Simons' cone and equivariant maximization of the first $p$-Laplace eigenvalue

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by  Sinan Ariturk

SIMONS’ CONE AND EQUIVARIANT MAXIMIZATION OF THE
FIRST p-LAPLACE EIGENVALUE
6
1
SINANARITURK
0
2
t Abstract. We consider an optimization problem for the first Dirichlet
c eigenvalue of the p-Laplacian on a hypersurface in R2n, with n ≥ 2. If
O p≥2n−1,thenamonghypersurfacesinR2nwhichareO(n)×O(n)-invariant
and have one fixed boundary component, there is a surface which maximizes
1
thefirstDirichleteigenvalueofthep-Laplacian. ThissurfaceiseitherSimons’
3 coneoraC1 hypersurface,dependingonpandn. Ifnisfixedandpislarge,

then the maximizing surface is not Simons’ cone. If p = 2 and n ≤ 5, then
]
P Simons’conedoesnotmaximizethefirsteigenvalue.
A
.
h
t
a 1. Introduction
m
In this article we consider an optimization problem for the first Dirichlet
[
eigenvalue of the p-Laplacian. This problem is motivated by Simons’ cone and by

2 the Faber-Krahn inequality. Simons’ cone was the first example of a singular area
v
minimizing cone. Almgren [2] showedthat the only areaminimizing hypercones in
9
R4 are hyperplanes. Simons [20] extended this to higher dimensions up to R7 and
9
9 established the existence of a singular stable minimal hypercone in R8, given by
0
0 (1.1) (x ,...,x ,y ,...,y ) R8 :x2+...+x2 =y2+...+y2 1
. 1 4 1 4 ∈ 1 4 1 4 ≤
1 (cid:26) (cid:27)
0 Bombieri, De Giorgi, and Giusti [5] showed that Simons’ cone is area minimizing.
6 That is, Simons’ cone has less volume than any other hypersurface in R8 with the
1
sameboundary. Lawson[12]andSimoes[19]gavemoreexamplesofareaminimizing
:
v hypercones.
Xi TheFaber-KrahninequalitystatesthatamongdomainsinRnwithfixedvolume,
the ball minimizes the first Dirichlet eigenvalue of the p-Laplacian for every
r
a 1<p< . The p-Laplacian ∆ is defined by
p
∞
(1.2) ∆ f =div f p−2 f
p
|∇ | ∇
(cid:16) (cid:17)
The Dirichlet eigenvalues of the p-Laplacian on a smoothly bounded domain Ω
in Rn are the numbers λ such that the equation ∆ ϕ = λϕp−2ϕ admits a
p
− | |
weak solution in W1,p(Ω). The p-Laplacian admits a smallest eigenvalue, denoted
0
λ (Ω). Lindqvist [14] showed this eigenvalue is simple on a connected domain,
1,p
meaningthe correspondingeigenfunctionisunique uptonormalization. IfLip (Ω)
0
is the set of Lipschitz functions f : Ω R which vanish on the boundary of Ω,
→
then λ (Ω) can be characterizedvariationally by
1,p
f p
(1.3) λ (Ω)=inf Ω|∇ | :f Lip (Ω)
1,p f p ∈ 0
(cid:26)R Ω| | (cid:27)
1
R
2 SINANARITURK
This characterization and the Polya-Szego inequality [18] imply the Faber-Krahn
inequality. Moreover, Brothers and Ziemer [6] proved a uniqueness result. In
particular, the ball is the only minimizer with smooth boundary.
We consider a similar optimization problem for the first Dirichlet eigenvalue
of the p-Laplacian on a hypersurface in R2n, for n 2. Let G = O(n) O(n)
and consider the usual action of G on R2n. Fix an o≥rbit of dimension ×2n 2.
Let be the set of all C1 immersed G-invariant hypersuOrfaces in R2n with−one
S
boundary component, given by . For a hypersurface Σ in , the immersion of Σ
intoR2n inducesacontinuousRiOemannianmetriconΣ. LetSLip (Σ)denotetheset
0
ofLipschitzfunctionsf :Σ Rwhichvanishon ,andletdV betheRiemannian
→ O
measure on Σ. Let λ (Σ) denote the first Dirichlet eigenvalue of the p-Laplacian,
1,p
which is given by
f pdV
(1.4) λ (Σ)=inf Σ|∇ | :f Lip (Σ)
1,p f pdV ∈ 0
(cid:26)R Σ| | (cid:27)
If istheproductoftwospheresofRthesameradiusR,thenletΓbeSimons’cone,
O
defined by
(1.5) Γ= (x ,...,x ,y ,...,y ) R2n :x2+...+x2 =y2+...+y2 R2
1 n 1 n ∈ 1 n 1 n ≤
(cid:26) (cid:27)
Let Lip (Γ) denote the set of Lipschitz functions f : Γ R which vanish on .
0 → O
Note that Γ 0 is a smooth immersed hypersurface. This immersion induces a
\{ }
Riemannian metric on Γ 0 . Let dV be the Riemannian measure. Then define
\{ }
f pdV
(1.6) λ (Γ)=inf Γ\{0}|∇ | :f Lip (Γ)
1,p (cid:26)R Γ\{0}|f|pdV ∈ 0 (cid:27)
The followingtheoremstates thatRif p 2n 1,then there is a hypersurface which
≥ −
maximizes the eigenvalue λ . This hypersurface is either Simons’ cone or a C1
1,p
hypersurface in .
S
Theorem 1.1. Fix n 2 and p 2n 1. If is the product of two spheres of
≥ ≥ − O
different radii, then there is a C1 embedded surface Σ∗ in such that
p S
(1.7) λ (Σ∗)=sup λ (Σ):Σ
1,p p 1,p ∈S
If is the product of two spheres of thensame radius andoif
O
(1.8) λ (Γ)<sup λ (Σ):Σ
1,p 1,p
∈S
n o
then there is a C1 embedded surface Σ∗ in such that
p S
(1.9) λ (Σ∗)=sup λ (Σ):Σ
1,p p 1,p ∈S
n o
A natural problem motivated by Theorem 1.1 is to determine if (1.8) holds, i.e.
ifSimons’conemaximizesλ . Forfixednandlargep,weshowthatSimons’cone
1,p
does not maximize λ . For the case p =2 and n 5, we also show that Simons’
1,p
≤
cone does not maximize λ .
1,2
Theorem 1.2. Assume is the product of two spheres of the same radius. For
O
each n, there is a value p such that if p p , then
n n
≥
(1.10) λ (Γ)<sup λ (Σ):Σ
1,p 1,p
∈S
n o
SIMONS’ CONE AND EQUIVARIANT MAXIMIZATION OF THE FIRST p-LAPLACE EIGENVALUE3
If n 5 and p=2, then
≤
(1.11) λ (Γ)<sup λ (Σ):Σ
1,2 1,2
∈S
n o
Inparticular,forthecasesn=4andn=5,Simons’coneisareaminimizing,but
doesnotmaximizetheeigenvalueλ . Thisisincontrasttotheinverserelationship
1,2
that the eigenvalue and the volume of a domain often exhibit. More accurately,
the eigenvalues of a domain Ω in Rn, are inversely related to the Cheeger constant
h(Ω), which is defined by
∂U
(1.12) h(Ω)=inf | | :U Ω
U ⊂
(cid:26) | | (cid:27)
HereU isasmoothlyboundedopensubsetofΩ,and ∂U isthe(n 1)-dimensional
| | −
volume of ∂U, while U is the n-dimensional volume of U. Cheeger’s inequality
| |
states that
p
h(Ω)
(1.13) λ (Ω)
1,p
≥ p
(cid:18) (cid:19)
Cheeger [7] first proved this inequality for the case p = 2. Lefton and Wei [13],
Matei [15], and Takeuchi [21] extended this inequality to the case 1 < p < .
∞
Moreover,Kawohland Fridman [11] showed that
(1.14) limλ (Ω)=h(Ω)
1,p
p→1
We observe that the relationship between the eigenvalue λ and the Cheeger
1,p
constant is strongest for small p. For large p, the eigenvalue λ is more strongly
1,p
related to the inradius of Ω, denoted inrad(Ω). Juutinen, Lindqvist, and Manfredi
[10] proved that
1/p 1
(1.15) lim λ (Ω) =
1,p
p→∞ inrad(Ω)
(cid:16) (cid:17)
Moreover, Poliquin [17] showed that for each p > n, there is a constant C ,
n,p
independent of Ω, such that
1/p C
n,p
(1.16) λ (Ω)
1,p
≥ inrad(Ω)
(cid:16) (cid:17)
Inlightof(1.15)and(1.16),itisnotsurprisingthatSimons’conedoesnotmaximize
λ for large p. We remark that Grosjean [9] established a result similar to (1.15)
1,p
onacompactRiemannianmanifold,withtheinradiusreplacedbyhalfthediameter
of the manifold. Valtorta [23] and Naber and Valtorta [16] obtained lower bounds
for the first eigenvalue of the p-Laplacian in terms of the diameter on a compact
Riemannian manifold.
A similar problem to the one described in Theorem 1.1 is to maximize the first
Dirichlet eigenvalue among surfaces of revolution in R3 with one fixed boundary
component. This problem has been studied for the case p = 2. It follows from a
resultofAbreuandFreitas[1]thatthediscmaximizesthefirstDirichleteigenvalue.
In fact the disc maximizes all of the Dirichlet eigenvalues [3]. Moreover, it follows
from a result of Colbois, Dryden, and El Soufi [8] that a flat n-dimensional ball in
Rn+1 maximizes the first Dirichlet eigenvalue among O(n)-invariant hypersurfaces
in Rn+1 with the same boundary. Among surfaces of revolution in R3 with two
fixed boundary components, there is a smooth surface which maximizes the first
Dirichlet eigenvalue [4].
4 SINANARITURK
The argument we use to prove Theorem 1.1 is a development of the argument
usedin[4]tomaximizeLaplaceeigenvaluesonsurfacesofrevolutioninR3. Forthe
case where p is large, the proof of Theorem 1.2 is a simple application of (1.15).
For the case where p = 2, we use a variational argument. In the next section, we
reformulate Theorem 1.1 and Theorem1.2 as statements about curves in the orbit
space R2n/G. In the third section, we prove a low regularity version of Theorem
1.1. In the fourth section, we complete the proof of Theorem 1.1. In the fifth
section, we prove Theorem 1.2.
2. Reformulation
In this section, we reformulate Theorem 1.1 and Theorem 1.2 as statements
about curves in the orbit space R2n/G. Identify R2n/G with a quarter plane
(2.1) R2n/G= (x,y) R2 :x 0,y 0
∈ ≥ ≥
(cid:26) (cid:27)
A point (x,y) is identified with the orbit
(2.2) (x ,...,x ,y ,...,y ) R2n :x2+...+x2 =x2,y2+...+y2 =y2
1 n 1 n ∈ 1 n 1 n
Letg bentheorbitaldistancemetriconR2n/G,i.e. g =dx2+dy2. Defineafuncotion
F : R2n/G R which maps an orbit to its (2n 2)-dimensional volume in R2n.
→ −
There is a constant c such that
n
(2.3) F(x,y)=c xn−1yn−1
n
·
Let (x ,y ) be the coordinates of the orbit . By symmetry, we may assume that
0 0
O
(2.4) x y >0
0 0
≥
For a C1 curve α : [0,1] R2n/G, let L (α) be the length of α with respect
g
to g. Let be the set of C1→curves α : [0,1] R2n/G which satisfy the following
propertiesC. First α(0) = (x ,y ) and α(1) is→in the boundary of R2n/G. Second
0 0
α(t)isintheinteriorofR2n/Gforeverytin[0,1). Third α′(t) =L (α)forevery
g g
t in [0,1]. Fourth α intersects the boundary of R2n/G a|way f|rom the origin, and
the intersection is orthogonal. If α is a curve in , let F =F α. Let Lip ([0,1])
be the set of Lipschitz functions w :[0,1] R wChich vanαish at◦zero. Then0define
→
1 |w′|pFα dt
(2.5) λ (α)=inf 0 |α′|pg−1 :w Lip ([0,1])
1,p  1RwpF α′ dt ∈ 0 
 0 | | α| |g 
For a function w in Lip ([0,1]),Rthe Rayleigh quotient of w is
0  
1 |w′|pFα dt
(2.6) 0 |α′|pg−1
1RwpF α′ dt
0 | | α| |g
Note that if α is in , then therRe is a corresponding surface Σ in such that α
parametrizes the proCjection of Σ in R2n/G. Moreover λ (Σ) = λS(α), because
1,p 1,p
the first eigenfunction on Σ is G-invariant. Furthermore, if Σ is a surface in and
S
λ (Σ) is non-zero,then Σ is connected and there is a curve α in corresponding
1,p
C
to Σ. In particular,
(2.7) sup λ (Σ):Σ =sup λ (α):α
1,p 1,p
∈S ∈C
n o n o
SIMONS’ CONE AND EQUIVARIANT MAXIMIZATION OF THE FIRST p-LAPLACE EIGENVALUE5
If x =y =R, then define a curve σ :[0,1] R2n/G by
0 0
→
(2.8) σ(t)=(1 t) (R,R)
− ·
Let F =F σ and define
σ
◦
1 |w′|pFσ dt
(2.9) λ (σ)=inf 0 |σ′|pg−1 :w Lip ([0,1])
1,p  1RwpF σ′ dt ∈ 0 
 0 | | σ| |g 
This curve corresponds to SimoRns’ cone Γ in R2n/G, and λ (Γ)=λ (σ).
 1,p  1,p
Lemma 2.1. Fix n 2 and p 2n 1. If x =y then there is a simple curve α
0 0
≥ ≥ − 6
in such that
C
(2.10) λ (α)=sup λ (β):β
1,p 1,p
∈C
If x =y and if n o
0 0
(2.11) λ (σ)<sup λ (β):β
1,p 1,p
∈C
then there is a simple curve α in suchnthat o
C
(2.12) λ (α)=sup λ (β):β
1,p 1,p
∈C
n o
Lemma 2.1 immediately yields Theorem 1.1. In the third section, we prove low
regularity versions of Lemma 2.1, replacing with larger spaces of curves. First,
we identify points on the boundary of R2n/CG to obtain a quotient space Q, and
we proveexistence in a space ofcurvesin Q whichcorrespondto hypersurfaces
h
in R2n of finite volume. TheRn we obtain a maximizing curve in a space + of
curves in R2n/G which project to curves in . Finally, we prove existencRehin a
h
space ∗ of curves in R2n/G which have finiRte length and intersect the boundary
Rg
transversally,awayfrom the singular point. In the fourth section, we complete the
proof of Lemma 2.1 by showing that a maximizing curve in ∗ must be in .
Rg C
Lemma 2.2. Assume x =y . For each n, there is a value p such that if p p ,
0 0 n n
≥
then
(2.13) λ (σ)<sup λ (α):α
1,p 1,p
∈C
If n 5 and p=2, then n o
≤
(2.14) λ (σ)<sup λ (α):α
1,2 1,2
∈C
n o
Lemma 2.2 immediately yields Theorem 1.2. We prove Lemma 2.2 in the fifth
sectionofthearticle. Forthecasewherepislarge,theproofisasimpleapplication
of (1.15). For the case where p=2, we use a variational argument.
3. Existence
In this section we prove a low regularity version of Lemma 2.1. We first extend
the defintion ofλ to low regularitycurves. Define a Riemannianmetric h onthe
1,p
interior of R2n/G by
(3.1) h=F2 g =c2 x2n−2y2n−2 dx2+dy2
· n·
The length of a curve in R2n/G with respect t(cid:16)o h is the(cid:17)(2n 1)-dimensional
volumeofthecorrespondingG-invarianthypersurfaceinR2n. Defi−neanequivalence
6 SINANARITURK
relation on R2n/G such that each point in the interior is only equivalent to itself,
andanytwopointsonthe boundaryareequivalent. LetQbe the quotientspaceof
R2n/Gwithrespecttothisequivalencerelation. LetQ betheimageoftheinterior
0
ofR2n/Gunderthequotientmap. LetQ denotetheremainingpointinQwhichis
B
theimageoftheboundaryofR2n/G. ThenQ=Q Q . WeviewQasametric
0 B
space, with distance function induced by h. The fu∪nc{tion}F :R2n/G R induces
→
afunctiononQ,whichwealsodenotebyF. Letα:[c,d] QbeaLipschitzcurve
→
such that α(t)=Q for all t in [c,d) and α(d)=Q . Let Lip ([c,d]) be the set of
Lipschitz funct6ionsBw :[c,d] R which vanish at c.BLet F =0F α and define
α
→ ◦
d |w′|pFαp dt
c |α′|p−1
(3.2) λ (α)=inf h :w Lip ([c,d])
1,p  Rd wp α′ dt ∈ 0 
 c | | | |h 
R
Iftheintegrandinthenumeratortakestheform0/0atsomepointin[c,d],thenwe
interprettheintegrandasbeingequaltozeroatthispoint. IftheRayleighquotient
takes the form 0/0, then we interpret the Rayleigh quotient as being infinite. Let
be the set of Lipschitz curves α : [0,1] Q such that α(0) = (x ,y ) and
h 0 0
R →
α(1) = Q and α(t) is in Q for every t in [0,1). Note that a curve in can be
B 0
identified with a curve in , by composing with the quotient map R2n/CG Q.
h
R →
Making this identification, the definitions (2.5) and (3.2) are the same.
For a Lipschitz curve γ : [c,d] Q, let L (γ) denote the length of γ. In the
h
→
followinglemma, we provethatreparametrizinga curve by arc length with respect
to h does not decrease the eigenvalue.
Lemma 3.1. Let γ :[c,d] Q be a Lipschitz curve such that γ(c)=(x ,y ) and
0 0
→
γ(d)=Q . Assume that γ(t)=Q for all t in [c,d). Define ℓ :[c,d] [0,1] by
B B h
6 →
1 t
(3.3) ℓ (t)= γ′(u) du
h h
L (γ) | |
h Zc
There is a curve β in such that β(ℓ (t)) = γ(t) for all t in [c,d]. Moreover
h h
R
β′(t) = L (β) for almost every t in [0,1], and L (β) = L (γ). Furthermore
h h h h
| |
λ (β) λ (γ) for every p 2n 1.
1,p 1,p
≥ ≥ −
Proof. Define η :[0,1] [c,d] by
→
(3.4) η(s)=min t [c,d]:ℓ (t)=s
h
∈
n o
Note that η may not be continuous,but β =γ η is in , and β(ℓ (t))=γ(t) for
h h
◦ R
all t in [c,d]. Also β′(t) = L (γ) for almost every t in [0,1], so L (β) = L (γ).
h h h h
| |
LetF =F γ andF =F β. LetwbeafunctioninLip ([0,1]). Definev =w ℓ .
γ ◦ β ◦ 0 ◦ h
Then v is in Lip ([c,d]), and changing variables yields
0
d |v′|pFγp dt 1 |w′|pFβp dt
c |γ′|p−1 0 |β′|p−1
(3.5) λ (γ) h = h
1,p ≤ Rd v p γ′ dt R1 wp β′ dt
c | | | |h 0 | | | |h
Since w is arbitrary, this implieRs that λ (γ) Rλ (β). (cid:3)
1,p 1,p
≤
In the following lemma, we bound the length L (γ) of a curve γ in in terms
h h
R
of the eigenvalue λ (γ).
1,p
SIMONS’ CONE AND EQUIVARIANT MAXIMIZATION OF THE FIRST p-LAPLACE EIGENVALUE7
Lemma 3.2. Fix p 2n 1. There is a constant C such that for any γ in ,
p h
≥ − R
C
p
(3.6) L (γ)
h
≤ λ (γ)
1,p
Proof. LetβbethereparametrizationgivenbyLemma3.1sothatλ (β) λ (γ)
1,p 1,p
≥
and β′(t) =L (γ)foralmosteverytin[0,1]. Letr >0beasmallnumber. Define
h h
w:[0|,1] | R by
→
Lh(γ) t 0 t r
(3.7) w(t)= r · ≤ ≤ Lh(γ)
(1 r t 1
Lh(γ) ≤ ≤
Let F = F β. Then there is a constant C , which is independent of γ and β,
β p
◦
such that
Lhr(γ) |w′|pFβp dt
(3.8) λ (γ) λ (β) 0 |β′|ph−1 Cp
1,p ≤ 1,p ≤ RL1hr(γ) |w|p|β′|hdt ≤ Lh(γ)
Toverifythelastinequality,notethatRifr issmall,thenthereisaconstantC′ such
p
that Fp C′ over [0,r/L (γ)]. Therefore,
β ≤ p h
(3.9) 0Lhr(γ) ||wβ′′||pph−Fβ1p dt Cp′
RL1hr(γ) |w|p|β′|hdt ≤ Lh(γ)rp−1(1− Lhr(γ))
If r is small, this estRablishes (3.8). (cid:3)
Thepurposeofthenextlemmaistoshowthatthereisaneigenvaluemaximizing
sequence of curves in whose images are contained in a fixed compact subset of
h
R
Q. Let ρ = x2+y2 and define
0 0 0
(3.10) p K = (x,y) R2n/G:x2+y2 ρ2
∈ ≤ 0
Let Q be the image of K nunder the quotient map R2n/Go Q.
K
→
Lemma 3.3. Let α be a curve in . There is a curve β in such that β(t) is
h h
R R
in Q for all t in [0,1] and λ (β) λ (α) for all p 2n 1.
K 1,p 1,p
≥ ≥ −
Proof. There are functions r : [0,1) R and θ : [0,1) [0,π/2] such that for
α α
→ →
all t in [0,1),
(3.11) α(t)= r (t)cosθ (t),r (t)sinθ (t)
α α α α
(cid:16) (cid:17)
Define a function r :[0,1) [0,ρ ] by
β 0
→
ρ2
(3.12) r (t)=min 0 ,r (t)
β α
r (t)
α
(cid:16) (cid:17)
Then define a curve β in so that β(1)=Q and for all t in [0,1),
h B
R
(3.13) β(t)= r (t)cosθ (t),r (t)sinθ (t)
β α β α
(cid:16) (cid:17)
Thenβ isin andβ(t)isinQ foralltin[0,1]. LetF =F αandF =F β.
h K α β
R ◦ ◦
For all p 2n 1 and for almost every t in [0,1],
≥ −
(F (t))p (F (t))p
α β
(3.14)
α′(t)p−1 ≤ β′(t)p−1
| |h | |h
8 SINANARITURK
To verify this, note that for all t in [0,1),
F (t) r2n−2
(3.15) α = α
Fβ(t) rβ2n−2
Also, for almost every t in [0,1],
r2n−1
(3.16) α′(t) = α β′(t)
| |h r2n−1 ·| |h
β
Therefore (3.14) follows, because p 2n 1. Also α′(t) β′(t) for almost
h h
every t in [0,1]. Therefore λ (α) ≥λ (β−) for all p| 2n| 1≥. | | (cid:3)
1,p 1,p
≤ ≥ −
We can now establish the existence of an eigenvalue maximizing curve in .
h
R
For p 2n 1, define
≥ −
(3.17) Λ =sup λ (α):α
p 1,p h
∈R
(cid:26) (cid:27)
Lemma 3.4. Fix p 2n 1. There is a curve α in such that λ (α) = Λ
h 1,p p
≥ − R
and α(t) is in Q for all t in [0,1].
K
Proof. Let γ be a sequence in such that
j h
{ } R
(3.18) lim λ (γ )=Λ
1,p j p
j→∞
By Lemma 3.3, we may assumethat γ (t) is in Q for everyj andeveryt in[0,1].
j K
Using Lemma 3.1, we may assume that γ′(t) = L (γ ) for every j and almost
| j |h h j
every t in [0,1]. By Lemma 3.2, the lengths L (γ ) are uniformly bounded. By
h j
passingtoasubsequence,wemayassumethatthelengthsL (γ )convergetosome
h j
positive number ℓ. The curves γ are uniformly Lipschitz. Therefore, by applying
j
the Arzela-Ascoli theorem and passing to a subsequence, we may assume that the
curves γ converge uniformly to a Lipschitz curve γ : [0,1] Q . Moreover
j K
→
γ′(t) ℓ for almost every t in [0,b]. For each j, define F =F γ . Also define
h j j
| | ≤ ◦
F =F γ. Define b in (0,1] by
γ
◦
(3.19) b=min t [0,1]:γ(t)=Q
B
∈
n o
Let w be in Lip ([0,b]). Define v to be a function in Lip ([0,1]) which agrees with
0 0
w over [0,b] and is constant over [0,1]. Then
1 |v′|pFjp dt b |w′|pFjp dt
(3.20) Λ = lim λ (γ ) liminf 0 Lh(γj)p−1 liminf 0 Lh(γj)p−1
p j→∞ 1,p j ≤ j→∞ R01|v|pLh(γj)dt ≤ j→∞ R0b|w|pLh(γj)dt
MoreoverFj convergesto Fγ uniformlRy over[0,b], because F isRcontinuous on QK.
Also L (γ ) converges to ℓ, so
h j
(3.21) lim 0b L|hw(′γ|jp)Fpj−p1 dt = 0b |wℓ′p|−pF1γp dt 0b ||wγ′′||pph−F1γp dt
j→∞ R0b|w|pLh(γj)dt R 0b|w|pℓdt ≤ R0b|w|p|γ′|hdt
Therefore R R R
b |w′|pFγp dt
0 |γ′|p−1
(3.22) Λ h
p ≤ Rb wp γ′ dt
0 | | | |h
R
SIMONS’ CONE AND EQUIVARIANT MAXIMIZATION OF THE FIRST p-LAPLACE EIGENVALUE9
Since w is arbitrary,
(3.23) Λ λ γ
p ≤ 1,p [0,b]
Letα be the reparametrizationofγ gi(cid:16)ven(cid:12) by(cid:17)Lemma 3.1. Thenα is in and
|[0,b] (cid:12) Rh
α(t) is in Q for all t in [0,1]. Moreover
K
(3.24) λ (α) λ (γ ) Λ
1,p 1,p [0,b] p
≥ | ≥
Therfore λ (α)=Λ . (cid:3)
1,p p
Let + bethesetofcontinuouscurvesα:[0,1] R2n/Gsuchthatcomposition
with thRehquotient map R2n/G Q yields a curve→in . We use (3.2) to define
h
λ (α) for α in +. In the ne→xt lemma we establishRexistence of an eigenvalue
1,p Rh
maximizing curve in +. We first introduce new coordinates functions on R2n/G.
Define u:R2n/G RRhand v :R2n/G [0, ) by
→ → ∞
1
(3.25) u(x,y)= (x2 y2)
2 −
and
(3.26) v(x,y)=xy
These coordinates can be used to identify R2n/G with a half-plane. A feature of
these coordinates is that the function F can be expressed as F =c vn−1. Define
n
·
a function r =√u2+v2. Then the metric h can be expressed as
c2 v2n−2
(3.27) h= n· du2+dv2
2r
(cid:16) (cid:17)
By (2.4), we have
(3.28) u(x ,y ) 0
0 0
≥
Lemma 3.5. Fix p 2n 1. There is a curve α in + such that λ (α) = Λ .
≥ − Rh 1,p p
Moreover u α is monotonically increasing over [0,1].
◦
Proof. By Lemma 3.4, there is a curve β in such that λ (β)=Λ and β(t) is
h 1,p p
in Q for all t in [0,1]. Define functions u :R[0,1) R and v :[0,1) [0, ) by
K β β
→ → ∞
u = u β and v = v β. Note that u is bounded over [0,1). Also v(t) > 0 for
β β β
◦ ◦
all t in [0,1) and
(3.29) limv (t)=0
β
t→1
Inparticularv has acontinuousextensionto [0,1]. Letv :[0,1] [0, )be this
β α
extension. Note that u (0) 0 by (3.28), and define u :[0,1] →R by ∞
β α
≥ →
(3.30) u (t)=sup u (s) :s [0,t)
α β
| | ∈
(cid:26) (cid:27)
Thenu ismonotonicallyincreasing,continuous,andbounded. Defineacontinuous
α
curve α : [0,1] R2n/G so that u α = u and v α = v . Then α is in +,
→ ◦ α ◦ α Rh
and u α is monotonically increasing over [0,1]. Let F = F α and F = F β.
α β
◦ ◦ ◦
Note that F = F over [0,1], and α′(t) β′(t) for almost every t in [0,1].
α β h h
Therefore λ (α) λ (β), hence λ| (α)|=≤Λ|. | (cid:3)
1,p 1,p 1,p p
≥
In the following lemma, we establish existence of a maximizing curve α in +
Rh
such that u α is monotonically increasing and r α is monotonically decreasing.
◦ ◦
10 SINANARITURK
Lemma 3.6. Fix p 2n 1. There is a curve α in + such that λ (α) = Λ .
≥ − Rh 1,p p
Moreover u α is monotonically increasing over [0,1] and r α is monotonically
◦ ◦
decreasing over [0,1].
Proof. By Lemma 3.5, there is a curve γ in + such that λ (γ)=Λ . Moreover
Rh 1,p p
u γ is monotonically increasing over [0,1]. Define r = r γ. Note that r is
γ γ
◦ ◦
non-vanishing over [0,1). There is a function θ :[0,1] [0,π/2] such that
γ
→
(3.31) u γ,v γ = r cosθ ,r sinθ
γ γ γ γ
◦ ◦
(cid:16) (cid:17) (cid:16) (cid:17)
Define a function r :[0,1] (0, ) by
β
→ ∞
(3.32) r (t)=min r (s):s [0,t]
β γ
∈
Define a curve β :[0,1] R2n/G suchnthat o
→
(3.33) u β,v β = r cosθ ,r sinθ
β γ β γ
◦ ◦
(cid:16) (cid:17) (cid:16) (cid:17)
Note that β is in + and r is monotonically decreasing over [0,1]. Define a set
Rh β
W by
(3.34) W = t [0,1]:β(t)=γ(t)
∈
The isolated points of W are conuntable, so β′(t) = γ′(ot) for almost every t in W.
Notethatthere arecountablymanydisjointintervals(a ,b ),(a ,b ),... suchthat
1 1 2 2
(3.35) [0,1) W = (a ,b )
j j
\
j
[
Moreoverr is constant oneach interval(a ,b ). For all p 2n 1 and for almost
β j j
≥ −
every t in [0,1],
(F (t))p (F (t))p
β γ
(3.36)
β′(t)p−1 ≥ γ′(t)p−1
| |h | |h
To verify this, note that for all t in [0,1),
F (t) r2n−2
γ γ
(3.37) =
Fβ(t) rβ2n−2
Also, for almost every t in [0,1],
r2n−1
(3.38) γ′ γ β′
| |h ≥ r2n−1 ·| |h
β
Therefore(3.36)follows,becausep 2n 1. Also β′(t) γ′(t) foralmostevery
h h
≥ − | | ≤| |
tin[0,1]. Thereforeλ (β) λ (γ),soλ (β)=Λ . Defineθ :[0,1] [0,π/2]
1,p 1,p 1,p p α
≥ →
by
θ (t) t W
γ
(3.39) θ (t)= ∈
α
(min θγ(s):s [aj,t] t (aj,bj)
∈ ∈
n o
The main difficulty in proving θ is Lipschitz continuous is to verify that for each
α
point b ,
j
(3.40) min θ (s):s [a ,b ] =θ (b )
γ j j γ j
∈
n o