 0, τ2 > 0 such that, for all Θ ∈ V(η2) with meas(Θ(Ω0)) = meas(Ω0), there exists a divergence-free vector
field XΘ ∈ C2,α(R2, R2) and mΘ ∈ C2,α(Tτ , R) such that XΘ = mΘ n on Tτ and the flow ΦΘ of XΘ maps
2
∂Ω0 onto Θ(∂Ω0) at time t = 1. Moreover, 2we have for all t ∈ [0, 1],
kΦΘ(t) − IdR
2
k2,α ≤ C kΘ − IdR
2
k2,α
.
(2.11)
Recall that the flow ΦΘ of the vector field XΘ is the solution of
(
∂tΦΘ(t, x) = XΘ(ΦΘ(t, x))
ΦΘ(0, x) = x.
Since divXΘ = 0, we have det ΦΘ(t, x) ≡ 1, so that, in particular, meas(ΦΘ(t, Ω0)) = meas(Ω0). We will use
this area-preserving path t → ΦΘ(t, Ω0) going from Ω0 to ΦΘ(1, Ω0) = Θ(Ω0) and control the variation of E(·)
along this path by studying the second derivative with respect to t of
eΘ(t) := E(ΦΘ(t, Ω0)).
Lemma 2.4. Assume Ω0 has a C4,α-boundary. Then, for all Θ ∈ V(η2), t → eΘ(t) is twice differentiable on
[0, 1].
A proof of this lemma can be found in [7] (see also [6, 16, 20, 21]). The main arguments are also given here
while computing e0Θ0 (t) (see Sect. 5).
We can now state our key-result.
Theorem 2.5. Assume ∂Ω0 is of class C4,α. Then there exist η0 > 0 and a function ω :]0, η0] → R with
lim ω(r) = 0, such that for all η ∈ (0, η0] and all Θ ∈ V(η) with meas(Θ(Ω0)) = meas(Ω0), we have for all
r↓0
t ∈ [0, 1],
2
|e0Θ0 (t) − eΘ00 (0)| ≤ ω(η) kmΘk
.
(2.12)
H1/2(∂Ω0)
It is easy to guess how Theorem 1 can be deduced from this theorem by application of Taylor formula (see the
end of this paper). The main point is that around the equilibrium shape Ω0, if the positivity condition (2.8)
holds, then the second derivative will actually be H1/2(∂Ω0)-coercive uniformly in a C2,α-neighborhood of Ω0.
The above property depends on the nature of the various terms which come out in the expression of the
second derivative. Obviously, it can be shown to be valid for many other similar functionals like those already
mentioned above (see [4,11]).
4,α
With respect to the regularity assumptions, the hypothesis that the critical point should be of class C
is
not a restriction since Henrot and the second author have shown in [13] that if a regular Jordan curve with a
2
C boundary is a critical point for this functional, then it is in fact analytic.

816
M. DAMBRINE AND M. PIERRE
3. Proofs of geometrical results
p
Let us recall the classical definitions of H¨older spaces and norms. Let D be an open subset of R , p ≥ 1 and
2
q
¯
let q ≥ 1. Then the C (D, R )-norm is defined as
kuk2 = kukL∞(D,R ) + sup k∂iukL∞(D,R )
q
q
+
sup k∂i2,jukL∞
q
.
(D,R )
1≤i≤n
1≤i,j≤n
2,α
q
2
¯
For α ∈ (0, 1), the Ho¨lder space C (D, R ) is the subspace of C -functions such that
k∂i2,ju(x) − ∂i2,ju(y)k
max
sup
< +∞,
α
1≤i,j≤n
(x,y)∈D2,
kx − yk
x=y
and the C2,α-norm is defined as
k∂i2,ju(x) − ∂i2,ju(y)k
kuk2,α = kuk2 + max
sup
·
α
1≤i,j≤n
Dkx − yk
(x,y)∈D2,
x=y
2,α
q
k,α
q
¯
¯
It gives to C (D, R ) a Banach space structure. A similar definition may be given for C (D, R ), k ≥ 0 by
replacing the second derivatives by the kth derivatives.
3.1. Proof of Lemma 2.2.
This is rather classical but we give here an elementary proof whose ingredients will be partly used later in
this paper.
2
Assume first that ∂Ω0 is of class C . Let us consider the mapping T from R × R into R2 defined by
1
T(s, τ) = γ(s) + τ n(s) which is of class C and L-periodic in s. The derivative of T at (s0, 0) is given by
T0(s0, 0) = [γ0(s0) n(s0)]
so that det T0(s0, 0) = −1. By the inverse mapping theorem, T is a local C -diffeomorphism.
1
Let us show that, if τ1 is small enough, T is also a global bijection from [0, L) × (−τ1, τ1) into Tτ .
1
Let τ(s0), η(s0) > 0 be such that T is a diffeomorphism from (s0 − η(s0), s0 + η(s0)) × (−τ(s0), τ(s0)) onto
a neighborhood ω(s0) of γ(s0). Up to still reducing τ(s0), one can also assume that
ꢁ
ꢂ
ω(s0) ∩ ∂Ω0 = T (s0 − η(s0), s0 + η(s0)) × {0} .
(3.1)
Let r(s0) be such that B(γ(s0), r(s0)) ⊂ ω(s0). One can find a finite covering of ∂Ω0 by the union of the
{∂Ω0 ∩ B(γ(si), 12 r(si)), si ∈ [0, L), i = 1...p}. Next we set:
r1 := min{r(si), i = 1...p}, τ1 := min{r1/5, τ(si), i = 1...p}.
Then the announced global bijection property holds. Indeed, assume
γ(s) + τ n(s) = γ(sˆ) + τˆ n(sˆ), with τ, τˆ ∈ (−τ1, τ1), s, sˆ ∈ [0, L).
(3.2)
Then
1
2
kγ(s) − γ(sˆ)k ≤ τ + τˆ ≤ 2 τ1 < r1.

ABOUT STABILITY OF EQUILIBRIUM SHAPES
Therefore, γ(s) and γ(sˆ) belong to the same ball B(γ(si), r(si)). Then, by (3.1), there exist
σ, σˆ ∈ (−η(si) + si, si + η(si)), σ = s mod (L), σˆ = sˆ mod (L),
such that
817
γ(s) = γ(σ), γ(sˆ) = γ(σˆ).
Then, (3.2) can be rewritten
γ(σ) + τ n(σ) = γ(σˆ) + τˆ n(σˆ), with τ, τˆ ∈ (−τ1, τ1), σ, σˆ ∈ (−η(si) + si, si + η(si)).
By the local bijection property of T, σ = σˆ, τ = τˆ. Since now γ is a bijection from [0, L) onto ∂Ω0, s, sˆ ∈ [0, L)
and γ(s) = γ(sˆ) imply finally that s = sˆ.
1
For the regularity, we know that (s˜(x), τ˜(x)) = T−1(x) is of class C on Tτ . We then easily check that
1
2
∇τ˜(x) = n(s˜(x)) (see 3.4 below). Therefore, τ˜0 is of class C and τ˜ is of class C (note that τ˜ is the distance
1
function to ∂Ω0 and it is classical that it has the regularity of ∂Ω0, see e.g. [10]). If γ is of class Ck,α, then T
and T−1 are of class Ck−1,α. So is τ˜0 so that τ˜ is of class C
.
k,α
2,α
Let us now consider Θ ∈ V(η1), η1 > 0; recall that Θ is then of class C
and that γ is assumed to be of class
k,α
2,α
C
with k = 2 or 3. The mapping Y (s) = Θ(γ(s)) is of class C
and, if η1 is small enough so that Y (s) ∈ Tτ
1
for all s, with the previous notations and ψ(s) := s˜(Y (s)), one can write
Y (s) = γ(ψ(s)) + τ˜(Y (s))n(ψ(s)).
(3.3)
k−1,α
For the regularity, ψ is of class C
and τ˜(Y ) of class C2,α. Let us check that ψ is invertible by proving that
its derivative does not vanish. It is given by
ψ0(s) = s˜ (Y (s))Y 0(s).
We can deduce the expression of s˜0 by inverting
T0(s, τ) = [γ0(s) + τn0(s) n(s)]
0
(3.4)
(3.5)
and we easily obtain
∇s˜(x) = −γ0(s˜(x))/det T0(x).
Since T is a diffeomorphism, det T0 does not vanish so that ψ0 vanishes if and only if 0 = γ0(s) · Y 0(s). It is not
the case if η1 is small enough since
|γ0(s) · Y 0(s) − 1| = |γ0(s) · (Y 0(s) − γ0(s))k ≤ kY 0(s) − γ0(s)k ≤ C kΘ − IdkC
1
where we used Y 0 = DΘ γ0 for the last inequality. Therefore ψ is invertible and is a bijection from [0, L) into
[ψ(0), ψ(0) + L). Plugging ψ−1(s) in place of s into (3.3) leads to
Y (ψ−1(s)) = γ(s) + dΘ(s)n(s)
(3.6)
k−1,α
where we set dΘ(s) = τ˜(Y (ψ−1(s))). We check that dΘ is of class C
.

818
M. DAMBRINE AND M. PIERRE
We finish the proof of Lemma 2.2 by obtaining the estimate on dΘ when k = 3. We will use the following
technical lemma whose proof is left to the reader:
2,α
Lemma 3.1. Let p, q ≥ 1, A : Rp → Rp of class C2,α, β : Rq → Rp of class C
and B : Rp → Rp of class
2,α
3,α
2,α
C
3,α. Then, there exists C1 = C1(kβkC ) and C2 = C2(kBkC , kAkC ) such that
2,α
2,α
kA ◦ βkC ≤ C1 kAkC
,
(3.7)
(3.8)
2,α
2,α
kB ◦ A − BkC ≤ C2 kA − IdkC
.
Remark. Note that C2 involves the third derivative of B.
We first apply twice the above lemma to bound the C2,α-norm of ψ = s˜(Θ(γ)) by a constant depending on
2,α
the data and on η1 (we use here that s˜ is of class C
and therefore that γ is of class C3,α). We deduce that
the C2,α-norm of ψ−1 is also bounded by a similar constant.
Next we apply the above lemma with A = dΘ(ψ) and β = ψ−1 to get
kdΘkC2,α ≤ kdΘ(ψ)kC2,α
.
Then we use
dΘ(ψ(s)) = τ˜ ◦ Θ ◦ γ(s) = τ˜ ◦ Θ ◦ γ(s) − τ˜ ◦ γ(s) = [τ˜ ◦ Θ − τ˜] ◦ γ,
and we apply again the first part of the above lemma with A = τ˜ ◦ Θ − τ˜ and β = γ to obtain
2,α
2,α
kdΘ ◦ ψkC ≤ C kτ˜ ◦ Θ − τ˜kC
.
Finally we use the second part of the above lemma to conclude (recall that by previous remarks, here τ˜ is of
class C3,α).
3.2. Proof of Proposition 2.3
We will construct the function mΘ from R2 into R so that XΘ = mΘn be divergence-free. Here, n denotes
the extension of the unit normal to Tτ as defined in (2.10). We will use the local coordinates (s, τ) introduced
1
in Lemma 1 to first define m in a neighborhood of ∂Ω0. Let us compute div(n) by differentiating
x = γ(s˜(x)) + τ˜(x)n(x)
with respect to x:
Id = Dγ(s˜(x))Ds˜(x) + n(x)Dτ˜(x) + τ˜(x)Dn(x).
We use the expressions of the derivatives of s˜, τ˜ obtained above (∇s˜ is given in (3.5) and ∇τ˜(x) = n(x)) and
we take the trace of the latter equality to get
2 = −(detT0(x))−1 + 1 + τ˜(x)div(n)(x).
By using also (see 3.4),
det T0(s, τ) = det(γ0, n)(s) + τdet(n0, n)(s) = −1 + τdet(n0, n)(s),
we deduce the expression of div(n):
det(n0, n)(s)
div(n)(s, τ) = τ−1(1 + (detT0)−1) =
·
(3.9)
τdet(n0, n)(s) − 1

ABOUT STABILITY OF EQUILIBRIUM SHAPES
819
From now on, we introduce the notation a(s) := det(n, n0)(s). Note that a(s) is exactly the curvature at γ(s)
2,α
of the curve ∂Ω0 seen from inside. It is of class C
since ∂Ω0 is assumed to be of class C4,α. An important
remark on (3.9) is that
∀(s, τ) ∈ Tτ , 1 + τa(s) > 0,
1
since det T0 = 0 on Tτ . We fix τ2 ∈ (0, τ1) (depending only on ∂Ω0 and τ1) such that
1
¯
∀(s, τ) ∈ Tτ , 1 + τa(s) ≥ cτ > 0.
(3.10)
(3.11)
2
2
For vector fields of the form X = m(s, τ)n, we have
a(s)m(s, τ)
1 + τa(s)
divX = (∇m, n) + mdivn = ∂τ m(s, τ) +
so that X = mn will be divergence-free if it satisfies:
,
ꢁ
ꢂ
1 + τa(s) ∂τ m(s, τ) + a(s)m(s, τ) = 0.
(3.12)
(3.13)
This leads to
f(s)
m(s, τ) =
1 + τa(s)
where f is to be determined as follows: let Θ ∈ V(η2) with η2 = min{η1, τ2} (where η1, τ2 are defined in Lemma 1
and 3.10); then f should be such that the following boundary conditions be satisfied
Φ(0, ∂Ω0) = ∂Ω0, Φ(1, ∂Ω0) = Θ(∂Ω0)
where Φ(t, .) is the flow of X, that is the solution of
(3.14)
ꢀ
∂tΦ(t, x)
Φ(0, x)
=
=
X(Φ(t, x)), t > 0,
x.
Since X = mn, the trajectories of Φ(t, x) starting at x = γ(s) ∈ ∂Ω0 are parallel to n, at least for small t, so
that
Φ(t, γ(s)) = γ(s) + τ(s, t)n(s).
The above system reduces to the scalar ordinary differential equation in τ(., .)
∂tτ(s, t) = m(s, τ(t, s)),
(3.15)
(3.16)
(3.17)
and the boundary conditions (3.14) become
τ(s, 0) = 0, τ(s, 1) = dΘ(s),
where dΘ is defined in Lemma 1. According to (3.13), the equation (3.16) is equivalent to the existence of a
function C(s) such that
1
2
a(s)τ(s, t)2 + τ(s, t) = f(s)t + C(s).
(3.18)

820
M. DAMBRINE AND M. PIERRE
Now f, C have to satisfy (3.17): the condition τ(s, 0) = 0 leads to C(s) = 0 and the other one to
ꢃ
ꢄ
1
2
f(s) = − a(s)dΘ(s)2 + dΘ(s) .
(3.19)
It remains to prove that, with this choice of f, the equation (3.18) does have a solution τ(s, t) for t ∈ [0, 1]. It
is indeed the case since it is a quadratic equation in τ(s, t) with discriminant
ꢃ
ꢄ
∆(t) = 1 + a(s)2dΘ(s)2 + 2a(s)dΘ(s) t.
This quantity is linear in t, nonnegative at t = 0 as well as at t = 1 since:
ꢃ
ꢄ
2
∆(1) = a(s)dΘ(s) + 1 .
Recall that, by (3.10) and η2 ≤ η1, 1 + a(s)dΘ(s) ≥ cτ . We deduce the existence on [0, 1] of a solution to (3.18)
2
given by
ꢀ
p
ꢃ
ꢄ
τ(s, t) = − 1 + 1 + t(a(s)2dΘ(s)2 + 2a(s)dΘ(s)) /a(s), if a(s) = 0
τ(s, t) = tdΘ(s), if a(s) = 0.
(3.20)
At this step, for all Θ ∈ V(η2), we have constructed a divergence-free vector-field X = mn in the neighbor-
hood Tτ of ∂Ω0 where m, given by the formulas (3.13, 3.19) is of class C2,α. We will now use it to define the
1
2
divergence-free vector-field XΘ on the whole space R (see also the remark at the end of this paragraph).
We denote by ζ a C0∞(R2, R)-function, identically equal to 1 on Tτ and with compact support in Tτ . Recall
2
1
3,α
that Φ([0, 1] × ∂Ω0) ⊂ Tτ . Since div(X) = 0 on Tτ , there exists ξ ∈ C
locally so that X = (ξy, −ξx). But,
as proved e.g. in [13], thi2s is valid globally on Tτ since
1
1
Z
Z
Z
L
0 =
(XΘ, n) =
m =
m(s, 0) ds,
(3.21)
∂Ω0
∂Ω0
0
as we check below. Then we set
XΘ := ((ξζ)y, −(ξζ)x).
Obviously XΘ coincides with X = mn on Tτ and, by construction of m, satisfies all the conclusions of
2
Proposition 2.3.
To obtain the final estimate on ΦΘ, recall first that, at least on T (η2)
f(s˜)
1 + τ˜ a(s˜)
1
2
XΘ
=
n(s˜), f = − ad2Θ − dΘ.
By using Lemma 3.1 and the estimate of Lemma 2.2, we deduce
2,α
2,α
2,α
kXΘkC ≤ C kdΘkC ≤ C kΘ − IdkC
.
Again by Lemma 3.1
ꢁ
ꢂ
2,α
2,α
2,α
kXΘ(ΦΘ(t))kC ≤ C kΦΘ(t)kC
kXΘkC
.
We then deduce the announced estimate on ΦΘ from the latter estimates and the fact that ΦΘ is the flow of XΘ.

ABOUT STABILITY OF EQUILIBRIUM SHAPES
821
It remains to check (3.21). This is coming from the assumption that meas(Θ(Ω0)) = meas(Ω0). Indeed, this
implies
Z
0 = L{ γ(s) + dΘ(s)n(s) ∧ γ(s) + dΘ(s)n(s) − γ(s) ∧ γ(s)0} ds
ꢁ
ꢂ
ꢁ
ꢂ0
0
which gives (we drop the s-dependence)
Z
L
0 =
dΘ(n ∧ γ0 + γ ∧ n0) + dΘ0 γ ∧ n + dΘ2 n ∧ n0
0
or after integrating the term in d0 by parts and using the notation a(s) introduced before:
Z
L
0 =
2dΘ + d2Θa(s).
0
But, according to (3.13, 3.19), this is exactly (3.21).
Remark. About the extension of X outside Tτ : the fact that divX = 0 on Tτ implies that, for any extension
1
1
ꢁ
ꢂ
of X to R2, t → meas Φ(t, Ω0) is linear in t since the second derivative vanishes. As Φ(1) is a diffeomorphism
ꢁ
ꢂ
ꢁ
ꢂ
and Φ(1, ∂Ω0) = Θ(∂Ω0), then Φ(1, Ω0) = Θ(Ω0). Since meas Θ(Ω0) = meas(Ω0), then t → meas Φ(t, Ω0) is
constant. Therefore, any extension of X would be convenient for our purpose. This remark is actually to be
used in the general case when ∂Ω0 is a finite number of Jordan’s curves and Tτ a neighborhood of ∂Ω0. Then,
1
if Ω0 is for instance not connected, we would not be able to extend X as a free-divergence vector field on R2.
3.3. Some more estimates
With the notations of the previous section, we have:
Proposition 3.2. Under the assumption of Proposition 2.3, there exists a constant C depending only on the
data such that for all Θ ∈ V(η2) and all t ∈ [0, 1]
L2(∂Ω0)
L2(∂Ω0)
kmΘ(ΦΘ(t)) − mΘk
≤ CkmΘk
kΘ − Idk2,α
.
(3.22)
kmΘ(ΦΘ(t)) − mΘk
≤ CkmΘk
kΘ − Idk2,α
.
(3.23)
H1/2(∂Ω0)
H1/2(∂Ω0)
Proof. Recall (see 3.13, 3.20) that, in terms of local coordinates and with the previous notations
mΘ(ΦΘ(t, γ(s))) − mΘ(γ(s)) = m(s, τ(s, t)) − m(s, 0) = f(s)[(1 + τ(s, t)a(s))−1 − 1],
(3.24)
(3.25)
ꢃ
ꢄ
1
2
f(s) = m(s, 0) = mΘ(γ(s)) = − a(s)dΘ(s)2 + dΘ(s) .
As a consequence
kmΘ(ΦΘ(t, .)) − mΘ(.)k
sup |(1 + τ(s, t)a(s))−1 − 1)|.
2
2
≤ kmΘ(.)k
L (∂Ω0)
L (∂Ω0)
s∈[0,L]
We know from (3.10) that 1 + τ(s, t)a(s) is bounded from below by cτ on {(s, t) ∈ [0, L] × [0, 1]}. Therefore,
2
since (1 + τa)−1 − 1 = −τa/(1 + τa)
sup |(1 + τ(s, t)a(s))−1 − 1)| ≤ cτ−1kak∞
sup
|τ(s, t)|.
2
s∈[0,L]
(s,t)∈[0,L]×[0,1]

822
M. DAMBRINE AND M. PIERRE
Using (3.20), we bound τ from above by
|τ(s, t)| ≤ |a(s)|dΘ(s)2 + 2|dΘ(s)| ≤ CkdΘk∞ ≤ CkΘ − Idk2,α
,
the latter inequality coming from (2.9). This proves the first estimate of Proposition 3.2. For the second one,
we use the following lemma:
1
Lemma 3.3. Let v ∈ H1/2(∂Ω0), w ∈ C (∂Ω0). Then, v w belongs to H1/2(∂Ω0) and
1
kv wkH1/2
≤ CkvkH1/2
kwkC (∂Ω )
(3.26)
(∂Ω0)
(∂Ω0)
0
for some constant C depending only on Ω0.
We postpone the proof of this lemma and continue the proof of the proposition. From this lemma and the
expression (3.24), we obtain:
≤ kmΘ(.)kH1/2(∂Ω )k(1 + τ(., t)a(.))−1 − 1)kC ([0,L])
.
1
kmΘ(ΦΘ(t, .)) − mΘ(.)k
H1/2(∂Ω0)
0
By differentiation with respect to s, we see that
k∂s(1 + τ(., t)a(.))−1kL∞
≤ c2 k∂s(a(.)τ(., t))k∞.
([0,L])
τ2
Using the expression of τ in (3.20), we obtain the existence of C such that
1
k∂s(a(.)τ(., t))k∞ ≤ CkdΘkC ≤ CkΘ − Idk2,α
.
This finishes the proof of the proposition.
1
Proof of Lemma 3.3. Let V be an harmonic extension of v to Ω0 and W a C -extension of w to Ω0 so that, for
some constant C depending only on Ω0,
kV kH1(Ω ) ≤ Ckvk
, kWkC (Ω )
1
≤ CkwkC (∂Ω )
1
.
H1/2(∂Ω0)
0
0
0
We then have
We now use
1
kv wkH1/2
≤ C kV WkH
.
(Ω0)
(∂Ω0)
2
2
∞
kV WkL (Ω ) ≤ kV kL (Ω )kWkL
,
(Ω0)
0
0
2
2
∞
2
2
∞(Ω0)
k∇V WkL (Ω ) ≤ k∇V kL (Ω )kWkL (Ω ), kV ∇WkL (Ω ) ≤ kV kL (Ω )k∇WkL
0
0
0
0
0
and the estimate (3.26) follows.
We finish this section by stating some more geometric estimates.
Proposition 3.4. There is a constant C > 0 such that for all Θ ∈ V(η2) and all t ∈ [0, 1]:
1. kDΦΘ(t) − IdR kL + kD2ΦΘ(t)kL ≤ C kΘ − IdR k2,α
∞
∞
2
2
ꢃ
ꢄ
2. kDΦΘ(t)−1 − IdR kL + kD DΦΘ(t)−1 kL ≤ C kΘ − IdR k2,α
3. if nt denotes the unit normal field to Ωt,
∞
∞
2
2
1
2
knt(ΦΘ(t)) − nk
≤ C kΘ − IdR k2,α
(3.27)
C (∂Ω0)

ABOUT STABILITY OF EQUILIBRIUM SHAPES
823
t
4. if J(t) = det DΦΘ(t) | DΦΘ(t)−1n| then
∞
2
kJ(t) − 1kL
≤ C kΘ − IdR k2,α
.
(3.28)
(∂Ω0)
Proof of Proposition 3.4. Note first that, thanks to the estimate (2.11), it is sufficient to bound each of the four
2
expressions in the Proposition by C kΦΘ(t) − IdR k2,α
.
The first estimate comes from the definition of the C2,α-norm applied to ΦΘ(t) − IdR
.
2
2
For the second one, use that for all x ∈ R
∞
X
DΦΘ(t)−1(x) = IdR
+
[IdR − DΦΘ(t)(x)]n.
(3.29)
2
2
n=1
It follows that (up to still reducing η2)
2
kDΦΘ(t)(x) − IdR
k
2
kDΦΘ(t)−1(x) − IdR
2
k ≤
≤ CkDΦΘ(t)(x) − IdR k.
2
1 − kDΦΘ(t)(x) − IdR k
For the estimate on the second derivative, we differentiate (3.29) and bound norms from above similarly.
For the third estimate, let us denote
z(s) := ∂s[ΦΘ(t, γ(s))] = DΦΘ(t, γ(s))γ0(s).
ꢁ
ꢂ
Since n(s) and nt ΦΘ(t, γ(s)) are respectively deduced from γ0(s) and z(s)/|z(s)| by a rotation of angle −π/2,
we have
z
1
1
knt(ΦΘ(t)) − nk
= k
− γ0kC (∂Ω )
.
C (∂Ω0)
t
|z|
We will estimate the right-hand side. We have
||z(s)| − 1| = ||z(s)| − |γ0(s)|| ≤ |z(s) − γ0(s)| ≤ kDΦΘ(t, γ(s)) − Idk ≤ kΦΘ(t) − IdR k2,α
If η2 is chosen small enough – and we assume it here – this implies
.
2
||z(s)| − 1| ≤ C kΘ − IdR k2,α ≤ 1/2, |z(s)| ≥ 1/2, ||z(s)|−1 − 1| ≤ C kΘ − IdR k2,α
2
2
ꢅ
ꢅ
ꢅ
ꢅ
ꢅ
ꢅ
ꢅ
ꢅ
z(s)
|z(s)|
0
−1
0
2
− γ (s) ≤ |z(s)| |z(s)| − 1 + |z(s) − γ (s)| ≤ C kΘ − IdR k2,α
,
whence the L∞-estimate of z|z|−1 − γ0. For the L∞-estimate of its derivative, note first that
z0(s) = D2ΦΘ(t, γ(s))(γ0(s), γ0(s)) + DΦΘ(t, γ(s))γ00(s)
so that
|z0(s) − γ00(s)| ≤ kD2ΦΘ(t)k∞ + kDΦΘ(t) − Idk|γ00(s)| ≤ C kΦΘ − Idk2,α
∂s(z(s)|z(s)|−1) = z0(s)|z(s)|−1 − z(s)|z(s)|−3hz(s), z0(s)i.
.
Now, we use

824
M. DAMBRINE AND M. PIERRE
We treat this expression as before, using in particular the estimate of ||z(s)|−1 − 1|, the only new term being
hz(s), z0(s)i which we estimate as follows:
|hz, z0i| = |hz, z0i − hγ0, γ00i| ≤ |hz − γ0, z0i + hγ0, z0 − γ00i|
2
and, as shown before, we obtain a bound from above by C kΘ − IdR k2,α
.
Finally, the variation of J(t) in (3.28) is then easily estimated.
4. Properties of uΩ
t
Here again we fix Θ ∈ V(η2). We consider the solution uΩ of the following Dirichlet problem on the moving
t
domain Ωt = ΦΘ(t, Ω0)
ꢀ
−∆uΩ = k in Ωt,
uΩ = t0
on ∂Ωt.
(4.1)
t
We will often write u(t) for uΩ . We will now make estimates on the “transported” solution u˜(t) defined on the
t
fixed domain Ω0 by
u˜(t) = uΩ ◦ ΦΘ(t) (= u(t) ◦ ΦΘ(t)).
t
Then u˜(t) is solution of a new problem on the fixed domain, namely
−L(Θ, t) u˜(t) = k ◦ ΦΘ(t) on Ω0, u˜(t) = 0 on ∂Ω0,
where L(Θ, t) is the differential operator explicitly given by:
(4.2)
L(Θ, t)
=
=
ai,j(t)Di,j + bi(t)Di
∂1Ψti∂1Ψjt + ∂2Ψti∂2Ψtj Di,j + ∂12,1Ψti + ∂22,2Ψti Di0
(4.3)
(4.4)
ꢃ
ꢄ
ꢃ
ꢄ
where Ψt = ΦΘ(t)−1. We then have the following main estimate.
Proposition 4.1. There exists a function ω : [0, 1] → R with ω(0+) = limr↓0 ω(r) = 0 (“modulus of continu-
ity”) such that:
ꢁ
ꢂ
sup ku˜(t) − uΩ kC (Ω ) ≤ ω kΘ − IdR
2
k2,α
.
(4.5)
2
¯
0
0
t∈[0,1]
Remark. The modulus of continuity ω depends on the regularity of the right-hand side k in (4.1). If we assume
2
2,α
2
k of class C , then one can choose ω(η) = C η and even have the C
norm instead of the C norm. More
comments and indications of the proof of this remark may be found after the proof of the proposition.
Proof of Proposition 4.1. The main idea is here to use Schauder’s estimates in H¨older spaces for the solution of
the Dirichlet problem governed by an uniformly elliptic operator with H¨older continuous coefficients. Indeed,
we have:
Lemma 4.2. Assume η2 is small enough. Then, there is a constant U depending only on the data and on the
C
2,α-norm of Θ such that for all t ∈ [0, 1]
ku˜(t)k2,α,Ω ≤ U.
(4.6)
0

ABOUT STABILITY OF EQUILIBRIUM SHAPES
825
Proof of Lemma 4.2. Let us first remark that the operators L(Θ, t) are uniformly elliptic. Indeed, the part of
second order of L(Θ, t) can be written matricially as
ꢆ
ꢇ ꢆ
ꢇ
∂1Ψ1t ∂2Ψ1t
∂1Ψ2t ∂2Ψ2t
∂1Ψ1t ∂1Ψ2t
∂2Ψ1t ∂2Ψ2t
t
A(Θ, t) =
= DΨt DΨt.
This proves that A(Θ, t) is a symmetric nonnegative matrix. Since det DΨt = 1 = 0, it is positive definite.
The smallest eigenvalue being a continuous function of t, it is uniformly bounded from below on the compact
interval [0, 1]. Actually the bound depends only on the C2,α-norm of Θ if η2 is small enough since
t
t
kDΨt DΨt − Idk
≤
≤
kDΨt DΨt − DΨtk + kDΨt − Idk
kΨt − Idk2,α ≤ C kΘ − Idk2,α ≤ C η2.
0,α
Now, since Ψt = ΦΘ(t)−1 is of class C2,α, the coefficients of L(Θ, t) are C
with a norm depending only on the
C
2,α-norm of ΦΘ(t)−1, that is also on the C2,α-norm of Θ. From classical Schauder’s estimates applied to the
2,α
equation (4.2) (see for example [10]), there is constant C depending only on the data and on the C
-norm of
Θ such that
ku˜(t)k2,α,Ω ≤ C kk ◦ ΦΘ(t)k0,α,Ω
.
0
0
0,α
0,α
Since k ∈ C
and ΦΘ(t) ∈ C2,α, then k ◦ ΦΘ(t) ∈ C
and the Lemma 4.2 is proved.
We can now finish the proof of Proposition 4.1. We define ω as
∀η ∈]0, η2[, ω(η) :=
sup
ku˜(t) − uΩ kC (Ω )
.
¯
0
2
0
Θ∈V(η),t∈[0,1]
Lemma 4.2 guarantees that this quantity is well defined, and we just need to prove that ω(0+) = 0. This follows
2,α
2
+
¯
from the compact embedding of C (Ω0) into C (Ω0). Indeed, if ω(0 ) = 0, there are a > 0 and sequences
tn ∈ [0, 1], Θn ∈ V(η2), Ωn = ΦΘ (tn, Ω0), u˜n = uΩ ◦ ΦΘ (tn) such that
n
n
n
2
kΘn − IdR k2,α ≤ 1/n,
(4.7)
(4.8)
ku˜n − uΩ kC (Ω ) ≥ a > 0.
2
¯
0
0
2,α
2
¯
¯
The sequence u˜n is bounded in C (Ω0) and by compactness, there is a subsequence converging in C (Ω0) to
2
¯
some ulim ∈ C (Ω0) and a subsequence of tn converging to some tlim ∈ [0, 1]. We then remark that
h
i
h
i
L(Θn, tn)u˜n − ∆ulim = L(Θn, tn) − ∆ u˜n + ∆ u˜n − ulim
.
Passing to the limit as n → ∞ shows that ulim is solution of the equation
∆u
u
=
=
k
0
in Ω0,
on ∂Ω0.
2
¯
This is in contradiction with (4.8) as the Dirichlet problem has a unique solution in C (Ω0) namely uΩ . This
0
proves that ω(0+) = 0.
2
¯
Remark. As a consequence of Proposition 4.1, u˜(t) is continuous into C (Ω0) at t0 = 0. Since t0 = 0 is
not particular for problem (4.1), the same is true for t → u(t) ◦ ΦΘ(t) ◦ ΦΘ(t0)−1 at t = t0 ∈ [0, 1] and, by
composition:
2
¯
t ∈ [0, 1] → u˜(t) ∈ C (Ω0) is continuous.
(4.9)

826
M. DAMBRINE AND M. PIERRE
Remark. Another approach to estimate a norm of the difference u˜(t) − uΩ would be the following. For
0
˜
simplicity, we set u0 := uΩ and k(t) := k ◦ ΦΘ(t). We have from (4.2) and from (4.1) with t = 0
0
˜
−∆(u˜(t) − u0) = k(t) − k + [L(Θ, t) − ∆](u˜(t)).
(4.10)
Since u˜(t) − u0 = 0 on ∂Ω0, we can apply the Schauder’s estimates again to (4.10)
˜
ku˜(t) − u0k2,α,Ω ≤ C kk(t) − kk0,α,Ω + k[L(Θ, t) − ∆](u˜(t))k0,α,Ω
0,α
.
0
0
0
2,α
Thanks to the C
estimate of Lemma 4.2, the C
norm of the second term of the right-hand side is easily
estimated by kΦΘ(t)−Idk2,α. But, even if k(t)−k is indeed in C0,α, its C0,α-norm cannot be estimated in terms
˜
of kΦΘ(t) − Idk2,α
.
What one can at least say is that:
α
˜
kk(t) − kk∞ ≤ kkk0,α kΦΘ(t) − Idk∞.
By interpolation, for ꢀ ∈ (0, α), we get an estimate of kk(t) − kk0,ꢀ with a modulus of continuity in ηα−ꢀ. This
˜
yields (4.1) and even a C2,ꢀ-estimate with an explicit modulus of continuity. One can also check that, if k is of
2
0,α
˜
class C , then, the C -norm (and even the Lipschitz-norm) of k(t) − k can be estimated by C kΦΘ(t) − Idk2,α
.
Then one obtains a C2,α-estimate of u˜(t) − uΩ
.
0
5. Proof of the theorems
ꢁ
ꢂ
Again, we fix Θ a diffeomorphism in V(η2) such that meas Θ(Ω0) = meas(Ω0). As in the previous sections,
we consider the function mΘ, the vector field XΘ = mΘ n, ΦΘ, the domains Ωt = ΦΘ(t, Ω0), the solutions
u(t) = uΩ , u˜(t) = u(t) ◦ ΦΘ(t). As announced in the introduction, we study the second derivatives of
t
t ∈ [0, t] → E(Ωt) + Λ meas(Ωt). Since, ∀t ∈ [0, 1], meas(Ωt) = meas(Ω0), it coincides with the second
derivative of t → E(Ωt). We denote
d
dt
2
eΘ(t) := E(Ωt) = − ku(t, .)kH
.
01(Ωt)
To compute the second derivative, we use the following classical lemma [6,16,21].
2
2
Lemma 5.1. Let H : [0, 1] × R → R be such that H, ∂tH, ∇H ∈ C([0, 1]; L1(R )). Then
Z
Z
ꢃ
ꢄ
d
dt
H(t, x)dx =
∂tH(t, x) + div{H(t, x)XΘ(x)} dx.
(5.1)
Ωt
Ωt
Remark. As usual in shape differentiation, this lemma may be extended to functions defined only on Ωt. If for
instance (this will be enough for us here),
1
0
0
1
¯
¯
H(t) ◦ ΦΘ(t) ∈ C ([0, 1]; C (Ω0)) ∩ C ([0, 1]; C (Ω0)),
(5.2)
0
¯
then we may define ∂tH(t) ∈ C (Ωt) in such a way that (5.1) remains valid. This may be done through a given
k
k
2
¯
linear continuous extension operator P from C (Ω0) into C (R ), k = 0, 1, 2. We set
ꢁ
ꢂ
H(t) := P H(t) ◦ ΦΘ(t) ◦ Φ−Θ1(t),
(5.3)
¯
¯
¯
¯
¯
and, by definition ∂tH(t) := ∂tH(t) on Ωt (note that H(t) and H(t) also coincide on Ωt). Applying Lemma 5.1
¯
as stated above to H yields the formula (5.1) for H.

ABOUT STABILITY OF EQUILIBRIUM SHAPES
827
2
¯
Here, u˜(t) = u(t) ◦ ΦΘ(t) is continuous into C (Ω0) (see (4.9)) so that its extension u¯ is well-defined. As we
will see below, its derivative ∂tu(= ∂tu¯) is well-defined and
0
1
¯
∂tu ◦ ΦΘ(t) ∈ C (0, 1; C (Ω0)).
(5.4)
(5.5)
We will use this in the next computations and we will often write u for u¯.
By differentiating formally (4.1) with respect to t, we see that ∂tu(t) should be solution of

−∆(∂tu(t)) = 0
in Ωt,
on ∂Ω0
on ∂Ωt.

∂tu(t, ΦΘ(t)) + h∇u(t, ΦΘ(t)), XΘ(ΦΘ(t))i = 0
or also ∂tu(t) + h∇u(t), XΘi = 0

Since ∇u(t) ∈ C1,α(∂Ωt), classical regularity results from Schauder theory (see e.g. [10]) ensure the existence of
a C1,α-solution to this equation with a C1,α-norm uniformly bounded for t ∈ [0, 1]. Moreover, its composition
1
¯
with ΦΘ(t) is continuous with values into C (Ω0) since it is the case of t → ∇u(t) ◦ ΦΘ(t). We easily deduce
that this solution is not hing but ∂tu and the formal computation is justified as well as (5.4).
Derivation of the magnetic energy
2
2
We apply Lemma 5.1 to H = |∇u| = |∇u¯| .
Z
Z
ꢃ
ꢄ
d
dt
2
2
ku(t, .)kH1(Ω ) = 2
h∇u(t, .), ∇∂tu(t, .)idx +
div |∇u(t, .)| XΘ dx.
t
0
Ωt
Ωt
The first term of this sum vanishes since by Green’s formula, we have:
Z
Z
Z
h∇u(t, .), ∇∂tu(t, .)idx = −
u(t, .)∆x∂tu(t, .)dx +
u(t, .)h∇∂tu(t, .), ntidσ.
Ωt
Ωt
∂Ωt
This is equal to zero since u(t) = 0 on ∂Ωt and ∆∂tu = 0 on Ωt; (recall that we denote by nt the unit normal
derivative to Ωt). So we obtain:
Z
ꢃ
ꢄ
d
dt
2
2
ku(t, .)kH1(Ω )
=
div |∇u(t, .)| XΘ dx.
(5.6)
t
0
Ωt
As a first consequence, using Green’s formula, we have
Z
d
2
[Λ meas(Ωt) + E(Ωt)] =
dtt=0
[Λ − |∇uΩ | ]hXΘ, nidσ.
0
∂Ω0
2
Since Ω0 is a critical shape, we see on this formula (valid for any vector field in place of XΘ) that Λ = |∇uΩ
on ∂Ω0 as announced in (2.4).
|
0
Next, applying Lemma 5.1 (at least formally, see below) to the expression (5.6) leads to the second derivative
Z
Z
d2
dt2
ꢃ
ꢄ
ꢃ
ꢄ
2
2
2
ku(t, .)kH1(Ω )
=
div ∂t|∇u(t, .)| XΘ dx +
div{div |∇u(t, .)| XΘ XΘ}dx.
(5.7)
t
0
Ωt
Ωt

828
M. DAMBRINE AND M. PIERRE
As such, these expressions may not be defined, but we integrate by parts and we set
e0Θ0 (t) = −2B(t) − A(t),
(5.8)
Z
B(t) =
h∇∂tu(t), ∇u(t)ihXΘ, ntidσ,
∂Ωt
∂Ωt
Z
2
A(t) =
h∇|∇u(t)| , XΘihXΘ, ntidσ,
where we used that div XΘ = 0 to simplify the expression of A(t). To justify this computation, we may apply
ꢃ
ꢄ
2
N
Lemma 5.1 to Hn(t) = div |∇Un(t)| XΘ where Un are regularized approximations by convolution in R of u¯.
Because of the regularity of u¯ at the boundary, the convergence under the integrals holds uniformly,
The study of A(t)
We first change variable in the integral on the moving boundary to get an integral on the fixed boundary.
The Jacobian we need is estimated in (3.28) of Proposition 3.4. We use also that
ꢃ
ꢄ−1
∇u(t) ◦ ΦΘ(t) =t DΦΘ(t) ∇u˜(t)
2
and the similar formula with |∇u(t)| in place of u(t).
˜
We will simply write Φt for ΦΘ(t), X for XΘ, m for mΘ, X(t) for XΘ ◦ ΦΘ(t), m˜ (t) for mΘ ◦ ΦΘ(t) and
n˜t(t) for nt ◦ ΦΘ(t). Note that by (2.10), (3.15), n ◦ ΦΘ(t) = n on T (η2). We obtain
Z
2
A(t)
=
=
=
h∇|∇u(t)| , XihX, ntidσ,
∂Ωt
∂Ω0
∂Ω0
Z
h DΦ−t 1∇| DΦt−1∇u˜(t)| , X(t)ihX(t), n˜t(t)iJ(t) dσ,
t
t
2
˜
˜
Z
m˜ (t)2h DΦ−t 1∇| DΦt−1∇u˜(t)| , nihn, n˜t(t)iJ(t) dσ.
t
t
2
The goal is now to estimate A(t) − A(0). We set
ꢀ
−1
−1
t
t
2
a0(t) := m˜ (t), a1(t) := h DΦt ∇| DΦt ∇u˜(t)| , ni
a2(t) := hn, n˜t(t)i, a3(t) := J(t).
We denote by C any constant depending only on the C2,α-norm of Θ and we set η := kΘ − IdR
k
2,α. We have
(5.9)
2
the following estimates for i in {2, 3} and for all t in [0, 1]
L∞(∂Ω0)
L∞(∂Ω0)
kai(t)k
≤ C, kai(t) − ai(0)k
≤ C η.
This is coming from (3.27) in Proposition 3.4. For i = 1, we will prove below the following for all t in [0, 1]
∞
∞
(∂Ω0)
ka1(t)kL
≤ C,
ka1(t) − a1(0)kL
≤ C ω(η),
(5.10)
(∂Ω0)
where ω is the modulus of continuity appearing in Proposition 4.1. Next, we also have for all t in [0, 1]
L2(∂Ω0)
L2(∂Ω0)
ka0(t)k
≤ CkmkL2(∂Ω )
,
ka0(t) − a0(0)k
≤ C η kmkL2(∂Ω )
.
(5.11)
0
0
This is the content of the first part of Proposition 3.2. Now, we write
Z
Z
Y
Y
Y
A(t) − A(0) =
[a0(t)2 − a0(0)2]
ai(t) +
a0(t)2[
ai(t) −
ai(0)].
(5.12)
∂Ω0
∂Ω0
i=1,2,3
i=1,2,3
i=1,2,3

ABOUT STABILITY OF EQUILIBRIUM SHAPES
829
Q
Q
2
The second integral is bounded by C kmkL2(∂Ω )
k
i=1,2,3 ai(t) − i=1,2,3 ai(0)k∞. We easily check that
0
Y
Y
X
k
ai(t) −
ai(0)k∞ ≤ C
kai(t) − ai(0)k∞ ≤ C (ω(η) + η),
i=1,2,3
i=1,2,3
i=1,2,3
the last inequality coming from (5.9, 5.10). Next we will assume for simplicity that ω(η) ≥ C η.
The first integral in (5.12) is bounded above by
Y
2
2
2
k
ai(t)k∞ka0(t) − a0(0)k
ka0(t) + a0(0)k
≤ C η kmk
,
L2(∂Ω0)
L (∂Ω0)
L (∂Ω0)
i=1,2,3
where we used (5.11) for the last inequality. Finally, for all t in [0, 1], we get
2
|A(t) − A(0)| ≤ C ω(η)kmkL2(∂Ω )
.
(5.13)
0
2
It remains to prove (5.10). The L∞-bound on a1(t) is obvious from the C -estimates of Lemma 4.2 on u˜(t). For
−1
t
2
the L∞-estimate on the difference, inserting ∇| DΦt ∇u˜(t)| , we obtain a first bound by
−1
−1
t
t
2
2
C k DΦt − Idk∞ + C k∇| DΦt ∇u˜(t)| − ∇|∇u(0)| k∞.
2
Using the C -estimates of Proposition 4.1 for the second term, we prove that this is bounded by C ω(η).
The study of B(t)
Recall that
Z
B(t) =
h∇∂tu(t), ∇u(t)ihXΘ, ntidσ.
∂Ωt
As u(t) is constant along ∂Ωt, its gradient is normal and therefore
∇u(t) = h∇u(t), ntint.
If we denote ∂n u := h∇u(t), nti, we deduce that
t
h∇∂tu(t), ∇u(t)i = ∂n u h∇∂tu(t), nti.
(5.14)
(5.15)
t
Since also XΘ = m n, the boundary condition for ∂tu(t) may be rewritten (see (5.5))
∂tu(t) + m ∂n uhnt, ni = 0 on ∂Ωt.
t
Let us now introduce the Steklov-Poincar´e operator Ct on Ωt defined from H1/2(∂Ωt) into H−1/2(∂Ωt) as
follows: if z ∈ H1/2(∂Ωt), we consider the harmonic extension Z of z to Ωt and we define Ct(z) := h∇Z, nti.
For the properties of this operator, one can refer for example to [5]. A simple computation shows that,
Z
Z
2
z Ct(z) =
|∇Z| ,
∂Ωt
Ωt
this, at least for regular enough functions z (in general, the first integral is to be replaced by hz, Ct(z)iH1/2×H−1/2 ).
Then, if we set z(t) := −m ∂n uhnt, ni, we have by (5.5, 5.14, 5.15)
t
h∇∂tu(t), nti = Ct(z(t)), h∇∂tu(t), ∇u(t)i = ∂n u Ct(z(t)).
t

830
M. DAMBRINE AND M. PIERRE
This gives a new expression for B(t):
Z
Z
2
B(t) = −
z(t)Ct(z(t)) = −
|∇Z(t)| ,
∂Ωt
Ωt
˜
where Z(t) is the harmonic extension of z(t) to Ωt. We also denote Z(t) := Z(t) ◦ Φt, z˜(t) := z(t) ◦ Φt. We then
have to estimate
Z
Z
| DΦ−t 1∇Z(t)| ,
Ω0
2
t
2
˜
B(t) − B(0) =
|∇Z(0)| −
Ω0
(recall that the Jacobian det DΦt is here equal to 1). We denote again by C any constant depending only on
the C2,α-norm of Θ and we set again η := kΘ − IdR k2,α
.
2
Lemma 5.2. We have the following main estimates: for all t ∈ [0, 1]
kz˜(t)kH1/2
≤ C kmkH1/2
≤ C kmk
, kz˜(t) − z(0)kH1/2
≤ C ω(η)kmkH1/2
(∂Ω0)
(∂Ω0)
(∂Ω0)
(∂Ω0)
˜
˜
1
1
kZ(t)kH
, kZ(t) − Z(0)kH (Ω ) ≤ C ω(η)kmk
.
H1/2(∂Ω0)
H1/2(∂Ω0)
(Ω0)
0
Assuming this lemma, we obtain
−1
−1
t
t
˜
˜
2
2
|B(t) − B(0)| ≤ k DΦt ∇Z(t) + ∇Z(0)kL k DΦt ∇Z(t) − ∇Z(0)kL
ꢃ
ꢄ
k DΦ−t 1∇Z(t) − DΦt ∇Z(0)kL2 + k DΦ−t 1∇Z(0) − ∇Z(0)kL2
t
t
−1
t
˜
≤ C kmkH1/2
≤ C kmkH1/2
(∂Ω0)
ꢃ
ꢄ
˜
2
2
k∇Z(t) − ∇Z(0)kL + C ηk∇Z(0)kL
(∂Ω0)
≤ C kmkH1/2(∂Ω )[ω(η)kmkH1/2(∂Ω ) + ηkmkH1/2(∂Ω )].
0
0
0
This together with (5.8, 5.13) finishes the proof of Theorem 2.5. It only remains to prove Lemma 5.2.
Proof of Lemma 5.2. Recall that
z(t) := −m ∂n uhnt, ni, z˜(t) := z(t) ◦ Φt.
t
From Lemma 3.3, we have
kz˜(t)kH1/2
≤ kmkH1/2
k(∂n u ◦ Φt)hn˜t, nik
1
.
C (∂Ω0)
(∂Ω0)
(∂Ω0)
t
From Propositions 3.4 and 4.1, we easily deduce the first estimate on z˜ in Lemma 5.2. For the difference, we
write
z˜(t) − z(0) = z˜(t) + mh∇u0, ni = m [h∇u0, ni − (∂n u ◦ Φt)hn˜t, ni].
t
1
Again, by Lemma 3.3, we have to estimate the C -norm of h∇u0, ni − (∂n u ◦ Φt)hn˜t, ni which is bounded by
t
the sum
1
˜
1
kh∇u0, ni(1 − hn˜t, ni)kC + khnt, ni[h∇u0, ni − ∂n u ◦ Φt]kC
.
t
1
The first term is estimated as expected by C η thanks to (3.27). The second one depends on the C -norm of
−1
t
h∇u0, ni − ∂n u ◦ Φt = h∇u0, ni − h DΦt ∇u˜(t), n˜(t)i. We use Propositions 4.1 and 3.4 to estimate it by ω(η)
t
and the part concerning z˜ in the Lemma is complete.

ABOUT STABILITY OF EQUILIBRIUM SHAPES
831
Since Z(0) is harmonic on Ω0 with value z(0) at the boundary, we have
1
kZ(0)kH (Ω ) ≤ C kz(0)k
≤ C kmkH1/2
.
(∂Ω0)
H1/2(∂Ω0)
0
1
˜
The estimate on the H -norm of Z(t) − Z(0) starts with the equation
˜
L(Θ, t)(Z(t)) = 0 on Ω0,
which we rewrite
˜
˜
−∆(Z(t) − Z(0)) = [L(Θ, t) − ∆](Z(t)) on Ω0.
This implies that
˜
˜
1
−1
(Ω0)
kZ(t) − Z(0)kH
≤ C [kz˜(t) − z(0)k
+ k[L(Θ, t) − ∆](Z(t))kH
.
(5.16)
H1/2(∂Ω0)
(Ω0)
The first term has just been estimated by C ω(η)kmkH1/2
. As we check below, for all w ∈ H1(Ω0), we have
(∂Ω0)
1
−1(Ω0)
(Ω0)
k[L(Θ, t) − ∆](w)kH
≤ C η kwkH
.
(5.17)
We now decompose
˜
˜
[L(Θ, t) − ∆](Z(t)) = [L(Θ, t) − ∆][Z(t) − Z(0)] + [L(Θ, t) − ∆](Z(0)),
to obtain
˜
˜
1
−1(Ω0)
k[L(Θ, t) − ∆](Z(t))kH
≤ C ηkZ(t) − Z(0)kH (Ω ) + C η kmk
.
(5.18)
H1/2(∂Ω0)
0
Together with (5.16), this implies
˜
1
kZ(t) − Z(0)kH (Ω )(1 − C η) ≤ C ω(η)kmk
H1/2(∂Ω0)
0
from which we deduce the last estimate of Lemma 5.2.
It remains to prove (5.17). We do it by duality as follows. Let ψ ∈ C0∞, then
Z
Z
X
X
ꢃ
ꢄ
[L(Θ, t) − ∆]w ψ =
(ai,j(t) − ai,j(0))Di,j w +
(bi(t) − bi(0))Diw ψ,
Ω0
Ω0
i,j
i
where ai,j, bi are the coefficients of L(Θ, t) (see 4.3). Now
Z
ꢅ
ꢅ
ꢅ
Z
ꢅ
ꢅ
ꢅ
ꢅ
ꢅ
ꢅ
ꢅ
ꢅ
(ai,j (t) − ai,j(0))Di,jw ψ
=
[Dj(ai,j(t) − ai,j(0)) ψ + (ai,j(t) − ai,j(0))D ψ]D w ,
ꢅ
j
i
Ω0
Ω0
1
1
1
≤
≤
C kai,j(t) − ai,j(0)kC kψkH kwkH
,
0
C η kψkH1 kwkH1
,
0
Z
ꢅ
ꢅ
ꢅ
ꢅ
ꢅ
ꢅ
∞
2
1
(b (t) − b (0))D w ψ
≤
≤
C kbi(t) − bi(0)kL kψkL kwkH
,
i
i
i
Ω0
2
1
C η kψkL kwkH
.
The estimate (5.17) follows.

832
M. DAMBRINE AND M. PIERRE
Proof of Theorem 2.1. Let Θ ∈ V(η) where η is small enough so that Theorem 2.5 applies. We write Taylor
2
formula at order 2 for t → eΘ(t) + Λ meas(Ωt) which is of class C . As meas(Ωt) is constant and Ω0 is assumed
to be a critical point for the constrained functional, e0Θ(0) = 0 and the Taylor’s formula writes
Z
1
eΘ(1) = eΘ(0) +
(1 − t)e0Θ0 (t)dt.
0
We have
Z
2
e0Θ0 (0) = −2B(0) − A(0) =
2z(0) C0(z(0)) − m2h∇|∇u0| , ni.
∂Ω0
2
2
Here z(0) = −m h∇u0, ni and |h∇u0, ni| = |∇u0| = Λ. On the other hand (see below),
2
h∇|∇u0| , ni = −2 Λ C
(5.19)
where C is the curvature of ∂Ω0 seen from inside. Therefore
Z
e0Θ0 (0) = 2 Λ
mC0(m) + C m2,
(5.20)
∂Ω0
R
2,α
which is the expression we announced in (2.7). Now, m is of class C
and satisfies (see (3.21))
m = 0. By
∂Ω0
assumption (2.8),
2
e0Θ0 (0) ≥ c kmk
.
(5.21)
H1/2(∂Ω0)
Recall that this occurs when Ω0 is convex for example. But by Theorem 2.5, we have for all t ∈ [0, 1],
2
e0Θ0 (t) ≥ e0Θ0 (0) − C ω(η) kmk
.
H1/2(∂Ω0)
Therefore, there exists η0 > 0 such that, for all Θ ∈ V(η0) and ∀t ∈ [0, 1],
2
e0Θ0 (t) ≥ C kmk
,
H1/2(∂Ω0)
and
2
eΘ(1) − eΘ(0) ≥ C kmk
,
H1/2(∂Ω0)
which is strictly positive if m = 0 whence the theorem.
Let us finally check (5.19) by the following elementary local computation (inspired from [7]) where we assume
that Ω0 is locally above the graph of the function f : (−ꢀ, +ꢀ) → R with f(0) = f0(0) = 0. For simplicity, we
write u instead of u0 that is: u0 = u(x, y). The function u is such that:
∀x ∈ (−ꢀ, +ꢀ), u(x, f(x)) = 0, h∇u0, ni = −uy(x, f(x)).
By differentiation, we have
0 = ux(x, f(x)) + uy(x, f(x))f0(x),
0 = uxx(x, f(x)) + 2uxy(x, f(x))f0(x) + uyy(x, f(x))f0(x)2 + uy(x, f(x))f00(x),

ABOUT STABILITY OF EQUILIBRIUM SHAPES
833
which gives at x = 0 : 0 = ux(0, 0), 0 = uxx(0, 0) + uy(0, 0)f00(0). We also have
2
h∇|∇u0| , ni = −∂y{(ux)2 + (uy)2} = −2 (uxuxy + uyuyy).
Recall now that the right-hand side k in (2.4) is compactly supported in Ω0. By regularity, we have uxx+uyy = 0
on the boundary. Therefore, at x = 0, we obtain
2
h∇|∇u0| , ni = 2 uy(0, 0)uxx(0, 0) = −2 (uy(0, 0))2 f00(0),
2
and (uy(0, 0))2 = |∇u0| = Λ. The formula (5.19) follows since f00(0) = C.
Remark. If we do not assume k to be compactly supported in Ω0, then we have to use uyy = −uxx − k instead
so that
2
h∇|∇u0| , ni = −2 uy[−uxx − k] = −2 (uy(0, 0))2 f00(0) + 2 k uy = −2 Λ C − 2 k h∇u0, ni.
We know that |h∇u0, ni| is constant on ∂Ω0 and equal to Λ1/2. Assume ∂Ω0 is connected. Then, by regularity,
h∇u0, ni itself is constant and equal to ꢀΛ1/2 with ꢀ = +1 or −1. The sign is determined by the relation
Z
Z
Z
k =
−∆u0 =
−h∇u0, ni = −ꢀ Λ1/2 length(∂Ω0),
Ω0
Ω0
∂Ω0
R
which shows that ꢀ = − sign(
k). Finally, the second derivative becomes
Ω0
Z
Z
ꢃ
ꢄ
e0Θ0 (0) = 2 Λ
mC0(m) + m2 C − k Λ−1/2sign(
k) .
(5.22)
∂Ω0
Ω0
6. About the coercivity of e0Θ0 (0)
If k is compactly supported in Ω0, then expression (5.20) is valid. Note that then the stability depends only
on the geometry of ∂Ω0. If moreover ∂Ω0 is convex, then the coercivity (5.21) holds. Obviously, this extends
2
to curves C -close to convex curves.
p
If k is identically equal to a positive constant, then any disk of radius R = S/π is a critical shape. We
have k S = Λ1/2 2πR and by (5.22)
Z
1
1
e0Θ0 (0) = 2 Λ
mC0(m) − m2 = 2 Λh[C0 − Id](m), miH−1/2
.
×H1/2
R
R
∂Ω0
We easily check that this vanishes for m = sin θ and m = cos θ so that e0Θ0 (0) does not satisfy (2.8) or (5.21).
This obviously corresponds to the fact that the disk remains a critical shape when moved by translation. One
can check that e0Θ0 (0) is however positive on the set of m0s orthogonal to the linear space spanned by {1, cos, sin}
and this is also a consequence of Theorem 2.5 (see e.g. [11,14]).
One may also compute what happens for radial functions k. Then the disk centered at the origin and of
p
R
−
radius R = S/π is a critical shape and (we assume for instance
k ≥ 0)
Ω0
Z
1
R
2πRk(R)
e0Θ0 (0) = 2 Λ
mC0(m) + m2[
].
R
k
∂Ω0
Ω0
We easily check that
Z
Z
1
R
mC0(m) ≥
m2.
∂Ω0
∂Ω0

834
M. DAMBRINE AND M. PIERRE
Therefore, for η positif and small
Z
2 m2
R
η
k(R)πR2
2
e0Θ0 (0) ≥ 2 Λ{η kmkH1/2
+
[1 −
−
] },
R
2
k
∂Ω0
Ω0
k
πR2
Ω
so that e0Θ0 (0) is coercive if k(R)h
·
0
More detailed studies of such quadratic forms may be found in [11] where more general functional involving
the perimeter of the shapes (i.e. surface tension) are considered. Moreover, the case of the less stable “exterior”
problem (or “exterior shaping problem”) is also treated where the Dirichlet problem is set in the exterior of the
shapes. The positivity is then more difficult to study. An extension of the results of this paper to the case with
surface tension can be found in [4] as well as N-dimensional situations.
Acknowledgements. We thank Michel Crouzeix for several helpful discussions and Jean Descloux for pointing out to us
this stability question.
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