Shape derivative of the Cheeger constant

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Shape derivative of the Cheeger constant

Enea Parini, Nicolas Saintier

To cite this version:

Enea Parini, Nicolas Saintier. Shape derivative of the Cheeger constant. ESAIM: Control, Optimi- sation and Calculus of Variations, EDP Sciences, 2014, 21 (2), pp.348-358. �10.1051/cocv/2014018�.

�hal-01067501�

ENEA PARINI AND NICOLAS SAINTIER

ABSTRACT. This paper deals with the existence of the shape derivative of the Cheeger constanth1(Ω)of a bounded domainΩ. We prove that ifΩadmits a unique Cheeger set, then the shape derivative ofh1(Ω)exists, and we provide an explicit formula. A counter- example shows that the shape derivative may not exist without the uniqueness assumption.

1. INTRODUCTION

LetΩ⊂Rⁿbe a bounded domain. TheCheeger constantofΩis defined as h₁(Ω):= inf

E⊂Ω

P(E;Rⁿ)

|E| .

HereP(E;Rⁿ)is the distributional perimeter ofEmeasured with respect toRⁿ, while|E|is then−dimensional Lebesgue measure ofE. A setC⊂Ωfor which the infimum is attained is called aCheeger set.

The problem of finding a Cheeger set for a given domainΩ has extensively received attention in the last decades, starting from the original work of Jeff Cheeger [5]. For an introductory survey on the Cheeger problem we refer to [18]; here we recall that for every bounded domainΩwith Lipschitz boundary there exists at least one Cheeger set. Unique- ness does not hold in general, but it is guaranteed if we assumeΩto be convex; in this case the Cheeger set turns out to be convex and of classC^1,1(see [1]). The Cheeger constant can be obtained as the limit for p→1 of the first eigenvalueλp(Ω)of the p−Laplacian under Dirichlet boundary conditions (see [12]), and corresponds to the first eigenvalue of the 1−Laplacian (see [14]).

Shape analysis roughly consists in studying the regularity and the optimisation of a functionalJ:Ω∈A →J(Ω)∈Rdefined over some classA of subsetsΩ⊂Rⁿ. Due to its physical relevance, a particularly important class of functionals are the ones defined in terms of the eigenvalues of some operator. A lot of works have been dedicated for instance to the study of the dependence of the eigenvalues of the Laplacian as functions of the domain under various boundary conditions. We refer for example to the monograph [11]

for an introduction to the field of shape analysis.

In order to optimize Jover A it is important to determine how sensitive isJ under perturbation of a given set Ω. Given a smooth vector fieldV ∈C^∞_c(Rⁿ;Rⁿ), defineF_t : Rⁿ→RⁿasF_t(x) = (Id+tV)(x). We then perturbΩin the directionV by considering the setsΩt=F_t(Ω). The shape derivative ofJin the directionV atΩis then defined as

J(Ω,V)^′:=lim

t→0

J(Ωt)−J(Ω)

t .

Date: September 23, 2014.

2010Mathematics Subject Classification. Primary: 49Q10; Secondary: 49Q20.

Key words and phrases.shape derivative, Cheeger constant, 1-Laplacian.

1

For instance the shape derivative of the first eigenvalueλ(Ω)of the Laplacian with Dirich- let boundary condition is

λ(Ω,V)^′=− Z

∂Ω

∂u

∂ ν

2

hV,νidHⁿ⁻¹,

whereuis the unique positive normalized eigenfunction inΩ andν is the unit exterior normal to∂Ω. This formula has been generalized in [8, 16] to the first eigenvalueλp(Ω) of thep-Laplacian (p>1):

(1) λp(Ω,V)^′=−(p−1)^Z

∂Ω

∂u_p

∂ ν

p

hV,νidHⁿ⁻¹, whereupis the unique positive normalized eigenfunction inΩ.

General results about the stability of the Cheeger constanth₁(Ω)as a function of Ω have been obtained in [10]. In particular the shape derivative was computed but only in the caseV(x) =λx,λ ∈R. The main purpose of this paper is to provide a formula for the shape derivative ofh₁(Ω)in the case of an arbitrary deformation fieldV. Notice that settingp=1 formally in (1) does not give any meaningful information. Indeed it is known that characteristic functions of Cheeger sets are, up to a multiplicative constant, normalized first eigenfunctions of the 1-Laplacian and they are obtained as limit of eigenfunctions of thep-Laplacian aspgoes to 1 (see Section 2). Therefore, ifCis a Cheeger set, the normal derivative should be thought as equal to−∞on∂Ω∩∂C, so that the integral in (1) would be infinite. This kind of problem has also been considered in [20] where the shape derivative of the best Sobolev constant for the embedding ofBV(Ω)intoL¹(∂Ω)was computed. Let us mention finally that the other extreme casep= +∞corresponding to the first eigenvalue of the∞-Laplacian has been recently studied in [17], [7] and [19] for Dirichlet, Steklov and Neumann boundary condition respectively.

The main result of our paper is the following.

Theorem 1.1. LetΩbe a bounded Lipschitz domain. Let V ∈C_c^∞(Rⁿ;Rⁿ), and let Ft : Rⁿ→Rⁿbe the one-parameter family of diffeomorphisms defined by Ft(x) = (Id+tV)(x).

SetΩt=F_t(Ω). Then

limt→0h₁(Ωt) =h₁(Ω).

If moreoverΩadmits a unique Cheeger set C then the shape derivative h1(Ω,V)^′=lim

t→0

h1(Ωt)−h1(Ω) t exists and is given by

h₁(Ω,V)^′= 1

|C| Z

∂^∗C

(div_∂CV−h₁(Ω)hV,νi)dHⁿ⁻¹, (2)

where∂^∗C is the reduced boundary of C,νis the unit exterior normal vector on∂^∗C, and div_∂ΩV(x) =divV(x)−(ν(x),DV(x)ν(x)), x∈∂^∗Ω, is the tangential divergence of V on

∂Ω.

In the case where∂Cis of classC^1,1, this formula can be simplified:

Corollary 1.2. IfΩadmits a unique Cheeger set C and∂C is of class C^1,1, then the shape derivative of h₁(Ω)is given by the formula

(3) h1(Ω,V)^′= 1

|C| Z

∂C∩∂Ω(κ−h1(Ω))hV,νidHⁿ⁻¹,

whereκ(x) =divνis the sum of the principal curvatures of∂Ωat the point x (i.e.(n−1) times the mean curvature), andνis the unit exterior normal to∂Ω.

The assumption in the Corollary is in particular satisfied for every dimensionnwhenΩ is convex (see [1]), or in dimensionn≤7 when∂Ω is of classC^1,1and admits a unique Cheeger setC(see [4]). We point out that the uniqueness hypothesis is necessary. Indeed, at the end of this paper we provide a counterexample of a domain admitting more than one Cheeger set, which is not shape differentiable for some choice ofV. However, it is interesting to observe that the bounded domainsΩadmitting a unique Cheeger set (and hence shape differentiable) are dense in theL¹topology (see [4]).

2. DEFINITIONS AND PRELIMINARY RESULTS

LetΩ⊂Rⁿbe an open set. Thetotal variationinΩof a functionu∈L¹(Ω)is defined as

|Du|(Ω):=sup _Z

Ω

udivϕ

ϕ∈C¹_c(Ω;Rⁿ),kϕk∞≤1

.

A functionu such that|Du|(Ω)<+∞is said to be ofbounded variation. The space of the functions of bounded variation will be denoted by BV(Ω). It can be easily proved that the total variation is lower semicontinuous with respect to the L¹-convergence (see [9]). Moreover, the following holds true. Suppose thatΩ is a Lipschitz domain, and let u∈BV(Ω); if we denote byuthe extension ofuby zero outsideΩ, thenu∈BV(Rⁿ), and

|Du|(Rⁿ) =|Du|(Ω) + Z

∂Ω|u|dHⁿ⁻¹, whereHⁿ−1is the(n−1)-dimensional Hausdorff measure on∂Ω.

Theperimeterof a setE⊂Ω(measured with respect toRⁿ) is defined as P(E;Rⁿ):=|DχE|(Rⁿ),

whereχEis the characteristic function ofE. TheCheeger constantofΩis h₁(Ω):= inf

E⊂Ω

P(E;Rⁿ)

|E| ,

where|E|stands for the n-dimensional Lebesgue measure ofE. A Cheeger setis a set C⊂Ωsuch that

P(C;Rⁿ)

|C| =h₁(Ω).

The existence of a Cheeger set for every bounded Lipschitz domainΩ is proved via the direct method of the Calculus of Variations. Uniqueness does not hold in general; however, any convex body has a unique Cheeger set (see [1]). IfCis a Cheeger set forΩ, then∂C∩Ω is analytic, up to a closed singular set of Hausdorff dimensionn−8; at the regular points of∂C∩Ω, the mean curvature is equal to ^h_n¹₋^(Ω)₁ (see e.g. [18, Proposition 4.2]). Morever, if∂Ωis of classC^1,1, then also∂Cenjoys the same regularity (see [4]); the same result holds ifΩis convex, as a consequence of the results in [21].

As an application of the coarea formula,h₁(Ω)can also be obtained as h1(Ω) = inf

u∈BV(Ω)\{0}

|Du|(Rⁿ) kuk¹ or equivalently

h1(Ω) =inf{|Du|(Rⁿ)|u∈BV(Ω),kuk¹=1}.

Therefore, h1(Ω) can be seen as the first eigenvalue of the 1-Laplacian with Dirichlet boundary condition, which is defined formally as

∆¹u=div ∇u

|∇u|

,

and the characteristic functions of Cheeger sets are corresponding eigenfunctions. We refer to [14] for a thorough analysis of this problem. Here we observe that ifΩadmits a unique Cheeger setC, thenu=_|C¹_|χC is the unique nonnegative normalized eigenfunction of the 1-Laplacian, since every level set of a first eigenfunction is a Cheeger set (see [3, Theorem 2]).

3. PROOF OF THE MAIN RESULTS

Recall that we are given a Lipschitz domainΩ⊂Rⁿthat we perturb in the direction of a smooth vector fieldV ∈C_c^∞(Rⁿ;Rⁿ)in the sense that we consider the perturbed domains

Ωt=F_t(Ω) with F_t(x) = (Id+tV)(x).

We leth=h1(Ω)andh_t=h1(Ωt). We also assume that any functionudefined inΩ(resp.

Ωt) is extended by 0 toRⁿ\Ω(resp.Rⁿ\Ωt). With the notation of the previous section this means thatu=u.¯

We recall the change of variable formula for BV functions (see [9, Lemma 10.1]). Let G_tbe the inverse ofF_t(which exists for smallt). For an arbitrary functionu∈BV(Ω), if we denote byvthe function ofBV(Ωt)defined byv(x) =u(Gt(x))we have the relations

Z

Ωt

v(x)dx= Z

Ωu(y)|detDF_t(y)|dy and

|Dv|(Rⁿ) = Z

Rⁿ|(DGt)^Tσ| · |detDF_t|d|Du|, whereσ comes from the polar decompositionDu=σ|Du|.

Proof of Theorem 1.1. Let u∈BV(Ω) be a nonnegative eigenfunction for h such that kuk¹=1 in the sense thatu is an extremal in (2) (which is known to exist). Consider the functionw_t∈BV(Ωt)defined asw_t=u◦G_t. Then

|Dw_t|(Rⁿ) = Z

Rⁿ|(DGt)^Tσ| · |detDF_t|d|Du|,

whereσ comes from the polar decompositionDu=σ|Du|. Since|σ|=1|∇u|-a.e., and DF_t→Iduniformly ast→0, so that|detDF_t| →1 uniformly, we have using (2) and the above change of variable formula that

h_t≤|Dwt|(Rⁿ) Z

Ωt

w_t

= Z

Rⁿ|(DGt)^T| · |detDF_t|d|Du| Z

Ωu(y)|detDF_t(y)|dy

= (1+o(1)) Z

Rⁿ

d|Du| Z

Ωu(y)dy .

It follows that

lim sup

t→0 h_t≤h

Letu_t ∈BV(Ωt)be a nonnegative extremal for h_t such thatku_tk¹=1. Consider the functionv_t∈BV(Ω)defined asv_t=u_t◦F_t. Then

|Dv_t|(Rⁿ) = Z

Rⁿ|(DFt)^Tσt| · |detDG_t|d|Du_t| ≤(1+o(1))^Z

Rⁿ

d|Du_t|

= (1+o(1))ht

≤h+o(1), (4)

and

(5) ^Z

Ω

v_tdx= Z

Ωt

u_t|detDF_t⁻¹|dx=1+o(1).

Therefore(vt)is bounded inBV(Rⁿ). Since the embedding ofBV(Rⁿ)intoL¹_loc(Rⁿ)is compact, it follows that there exists a functionv∈BV(Rⁿ)such that (up to a subsequence), v_t→va.e.. We deduce first thatv=0 inRⁿ\Ω, then, using (5), that

Z

Ω

v dx=lim

t→0 Z

Ω

v_tdx=1, and eventually according to (4), that

|Dv|(Rⁿ)≤lim inf

t→0 |Dv_t|(Rⁿ)≤h.

Lettingv=v_|Ω, it follows that^R_Ωv dx=1, and h≤ |Dv|(Rⁿ)≤lim inf

t→0 |Dv_t|(Rⁿ) =h.

It follows that

limt→0h_t=h, and thatvis an extremal forh.

We assume from now on thatΩadmits a unique Cheeger setC⊂Ω. As a consequence, the only nonnegative normalized extremal for h is|C|⁻¹χC; this follows from the fact that every level set of an extremal is a Cheeger set (see [3, Theorem 2]). In particular u=v=|C|⁻¹χC. Thereforev_t→uinL¹(Ω)and

limt→0|Dv_t|(Rⁿ) =|Du|(Rⁿ).

By [2, Proposition 3.13], this implies that limt→0

Z

Rⁿφd|Dv_t|= Z

Rⁿφd|Du| for anyφ∈C_c(Rⁿ).

Let us prove the differentiability. Usingw_t=u◦G_t as a test-function forh_t, we obtain

h_t−h≤ Z

Rⁿ|(DGt)^Tσ| · |detDF_t|d|Du| Z

Ω

u(y)|detDF_t(y)|dy

−h.

Observe that

|detDF_t(y)|=1+t.divV(y) +o(t), and

|(DGt(y))^Tσ(y)|=|σ(y)| −thσ(y),DV(y).σ(y)i+o(t),

whereo(t)is uniform iny. Therefore

ht−h≤ h+t

Z

Rⁿ(divV− hσ,DVσi)d|Du|+o(t) 1+t

Z

Ω

udivV+o(t) −h

= t

Z

Rⁿ(divV− hσ,DVσi)d|Du| −h Z

Ω

udivV

1+t Z

Ω

udivV+o(t)

.

We used the fact that|σ|=1|Du|-a.e. anduis a normalized extremal forh. It follows that lim sup

t→0⁺

ht−h

t ≤

Z

Rⁿ(divV− hσ,DVσi)d|Du| −h Z

Ω

udivV, and

lim inf

t→0⁻

h_t−h

t ≥

Z

Rⁿ

(divV− hσ,DVσi)d|Du| −h Z

Ω

udivV.

Let us now prove the opposite inequality. We usev_t as a test-function forh, and we obtain

h_t−h= Z

Rⁿ

d|Du_t| −h≥ Z

Rⁿ|(DGt)^Tσt| · |detDF_t|d|Dv_t| − R

Rⁿd|Dvt| R

Ωv_t , whereσtis such thatDu_t=σt|Du_t|. We can also write

h_t−h≥ Z

Rⁿ

d|Dv_t|+t Z

Rⁿ(divV− hσt,DVσti)d|Dv_t| − R

Rⁿd|Dv_t| R

Ωv_t +o(t).

Since divV∈C_c(Rⁿ), we have limt→0

Z

Rⁿ

divV d|Dv_t|= Z

Rⁿ

divV d|Du|. Observe also that

Z

Ω

v_t=1−t Z

Rⁿ

u_tdivV+o(t) =1−t Z

Rⁿ

udivV+o(t).

so that

R

Rⁿd|Dv_t| R

Ωv_t = Z

Rⁿ

d|Dv_t|+t Z

Rⁿ

d|Dv_t| Z

Ω

udivV

+o(t)

= Z

Rⁿ

d|Dv_t|+th Z

Ω

udivV+o(t), where we used the fact that|Dv_t|(Rⁿ) =h+o(1). Hence,

h_t−h≥t Z

Rⁿ

divV d|Du| −h Z

Ω

udivV− Z

Rⁿhσt,DVσtid|Dv_t|

+o(t) SinceDv_t⇀^∗Duand|Dv_t|(Rⁿ)→ |Du|(Rⁿ), we have, according to Reshetnyak’s Theorem (see [2, Theorem 2.39]), that

limt→0 Z

Rⁿ

f(x,σt(x))d|Dv_t|= Z

Rⁿ

f(x,σ(x))d|Du| for anyf ∈C_b(Rⁿ×Sⁿ⁻¹).

It follows in particular that

tlim→0 Z

Rⁿhσt,DVσtid|Dv_t|= Z

Rⁿhσ,DVσid|Du|.

We thus obtain lim sup

t→0⁺

ht−h

t ≥

Z

Rⁿ(divV− hσ,DVσi)d|Du| −h Z

Ω

udivV and

lim inf

t→0⁻

h_t−h

t ≤

Z

Rⁿ

(divV− hσ,DVσi)d|Du| −h Z

Ω

udivV.

Therefore

h1(Ω,V)^′= lim

t→0⁺

h_t−h t exists, and

h₁(Ω,V)^′= Z

Rⁿ

(divV− hσ,DVσi)d|Du| −h Z

Ω

udivV.

Sinceu=|C|⁻¹χC, we have that|Du|=|C|⁻¹Hⁿ−1

|∂^∗C as a measure. We can thus rewrite the previous formula as

h1(Ω,V)^′= 1

|C| _Z

∂^∗C(divV− hσ,DVσi)dHⁿ−1

−h Z

CdivV

= 1

|C| Z

∂^∗C(divV− hσ,DVσi −hhV,νi)dHⁿ⁻¹,

whereν is the unit exterior normal to∂^∗C, andσ is given byDu=σ|Du|. We observe thatσ=−νHⁿ−1- a.e. on∂^∗C. Recall that

divV(x)−(ν(x),DV(x)ν(x)) =div_∂CV(x), x∈∂^∗C,

is the tangential divergence ofV on∂^∗C(see e.g. [11, Definition 5.4.6]). We thus obtain that

(6) h₁(Ω,V)^′= 1

|C| Z

∂^∗C

(div_∂CV−hhV,νi)dHⁿ⁻¹

which ends the proof of Theorem 1.1.

Proof of Corollary 1.2. Suppose thatΩadmits a unique Cheeger setCwhich isC^1,1. The unit exterior normal vectorνto∂Cis thus defined at every point and is Lipschitz continu- ous. Its components are thus differentiable atHⁿ−1almost every point of∂C; moreover, the quantityκ:=div_∂Cνbelongs toL^∞(∂C)and it can be seen as the distributional curva- ture of∂C. Indeed one can easily adapt [11, Section 5.4.3] to the case ofC^1,1domains to obtain

div_∂CV =div_∂CV_∂C+κ(V,ν) Hⁿ⁻¹−a.e., whereV_∂C=V−(V,ν)νis the tangential part ofV, and

Z

∂C

div_∂CV_∂CdHⁿ⁻¹=0.

Therefore it holds

Z

∂Cdiv_∂CV = Z

∂CκhV,νi

and we can rewrite (6) as

h₁(Ω,V)^′= 1

|C| Z

∂C(div_∂CV−h₁(Ω)hV,νi)dHⁿ⁻¹

= 1

|C| Z

∂C(κ−h1(Ω))hV,νidHⁿ⁻¹

= 1

|C| Z

∂C∩∂Ω(κ−h1(Ω))hV,νidHⁿ⁻¹

sinceκ=h1(Ω)in∂C∩Ω. We then deduce (3).

We complete this section providing some explicit examples of computation of shape derivatives.

Example 3.1(The ball). LetΩ=BRbe the ball of radiusR, andV is a vector field such thatV(x) =ν(x)on∂B_R, we have that ^dh_dt^t(0) =_d

drh₁(Br)(R). Sinceh₁(Br) = ⁿ_r, we obtain using (3) that

h₁(Ω,V)^′=nωnRⁿ⁻¹ ωnRⁿ ·

n−1 R −n

R

=− n R²

as expected. Now letV be a volume-preserving perturbation; formula (3) becomes h₁(Ω,V)^′=− 1

|Ω|

Z

∂ΩhV,νidHⁿ⁻¹=− 1

|Ω|

Z

Ω

divV=0

in accordance with the well-known fact that the ball minimizesh₁(Ω)among all bounded domains with fixed volume.

Example 3.2(The annulus). As another simple example takeΩ=A_r,R=B_R\B¯_r, the an- nulus{r<|x|<R},r<R. According to [6] and [13],A_r,Rcoincides with its Cheeger set so that

h1(Ar,R) =|∂A_r,R|

|A_r,R| =nRⁿ⁻¹+rⁿ⁻¹ Rⁿ−rⁿ . TakingV(x) =ν(x), we have by direct computation that

d

dth₁(Ar−t,R+t)_|t=0

=n−R²ⁿ⁻²−r²ⁿ⁻²−(n−1)rⁿ⁻²Rⁿ−(n−1)Rⁿ⁻²rⁿ−2n(rR)ⁿ⁻¹

(Rⁿ−rⁿ)² ,

which coincides with formula (3):

h₁(Ω,V)^′=n−1

R −h₁(A_r,R)|∂B_R|

|A_r,R| −n−1

r +h1(A_r,R)|∂B_r|

|A_r,R|. In dimension 2 this example can be generalized to curved annulus:

Example 3.3(Curved annulus in the plane). LetΓbe a smooth planar closed curve with no self-intersection, andΩ=ΣΓ,a={x∈R²,dist(x,Γ)<a}its tubular neighborhood of widtha. We takeaso small thatΩhas no self-intersection. According to [15],h1(Ω) =¹_a andΩitself is the unique Cheeger set. We takeV =ν. ThenΩt=ΣΓ,a+t andh(Ω,V)^′=

−_a¹2 =−h₁(Ω)²which coincides with formula (3):

h1(Ω,V)^′= 1

|Ω|

Z

∂Ω(κ−h1(Ω))dHⁿ⁻¹ since^R_∂Ωκ=2π χ(Ω) =0 according to the Gauss-Bonnet formula.

Example 3.4(The square). We eventually provide an example where the Cheeger set is a proper subset ofΩ. According to [13] a rectangleR_a,b⊂R²of edges 2aand 2bhas a unique Cheeger setCwith

(7) h1(Ra,b) = 4−π

2(a+b)−2p

(a−b)²+πab

(see e.g. one of the two squares in figure 4). We takeΩ= [0,1]×[0,1] =R_1/2,1/2and V(x,y) = (η(x),0)withη:R→[0,1]smooth with compact support in(1−δ,1+δ),δ small, andη(x) =1 forx∈(1−δ/2,1+δ/2). ThenΩt= (0,1+t)×(0,1)for sufficiently smallt. It follows by direct computations from (7) that

h1(Ω,V)^′=−1 2h1(Ω).

Since∂C∩Ωis made of arc of circle of radius 1/h1(Ω), it is easily seen that

|C|=1− 4−π h₁(Ω)²=4√

π−2π 4−π , H¹(∂C∩S) =1− 2

h₁(Ω)=2√ π−π 4−π , whereS:={1} ×[0,1]. It follows that

h1(Ω,V)^′=−h1(Ω)H¹(∂C∩S)

|C| ,

which is formula (3) sinceκ=0 on∂C∩∂Ω,hV,νi=1 onSandhV,νi=0 on∂Ω\S.

4. ACOUNTER-EXAMPLE TO THE DIFFERENTIABILITY OFh₁(Ω)

IfΩdoes not admit a unique Cheeger set, thenh₁(Ω)is in general not differentiable.

As a counterexample, we consider the “barbell domain”, made of two equal rectanglesR₁ andR₂linked by a thin strip (see Figure 4), defined as

Ω= ([0,1]×[0,1])∪([1,2]×[0,ε])∪([2,3]×[0,1]), whereε>0 is sufficiently small. LetV be a smooth vector field such that:

• Vis supported in[3−δ,3+δ]×[−δ,1+δ]for some smallδ;

• V(x,y) =f(x,y)−→e₁for some smooth nonnegative functionf satisfying f(3,y) =1 fory∈[0,1].

In other words,V shifts the far right edge of Ω to the right. For small positive values oft, h1(Ωt)behaves like the Cheeger constant of a rectangle obtained by enlargingR2. Recalling formula (7) which gives the Cheeger constant of a rectangleR_a,bof edges 2aand 2b, we see that_∂^∂_bh1(Ra,b)<0. Therefore

lim

t→0⁺

h₁(Ωt)−h₁(Ω) t <0.

For small negative values oft,h1(Ωt) =h1(R1) =h1(Ω)so that lim

t→0⁻

h₁(Ωt)−h₁(Ω)

t =0.

It follows thath1(Ω)is not differentiable att=0.

C

l

l

C

F^IGURE1. Ifl1=l2, the Cheeger sets are given byC1,C2andC1∪C2.

C

l

l

F^IGURE2. Ifl1>l2, the only Cheeger set is given byC1.

C

l

l

F^IGURE3. Ifl2>l1, the only Cheeger set is given byC2.

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