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Conformal bounds for the first eigenvalue of the p -Laplacian
Nonlinear Analysis 80 (2013) 88–95
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Conformal bounds for the first eigenvalue of the p-Laplacian Ana-Maria Matei ∗ Department of Mathematical Sciences, Loyola University New Orleans, 6363 St Charles Ave., New Orleans, LA 70118, United States
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info
Article history: Received 22 October 2012 Accepted 30 November 2012 Communicated by Enzo Mitidieri MSC: primary 58C40 secondary 53C21
abstract Let M be a compact, connected, m-dimensional manifold without boundary and p > 1. For 1 < p ≤ m, we prove that the first eigenvalue λ1,p of the p-Laplacian is bounded on each conformal class of Riemannian metrics of volume one on M. For p > m, we show that any conformal class of Riemannian metrics on M contains metrics of volume one with λ1,p arbitrarily large. As a consequence, we obtain that in two dimensions λ1,p is uniformly bounded on the space of Riemannian metrics of volume one if 1 < p ≤ 2, respectively unbounded if p > 2. © 2012 Elsevier Ltd. All rights reserved.
Keywords: p-Laplacian Eigenvalue Conformal volume
1. Introduction Let M be a compact m-dimensional manifold. All through this paper we will assume that M is connected and without boundary. The p-Laplacian (p > 1) associated to a Riemannian metric g on M is given by
∆p u = δ(|du|p−2 du), where δ = −divg is the adjoint of d for the L2 -norm induced by g on the space of differential forms. This operator can be viewed as an extension of the Laplace–Beltrami operator which corresponds to p = 2. The real numbers λ for which the nonlinear partial differential equation
∆p u = λ|u|p−2 u has nontrivial solutions are the eigenvalues of ∆p , and the associated solutions are the eigenfunctions of ∆p . Zero is an eigenvalue of ∆p , the associated eigenfunctions being the constant functions. The set of nonzero eigenvalues is a nonempty, unbounded subset of (0, ∞) [1]. The infimum λ1,p of this set is itself a positive eigenvalue, the first eigenvalue of ∆p , and has a Rayleigh type variational characterization [2]:
|du|p νg p−2 M u ∈ W 1,p (M ) \ {0}, λ1,p (M , g ) = inf | u | u ν = 0 , g |u|p νg M M where νg denotes the Riemannian volume element associated to g.
∗
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0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.11.026
A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95
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The first eigenvalue of ∆p can be viewed as a functional on the space of Riemannian metrics on M: g → λ1,p (M , g ). p
Since λ1,p is not invariant under dilatations (λ1,p (M , cg ) = c − 2 λ1,p (M , g )), a normalization is needed when studying the uniform boundedness of this functional. It is common to restrict λ1,p to the set M (M ) of Riemannian metrics of volume one on M. In the linear case p = 2 this problem has been extensively studied in various degrees of generality. The functional λ1,2 was shown to be uniformly bounded on M (M ) in two dimensions [3–5], and unbounded in three or more dimensions [6–11]. However, λ1,2 becomes uniformly bounded when restricted to any conformal class of Riemannian metrics in M (M ) [12]. In the general case p > 1, the functional λ1,p is unbounded on M (M ) in three or more dimensions [13]. In this paper we study the existence of uniform upper bounds for the restriction of λ1,p to conformal classes of Riemannian metrics in M (M ):
• for 1 < p ≤ m we extend the results from the linear case and obtain an explicit upper bound for λ1,p in terms of p, the dimension m and the Li–Yau n-conformal volume.
• for p > m, we consider first the case of the unit sphere S m and we construct Riemannian metrics in M(S m ), conformal to the standard metric can and with λ1,p arbitrarily large. We use then the result on spheres to show that any conformal class of Riemannian metrics on M contains metrics of volume one with λ1,p arbitrarily large. As a consequence, we obtain that in two dimensions, λ1,p is uniformly bounded on M (M ) when 1 < p ≤ 2, and unbounded when p > 2. 2. The case 1 < p ≤ m: Li–Yau type upper bounds Let g be a Riemannian metric on M and denote by [g ] = {fg |f ∈ C ∞ (M ), f > 0} the conformal class of g. Let G(n) = {γ ∈ Diff(S n ) | γ ∗ can ∈ [can]} denote the group of conformal diffeomorphisms of (S n , can). For n big enough, the Nash–Moser Theorem ensures (via the stereographic projection) that the set In (M , [g ]) = φ : M
→ S n | φ ∗ can ∈ [g ] of conformal immersions from (M , g ) to (S n , can) is nonempty. The n-conformal volume of [g ] is defined by [5]: Vnc (M , [g ]) =
sup Vol M , (γ ◦ φ)∗ can ,
inf
φ∈In (M ,[g ]) γ ∈G(n)
where Vol (M , (γ ◦ φ)∗ can) denotes the volume of M with respect to the induced metric (γ ◦ φ)∗ can. By convention, Vnc (M , [g ]) = ∞ if In (M , [g ]) = ∅. Theorem 2.1. Let M be an m-dimensional compact manifold and 1 < p ≤ m. For any metric g ∈ M (M ) and any n ∈ N we have p p p λ1,p (M , g ) ≤ m 2 (n + 1)| 2 −1| Vnc (M , [g ]) m .
Remark 2.2. In the linear case p = 2, this result was proved by Li and Yau [5] for surfaces and by El Soufi and Ilias [12] for higher dimensional manifolds. Remark 2.3. Theorem 2.1 gives an explicit upper bound for λ1,p , 1 < p ≤ m, in the case of some particular manifolds: the sphere S m , the real projective space RPm , the complex projective space C Pd , the equilateral torus T2eq , the generalized Clifford torus S r
√
r /r + q × S q
(M , [can]) = Vol(M ,
λ1,2 m
√
q/r + q , endowed with their canonical metrics. For these manifolds we have [12]: Vnc
can) for n + 1 greater or equal to the multiplicity of λ1,2 .
Using the relationships between the conformal volume and the genus of a compact surface [5,14] we obtain: Corollary 2.4. Suppose m = 2 and 1 < p ≤ 2. Then for any metric g ∈ M (M )
λ1,p (M , g ) ≤ kp
genus(M ) + 3 2
2p
,
p p p p where [ ] denotes the integer part, kp = 3| 2 −1| (8π ) 2 if M is orientable and kp = 5| 2 −1| (24π ) 2 if not.
Remark 2.5. In the case p = 2 and M = S 2 , this result is the well known Hersch inequality [3]. For higher genus surfaces, the upper bound of λ1,2 in terms of the genus is due to Yang and Yau [4] (see also [14]). In order to prove Theorem 2.1 we need two lemmas:
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A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95
Lemma 2.6. Let φ : (M , g ) → (S n , can) be a smooth map whose level sets are of measure zero in (M , g ). Then for any p > 1 there exists γ ∈ G(n) such that
|(γ ◦ φ)i |p−2 (γ ◦ φ)i νg = 0,
1 ≤ i ≤ n + 1.
M
Proof of Lemma 2.6. Let a ∈ S n and denote by πa the stereographic projection of pole a. Let t ∈ (0, 1] and H 1−t = e (i.e. H 1−t is the linear dilatation of Rn of factor e
1−t t
t
). Let γta ∈ G(n), γta (x) =
t
πa−1 ◦ H 1−t ◦ πa (x)
if x ∈ S \ {a}
a
if x = a
t
1−t t
· IdRn
n
and consider
the continuous map F : (0, 1] × S n → Rn+1
1
F (t , a) =
Vol(M , g )
|(γ ◦ φ)1 |
p−2
a t
(γ ◦ φ)1 νg , . . . , a t
M
|(γ ◦ φ)n+1 | a t
p−2
(γ ◦ φ)n+1 νg . a t
M
For any x ∈ M \ {φ −1 (−a)} we have limt →0+ γta ◦ φ(x) = a. Since φ −1 (−a) is of measure zero in M, we can extend F into a continuous function on [0, 1] × S n by setting F (0, a) = |a1 |p−2 a1 , . . . , |an+1 |p−2 an+1 .
The map a → F (0, a) is odd on S n , and since γ1a = IdS n , the map a → F (1, a) is constant. Assume ∥F (t , a)∥ ̸= 0 for any (t , a) ∈ [0, 1] × S n . Then the map G : [0, 1] × S n → S n F (t , a) G(t , a) = ∥F (t , a)∥ gives a homotopy between the odd map a → G(0, a) and the constant map a → G(1, a), and this is impossible. Hence there exists (t , a) ∈ [0, 1] × S n such that ∥F (t , a)∥ = 0, i.e. M | γta ◦ φ i |p−2 γta ◦ φ i νg = 0, 1 ≤ i ≤ n + 1. Lemma 2.7. Suppose g ∈ M (M ) and let φ : (M , g ) → (S n , can) be a smooth map whose level sets are of measure zero in (M , g ). Then there exists γ ∈ G(n) such that p λ1,p (M , g ) ≤ (n + 1)| 2 −1| |d(γ ◦ φ)|p νg , M
where |d(γ ◦ φ)| denotes the Hilbert–Schmidt norm of d(γ ◦ φ). Proof of Lemma 2.7. Lemma 2.6 implies there exists γ ∈ G(n) such that ψ = γ ◦ φ : M → S n verifies 1 ≤ i ≤ n + 1. The variational characterization for λ1,p (M , g ) implies that λ1,p (M , g ) ≤ +1 n M
λ1,p (M , g ) ≤
M
|ψi |p−2 ψi νg = 0,
, 1 ≤ i ≤ n + 1. Then
|dψi |p νg
i=1
+1 n M
|dψi |p νg M p M |ψi | νg
.
(2.1)
|ψi |p νg
i =1
• Case 1: p ≥ 2. It is straightforward that n +1
|dψi | = p
i =1
n+1
p
(|dψ | ) ≤ 2 2 i
n +1
2p |dψi |
2
= |dψ|p .
(2.2)
i =1
i=1
On the other hand n+1
|ψi | ≥ (n + 1) p
p
1− 2
i=1
n +1
2p 2
|ψi |
p
= (n + 1)1− 2 ,
(2.3)
i =1 p
where we have used the fact that x → x 2 is convex and that p
λ1,p (M , g ) ≤ (n + 1) 2 −1
|dψ|p νg . M
n +1 i=1
|ψi |2 = 1. Replacing (2.2) and (2.3) in (2.1) we obtain
A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95
91
• Case 2: 1 < p < 2. Since |ψi | ≤ 1 we have |ψi |2 ≤ |ψi |p and n +1 n+1 1 = Vol(M , g ) = |ψi |2 νg ≤ |ψi |p νg . M i =1
(2.4)
M i=1
On the other hand n +1
n+1 p p |dψi | = (|dψi |2 ) 2 ≤ (n + 1)1− 2
p
i =1
n +1
i=1
2p |dψi |
2
p
= (n + 1)1− 2 |dψ|p ,
(2.5)
i=1 p
where the inequality follows from the concavity of x → x 2 . Replacing (2.4) and (2.5) in (2.1) we obtain p
λ1,p (M , g ) ≤ (n + 1)1− 2
|dψ|p νg . M
Proof of Theorem 2.1. Let φ : (M , g ) → (S n , can) be a conformal immersion. From Lemma 2.7 we have that there exists
γ ∈ G(n) such that
p λ1,p (M , g ) ≤ (n + 1)| 2 −1|
|d(γ ◦ φ)|p νg . M
Since g ∈ M (M ), Hölder’s inequality implies
|d(γ ◦ φ)| νg ≤ p
M
|d(γ ◦ φ)| νg m
mp
.
M
On the other hand since γ ◦ φ : (M , g ) → (S n , can) is a conformal immersion, (γ ◦ φ)∗ can =
m
|d(γ ◦φ)|2 m
g and we have
m
|d(γ ◦ φ)|m νg = m 2 Vol(M , (γ ◦ φ)∗ can) ≤ m 2 sup Vol(M , (γ ◦ φ)∗ can). γ ∈G(n)
M
Combining the inequalities above we obtain: p 2
p λ1,p (M , g ) ≤ m (n + 1)| 2 −1|
mp
sup Vol(M , (γ ◦ φ) can) ∗
γ ∈G(n)
.
Taking the infimum over all φ ∈ In (M , [g ]) we obtain the desired inequality.
Proof of Corollary 2.4. Li and Yau [5] have noted that the n-conformal volume is bounded above by a constant depending only on the genus of the surface. Their constant has been improved by El-Soufi and Ilias [14]. If M is orientable we have Vnc
(M , [g ]) ≤ 4π
genus(M ) + 3
2
for n ≥ 2.
If M is non orientable, Vnc
(M , [g ]) ≤ 12π
genus(M ) + 3 2
for n ≥ 4.
p p p p Theorem 2.1 implies now the desired result with kp = 3| 2 −1| (8π ) 2 when M is orientable and kp = 5| 2 −1| (24π ) 2 when M is non orientable.
3. The case p > m For the sake of self-containedness we include here the variational characterizations for the first eigenvalues for the Dirichlet and the Neumann problems for ∆p . Let Ω be a domain in M and consider the Dirichlet problem:
∆p u = λ |u|p−2 u in Ω u = 0 on ∂ Ω .
The infimum λD1,p (Ω , g ) of the set of eigenvalues for this problem is itself a positive eigenvalue with the variational characterization
|du|p νg 1,p Ω λ (Ω , g ) = inf u ∈ W0 (Ω ) \ {0} . |u|p νg Ω D 1,p
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A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95
Consider now the Neumann problem on Ω :
∆p f = |f |p−2 f in Ω df (η) = 0 on ∂ Ω ,
where η denotes the exterior unit normal vector field to ∂ Ω . Here too, the infimum λN1,p (Ω , g ) of the set of nonzero eigenvalues is a positive eigenvalue with the variational characterization
|df |p νg p−2 f ∈ W 1,p (Ω , g ) \ {0}, λ (Ω , g ) := inf Ω | f | f ν = 0 . g |f |p νg Ω Ω N 1 ,p
We consider first the case of (S m , [can]): Theorem 3.1. For any p > m, S m carries Riemannian metrics of volume one, conformal to the standard metric can, with λ1,p arbitrarily large. Proof of Theorem 3.1. Let r ∈ [0, π], denote the geodesic distance on (S m , can) w.r.t. a point x0 ∈ S m . Let ε > 0 and define a radial function fε : S m → R by 4p
fε (r ) = ε m(p−m) · χ[0, π −ε]∪[ π +ε,π ] (r ) + χ( π −ε, π +ε) (r ). 2
2
2
(3.1)
2
Let
m−p 2 p | du | f ν m ε m can 1,p m p−2 2 ν λ1,p (ε) = inf Rε (u) := S u ∈ W ( S ) \ { 0 }, | u | u | f | = 0 . ε can m pf 2 ν Sm | u | ε m can S
We will show first that p
lim sup λ1,p (ε) · ε m = ∞.
(3.2)
ε→0
Classical density arguments imply that there exists uε ∈ W 1,p (S m ) \ {0} with Rε (uε ). Let u¯ ε : S m → R be a radial function defined by u¯ ε (r ) = p
1 V
S m−1
Sm
m
|uε |p−2 uε fε 2 νcan = 0 such that λ1,p (ε) =
|uε (r , ·)|p νcan ,
(3.3)
where V = Vol(S m−1 , can). Differentiating w.r.t. r we obtain p
pu¯ pε−1 u¯ ′ε =
V
S m−1
|uε |p−2 uε
∂ uε νcan . ∂r
By Hölder’s inequality we obtain u¯ pε−1 |¯u′ε | ≤
p−p 1 1p ∂ uε p ∂ uε p νcan ≤ 1 |uε |p−1 | u | ν · ν ε can ∂ r can . V S m−1 ∂r V S m−1 S m−1 1
It follows that
|¯u′ε |p ≤
∂ uε p νcan ≤ 1 |duε |p νcan . V S m−1 ∂ r V S m−1
1
On the other hand
π m m |¯uε |p fε 2 νcan = V · |¯uε |p fε 2 sin r m−1 dr 0 Sm m π p = |uε | νcan fε 2 sin r m−1 dr 0 S m−1 m = |uε |p fε 2 νcan , Sm
where the second equality follows from (3.3). Similarly (3.4) implies
m−p 2
Sm
|¯u′ε |p fε
νcan ≤
m−p 2
Sm
|duε |p fε
νcan .
(3.4)
A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95
93
In particular, we obtain that u¯ ε ∈ W 1,p (S m ) and
Sm
λ1,p (ε) = Rε (uε ) ≥
m−p 2
|¯u′ε |p fε
νcan
m 2
|¯uε | fε νcan Sm m−p m−p S m |¯u′ε |p fε 2 νcan S m |¯u′ε |p fε 2 νcan , − ≥ min + , m m m |¯uε |p fε 2 νcan |¯uε |p fε 2 νcan S Sm p
+
−
where S+ , S− denote the hemispheres centered at x0 , respectively −x0 . Without loss of generality we may assume that m
m
m S+
λ1,p (ε) ≥
m−p 2
|¯u′ε |p fε
m S+
νcan
m 2
|¯uε |p fε νcan
.
(3.5) π − ε] π 2 π and vε on ( − ε, ]
Let wε ∈ W
1,p
u¯ ε
(S+ ), wε = m
u¯ ε (
π 2
on [0,
− ε)
2
= u¯ ε − wε . Then vε = 0 on [0, π2 − ε] and wε′ = 0 on ( π2 − ε, π2 ).
2
Since vε′ and wε′ have disjoint supports, we have |¯u′ε |p = |vε′ |p + |wε′ |p . On the other hand |¯uε |p = |vε + wε |p ≤ 2p−1 (|vε |p + |wε |p ). Then (3.5) and (3.1) imply
m S+
λ1,p (ε) ≥ 21−p
m−p 2
(|vε′ |p + |wε′ |p )fε
νcan
m 2
(|vε |p + |wε |p )fε νcan − 2p ′ p ′ p m |vε | νcan + ε m m |wε | νcan S+ 1−p S+ =2 . m p p 2 m |vε | νcan + m |wε | fε νcan S S m S+
+
+
Quite to multiply u¯ ε by a constant we may assume
λ1,p (ε) ≥ 21−p
m
S+
• Case 1: lim supε→0
2p
m S+
|vε′ |p νcan + ε − m
m S+
|vε |p νcan +
m
m S+
|wε |p fε 2 νcan = 1 and the inequality above becomes
|wε′ |p νcan .
m
S+
(3.6)
|wε′ |p νcan > 0. p
p
Inequality (3.6) implies that λ1,p (ε)ε m ≥ 21−p ε − m
m S+
|wε′ |p νcan , and therefore (3.2) is verified.
|wε′ |p νcan = 0. Then we may find a sequence εn → 0 such that wεn → c strongly in Lp (M ), where c is a constant. In particular since p > m m m, {fεn } is uniformly bounded and we have limn→∞ S m fεn2 νcan = 0. It follows that limn→∞ S m |wεn |p fεn2 νcan = limn→∞ + + m m m 2 2 p p p p |wεn |p fε 2 νcan ≥ 12 m (|wεn | −|c | )fεn νcan +|c | limn→∞ m fεn νcan = 0. Hence for εn small enough, m |vεn | νcan = 1 − Sm S S S • Case 2: limε→0
m S+
+
+
+
+
and (3.6) implies p 1− 2
λ1,p (εn ) ≥ 2
− 2p
≥2
− 2p
m S+
sin
|vε′ n |p νcan ≥ 2 π 2
− εn
m−1
m S+
|vε′ n |p νcan
m S+
|vεn |p νcan
π
2
|vε′ n |p dr
2
|vεn |p dr
π −ε n 2 π π −ε n 2
− 2p
=2
π
2
|vε′ n |p sin r m−1 dr
2
|vεn |p sin r m−1 dr
π −ε n 2 π π −ε n 2
.
(3.7)
1,p Let v¯ εn ∈ W0 (−εn , εn ) be an even function such that v¯ εn (s) = vεn (s + π2 − εn ) for 0 ≤ s ≤ εn . We have then
π
2
π −ε n 2 π 2
π −ε n 2
|vε′ n |p dr p dr
|vεn |
εn 0
|¯vε′ n |p dr
0
|¯vεn |p dr
= εn
εn −ε = εn n
|¯vε′ n |p dr
vεn |p dr −εn |¯
≥ λD1,p (−εn , εn ) = εn−p λD1,p (−1, 1).
(3.8)
94
A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95 p
Inequalities (3.7), (3.8) imply λ1,p (εn ) ≥ 2− 2 [sin( π2 − εn )]m−1 εn λD1,p (−1, 1) and it is straightforward that (3.2) is verified in this case too. Fix now ε > 0 and let f˜ε ∈ C ∞ (S m ), radial with respect to x0 and such that f˜ε ≤ fε , f˜ε (r ) = fε (r ) = 1 on [ π2 − 2ε , π2 + 2ε ] −p
and f˜ε (π − r ) = f˜ (r ). Then
Vol(S m , f˜ε can) =
m
f˜ε 2 νcan =
Sm
π +ε 2 2
>V
π −ε 2 2
π
m
2
f˜ε 2 sin r m−1 dr νcan
− π2
S m−1
sin r m−1 dr
π m−1 > εV sin −ε ,
where V = Vol(S m−1 , can).
2
(3.9)
˜− We will compare now λ1,p (S m , f˜ε can) and λ1,p (ε). Let u˜ ε be an eigenfunction for λ1,p (S m , f˜ε can) and denote by u˜ + ε ,u ε the positive, respectively, the negative part of u˜ ε . Then [15] m−p 2
p ˜ |du˜ + ε | fε
Sm
λ1,p (S , f˜ε can) = m
Sm
m−p 2
p ˜ |du˜ − ε | fε
νcan
. p ˜ |˜u− ε | fε νcan m ˜− = t u˜ + uε,t0 |p−2 u˜ ε,t0 fε 2 νcan = 0 and the equation above implies ε +u ε . Then there is t0 such that S m |˜
Sm
Let t ∈ R and u˜ ε,t
νcan
m−p 2
Sm
Sm
=
p ˜ |˜u+ ε | fε νcan
|du˜ ε,t0 |p f˜ε
λ1,p (S , f˜ε can) = m
m 2
m 2
Sm
νcan
m 2
Sm
≥
|˜uε,t0 |p f˜ε νcan
m−p 2
|du˜ ε,t0 |p fε
Sm
νcan
m 2
|˜uε,t0 |p fε νcan
≥ λ1,p (ε),
(3.10)
where the first inequality follows from the fact that f˜ε ≤ fε and the second from the variational characterization for λ1,p (ε). Inequalities (3.9), (3.10) and (3.2) yield p
p
p
lim sup λ1,p (S m , f˜ε can)Vol(S m , f˜ε can) m ≥ V m · lim sup λ1,p (ε) · ε m = ∞. ε→0
ε→0
2 −m
Finally, let hε = Vol(S m , f˜ε can) Vol(S m , hε can) = 1
f˜ε . We have then lim sup λ1,p (S m , hε can) = ∞.
and
ε→0
We will extend the construction from (S m , [can]) to (M , [g ]) by means of the first eigenvalue for the Neumann problem for ∆p on a domain Ω in M. Theorem 3.2. Let (M , g ) be a compact Riemannian manifold of dimension m. Then for any p > m, [g ] contains Riemannian metrics of volume one with λ1,p arbitrarily large. Proof of Theorem 3.2. Let r denote the geodesic distance on (S m , can) w.r.t. a point x0 . Let f ∈ C ∞ (S m ) be a function radial m denote the hemisphere centered at x0 . Let v w.r.t. x0 , such that f (r ) = f (π − r ) and Vol(S m , fcan) = 1. As before, let S+
be an eigenfunction for λ
N 1 ,p
2
m S+
(S+ , fcan) and let w ∈ W m
1,p
(S ), w(r ) = m
v(r )
if 0 ≤ r ≤
v(π − r )
if
m
π 2
π
2
. Then
m
Sm
|w|p−2 wf 2 νcan =
|v|p−2 v f 2 νcan = 0 and the variational characterization for λ1,p (S m , fcan) implies λ1,p (S , fcan) ≤ m
Sm
|dw|p f
Sm
|w|
m−p 2 m
pf 2
νcan
νcan
m S+
=
|dv|p f m
S+
m−p 2 m
pf 2
|v|
νcan
νcan
= λN1,p (S+m , f can).
(3.11)
m Let Ω be a domain in M such that there exists a diffeomorphism Φ : Ω → S+ . We may assume Ω is included in the open
region of a local chart of M. In this chart we have νg = det(gij )dx1 ∧ dx2 ∧ · · · ∧ dxm and νΦ ∗ can = dx2 ∧ · · · ∧ dxm . There exist positive constants c1 , c2 such that
c1
det(gij ) ≤
det((Φ ∗ can)ij ) ≤ c2
We will compare now λ m and (S+ , fcan) we have
N 1 ,p
det(gij ) on Ω .
det((Φ ∗ can)ij )dx1 ∧
(3.12)
(S+ , f can) and λ (Ω , (f ◦ Φ )g ). Note first that since Φ is an isometry between (Ω , (f ◦ Φ )Φ ∗ can) m
N 1 ,p
λN1,p (S+m , fcan) = λN1,p (Ω , (f ◦ Φ )Φ ∗ can).
(3.13)
A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95
95
Let u be an eigenfunction for λN1,p (Ω , (f ◦ Φ )g ) and denote by u+ , u− the positive, respectively, the negative part of u. Then
|dus |p (f ◦ Φ )
Ω
λN1,p (Ω , (f ◦ Φ )g ) =
Ω
c1
≥
c2
m−p 2 m 2
νg
|us | (f ◦ Φ ) νg p
≥
c1
c2
m
there is s ∈ R such that the function us = su+ + u− verifies
Ω
|us |p−2 us (f ◦ Φ ) 2 νΦ ∗ can = 0. Furthermore
|dus |p (f ◦ Φ )
Ω
Ω
m−p 2
νΦ ∗ can
m 2
|us | (f ◦ Φ ) νΦ ∗ can p
λN1,p (Ω , (f ◦ Φ )Φ ∗ can),
(3.14)
where the first inequality follows from (3.12) and the second from the variational characterization of λN1,p (Ω , (f ◦ Φ )Φ ∗ can). From (3.11), (3.13) and (3.14) we obtain c1
λN1,p (Ω , (f ◦ Φ )g ) ≥
c2
λ1,p (S m , fcan).
(3.15)
Let now δ > 0; there is an extension f ◦ Φ of f ◦ Φ on the entire manifold M such that the metric g˜ = f ◦ Φ g verifies [16]: N ˜ λ1,p (M , g ) > λ1,p (Ω , (f ◦ Φ )g ) − δ . Inequality (3.15) implies
λ1,p (M , g˜ ) >
c1 c2
λ1,p (S m , fcan) − δ.
(3.16)
On the other hand Vol(M , g˜ ) > Vol(Ω , (f ◦ Φ )g ) ≥
=
1 c2
m Vol(S+ , fcan) =
1 c2 1
2c2
Vol(Ω , (f ◦ Φ )Φ ∗ can) Vol(S m , fcan) =
1 2c2
.
(3.17) p
p
Let K > 0; from the proof of Theorem 3.1 we may assume that f is chosen such that λ1,p (S m , fcan) > 2 m +1 c1−1 c2m
p
2 −m
Finally, let h = Vol(M , g˜ )
K . For
δ < K , inequalities (3.16) and (3.17) imply p c1 ≥ λ1,p (S m , fcan) − δ (2c2 )− m > K .
δ small enough such that (2c2 ) λ1,p (M , g˜ )Vol(M , g˜ ) m
+1
p −m
c2
g˜ . Then h ∈ [g ], Vol(M , h) = 1 and λ1,p (M , h) > K .
References [1] J.P. Garzia Azorero, I. Peral Alonso, Existence and nonuniqueness for the p-Laplacian eigenvalues, Comm. Partial Differential Equations 12 (1987) 1389–1430. [2] L. Veron, Some existence and uniqueness results for solution of some quasilinear elliptic equations on compact Riemannian manifolds, in: Colloquia Mathematica Societatis Janos Bolyai, 62, P. D. E, Budapest, 1991, pp. 317–352. [3] J. Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris 270 (1980) 1645–1648. [4] P.C. Yang, S.T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci., serie 4 7 (1980) 55–63. [5] P. Li, S.T. Yau, A new conformal invariant and its application to the Wilmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982) 269–291. [6] S. Tanno, The first eigenvalue of the Laplacian on spheres, Tôhoku Math. J. 31 (1979) 179–185. [7] H. Urakawa, On the least positive eigenvalue of the Laplacian for compact group manifold, J. Math. Soc. Japan 31 (1979) 209–226. [8] H. Muto, The first eigenvalue of the Laplacian on even dimensional spheres, Tohoku Math. J. 32 (1980) 427–432. [9] L. Bérard Bergery, J.-P. Bourguignon, Laplacians and riemannian submersions with totally geodesic fibres, Illinois J. Math. 26 (1982) 181–200. [10] B. Colbois, J. Dodziuk, Riemannian metrics with large λ1 , Proc. Amer. Math. Soc. 122 (3) (1994) 905–906. [11] J. Dodziuk, Nonexistence of universal upper bounds for the first positive eigenvalue of the Laplace–Beltrami operator, in: Geometry of the Spectrum (Seatle, Wa, 1993), in: Contemp. Math., vol. 173, Amer. Math. Soc., Providence RI, 1994, pp. 109–114. [12] A. El Soufi, S. Ilias, Immersion minimales, première valeur propre du laplacien et volume conforme, Math. Ann. 275 (1986) 257–267. [13] A.-M. Matei, Boundedness of the first eigenvalue of the p-Laplacian, Proc. Amer. Math. Soc. 133 (7) (2005) 2183–2192. [14] A. El Soufi, S. Ilias, Le volume conforme et ses applications d’apres Li et Yau, in: Sém, Théorie Spectrale et Géométrie, in: Inst. Fourier année 1983–1984, vol. VII, 1984. [15] A.-M. Matei, First eigenvalue for the p-Laplace operator, Nonlinear Anal. 39 (2000) 1051–1068. [16] A.-M. Matei, The effect of perturbations on the first eigenvalue of the p-Laplacian, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 255–258.
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Conformal bounds for the first eigenvalue of the p-Laplacian Ana-Maria Matei ∗ Department of Mathematical Sciences, Loyola University New Orleans, 6363 St Charles Ave., New Orleans, LA 70118, United States
article
info
Article history: Received 22 October 2012 Accepted 30 November 2012 Communicated by Enzo Mitidieri MSC: primary 58C40 secondary 53C21
abstract Let M be a compact, connected, m-dimensional manifold without boundary and p > 1. For 1 < p ≤ m, we prove that the first eigenvalue λ1,p of the p-Laplacian is bounded on each conformal class of Riemannian metrics of volume one on M. For p > m, we show that any conformal class of Riemannian metrics on M contains metrics of volume one with λ1,p arbitrarily large. As a consequence, we obtain that in two dimensions λ1,p is uniformly bounded on the space of Riemannian metrics of volume one if 1 < p ≤ 2, respectively unbounded if p > 2. © 2012 Elsevier Ltd. All rights reserved.
Keywords: p-Laplacian Eigenvalue Conformal volume
1. Introduction Let M be a compact m-dimensional manifold. All through this paper we will assume that M is connected and without boundary. The p-Laplacian (p > 1) associated to a Riemannian metric g on M is given by
∆p u = δ(|du|p−2 du), where δ = −divg is the adjoint of d for the L2 -norm induced by g on the space of differential forms. This operator can be viewed as an extension of the Laplace–Beltrami operator which corresponds to p = 2. The real numbers λ for which the nonlinear partial differential equation
∆p u = λ|u|p−2 u has nontrivial solutions are the eigenvalues of ∆p , and the associated solutions are the eigenfunctions of ∆p . Zero is an eigenvalue of ∆p , the associated eigenfunctions being the constant functions. The set of nonzero eigenvalues is a nonempty, unbounded subset of (0, ∞) [1]. The infimum λ1,p of this set is itself a positive eigenvalue, the first eigenvalue of ∆p , and has a Rayleigh type variational characterization [2]:
|du|p νg p−2 M u ∈ W 1,p (M ) \ {0}, λ1,p (M , g ) = inf | u | u ν = 0 , g |u|p νg M M where νg denotes the Riemannian volume element associated to g.
∗
Tel.: +1 504 865 3340; fax: +1 504 865 2051. E-mail address: [email protected].
0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2012.11.026
A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95
89
The first eigenvalue of ∆p can be viewed as a functional on the space of Riemannian metrics on M: g → λ1,p (M , g ). p
Since λ1,p is not invariant under dilatations (λ1,p (M , cg ) = c − 2 λ1,p (M , g )), a normalization is needed when studying the uniform boundedness of this functional. It is common to restrict λ1,p to the set M (M ) of Riemannian metrics of volume one on M. In the linear case p = 2 this problem has been extensively studied in various degrees of generality. The functional λ1,2 was shown to be uniformly bounded on M (M ) in two dimensions [3–5], and unbounded in three or more dimensions [6–11]. However, λ1,2 becomes uniformly bounded when restricted to any conformal class of Riemannian metrics in M (M ) [12]. In the general case p > 1, the functional λ1,p is unbounded on M (M ) in three or more dimensions [13]. In this paper we study the existence of uniform upper bounds for the restriction of λ1,p to conformal classes of Riemannian metrics in M (M ):
• for 1 < p ≤ m we extend the results from the linear case and obtain an explicit upper bound for λ1,p in terms of p, the dimension m and the Li–Yau n-conformal volume.
• for p > m, we consider first the case of the unit sphere S m and we construct Riemannian metrics in M(S m ), conformal to the standard metric can and with λ1,p arbitrarily large. We use then the result on spheres to show that any conformal class of Riemannian metrics on M contains metrics of volume one with λ1,p arbitrarily large. As a consequence, we obtain that in two dimensions, λ1,p is uniformly bounded on M (M ) when 1 < p ≤ 2, and unbounded when p > 2. 2. The case 1 < p ≤ m: Li–Yau type upper bounds Let g be a Riemannian metric on M and denote by [g ] = {fg |f ∈ C ∞ (M ), f > 0} the conformal class of g. Let G(n) = {γ ∈ Diff(S n ) | γ ∗ can ∈ [can]} denote the group of conformal diffeomorphisms of (S n , can). For n big enough, the Nash–Moser Theorem ensures (via the stereographic projection) that the set In (M , [g ]) = φ : M
→ S n | φ ∗ can ∈ [g ] of conformal immersions from (M , g ) to (S n , can) is nonempty. The n-conformal volume of [g ] is defined by [5]: Vnc (M , [g ]) =
sup Vol M , (γ ◦ φ)∗ can ,
inf
φ∈In (M ,[g ]) γ ∈G(n)
where Vol (M , (γ ◦ φ)∗ can) denotes the volume of M with respect to the induced metric (γ ◦ φ)∗ can. By convention, Vnc (M , [g ]) = ∞ if In (M , [g ]) = ∅. Theorem 2.1. Let M be an m-dimensional compact manifold and 1 < p ≤ m. For any metric g ∈ M (M ) and any n ∈ N we have p p p λ1,p (M , g ) ≤ m 2 (n + 1)| 2 −1| Vnc (M , [g ]) m .
Remark 2.2. In the linear case p = 2, this result was proved by Li and Yau [5] for surfaces and by El Soufi and Ilias [12] for higher dimensional manifolds. Remark 2.3. Theorem 2.1 gives an explicit upper bound for λ1,p , 1 < p ≤ m, in the case of some particular manifolds: the sphere S m , the real projective space RPm , the complex projective space C Pd , the equilateral torus T2eq , the generalized Clifford torus S r
√
r /r + q × S q
(M , [can]) = Vol(M ,
λ1,2 m
√
q/r + q , endowed with their canonical metrics. For these manifolds we have [12]: Vnc
can) for n + 1 greater or equal to the multiplicity of λ1,2 .
Using the relationships between the conformal volume and the genus of a compact surface [5,14] we obtain: Corollary 2.4. Suppose m = 2 and 1 < p ≤ 2. Then for any metric g ∈ M (M )
λ1,p (M , g ) ≤ kp
genus(M ) + 3 2
2p
,
p p p p where [ ] denotes the integer part, kp = 3| 2 −1| (8π ) 2 if M is orientable and kp = 5| 2 −1| (24π ) 2 if not.
Remark 2.5. In the case p = 2 and M = S 2 , this result is the well known Hersch inequality [3]. For higher genus surfaces, the upper bound of λ1,2 in terms of the genus is due to Yang and Yau [4] (see also [14]). In order to prove Theorem 2.1 we need two lemmas:
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A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95
Lemma 2.6. Let φ : (M , g ) → (S n , can) be a smooth map whose level sets are of measure zero in (M , g ). Then for any p > 1 there exists γ ∈ G(n) such that
|(γ ◦ φ)i |p−2 (γ ◦ φ)i νg = 0,
1 ≤ i ≤ n + 1.
M
Proof of Lemma 2.6. Let a ∈ S n and denote by πa the stereographic projection of pole a. Let t ∈ (0, 1] and H 1−t = e (i.e. H 1−t is the linear dilatation of Rn of factor e
1−t t
t
). Let γta ∈ G(n), γta (x) =
t
πa−1 ◦ H 1−t ◦ πa (x)
if x ∈ S \ {a}
a
if x = a
t
1−t t
· IdRn
n
and consider
the continuous map F : (0, 1] × S n → Rn+1
1
F (t , a) =
Vol(M , g )
|(γ ◦ φ)1 |
p−2
a t
(γ ◦ φ)1 νg , . . . , a t
M
|(γ ◦ φ)n+1 | a t
p−2
(γ ◦ φ)n+1 νg . a t
M
For any x ∈ M \ {φ −1 (−a)} we have limt →0+ γta ◦ φ(x) = a. Since φ −1 (−a) is of measure zero in M, we can extend F into a continuous function on [0, 1] × S n by setting F (0, a) = |a1 |p−2 a1 , . . . , |an+1 |p−2 an+1 .
The map a → F (0, a) is odd on S n , and since γ1a = IdS n , the map a → F (1, a) is constant. Assume ∥F (t , a)∥ ̸= 0 for any (t , a) ∈ [0, 1] × S n . Then the map G : [0, 1] × S n → S n F (t , a) G(t , a) = ∥F (t , a)∥ gives a homotopy between the odd map a → G(0, a) and the constant map a → G(1, a), and this is impossible. Hence there exists (t , a) ∈ [0, 1] × S n such that ∥F (t , a)∥ = 0, i.e. M | γta ◦ φ i |p−2 γta ◦ φ i νg = 0, 1 ≤ i ≤ n + 1. Lemma 2.7. Suppose g ∈ M (M ) and let φ : (M , g ) → (S n , can) be a smooth map whose level sets are of measure zero in (M , g ). Then there exists γ ∈ G(n) such that p λ1,p (M , g ) ≤ (n + 1)| 2 −1| |d(γ ◦ φ)|p νg , M
where |d(γ ◦ φ)| denotes the Hilbert–Schmidt norm of d(γ ◦ φ). Proof of Lemma 2.7. Lemma 2.6 implies there exists γ ∈ G(n) such that ψ = γ ◦ φ : M → S n verifies 1 ≤ i ≤ n + 1. The variational characterization for λ1,p (M , g ) implies that λ1,p (M , g ) ≤ +1 n M
λ1,p (M , g ) ≤
M
|ψi |p−2 ψi νg = 0,
, 1 ≤ i ≤ n + 1. Then
|dψi |p νg
i=1
+1 n M
|dψi |p νg M p M |ψi | νg
.
(2.1)
|ψi |p νg
i =1
• Case 1: p ≥ 2. It is straightforward that n +1
|dψi | = p
i =1
n+1
p
(|dψ | ) ≤ 2 2 i
n +1
2p |dψi |
2
= |dψ|p .
(2.2)
i =1
i=1
On the other hand n+1
|ψi | ≥ (n + 1) p
p
1− 2
i=1
n +1
2p 2
|ψi |
p
= (n + 1)1− 2 ,
(2.3)
i =1 p
where we have used the fact that x → x 2 is convex and that p
λ1,p (M , g ) ≤ (n + 1) 2 −1
|dψ|p νg . M
n +1 i=1
|ψi |2 = 1. Replacing (2.2) and (2.3) in (2.1) we obtain
A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95
91
• Case 2: 1 < p < 2. Since |ψi | ≤ 1 we have |ψi |2 ≤ |ψi |p and n +1 n+1 1 = Vol(M , g ) = |ψi |2 νg ≤ |ψi |p νg . M i =1
(2.4)
M i=1
On the other hand n +1
n+1 p p |dψi | = (|dψi |2 ) 2 ≤ (n + 1)1− 2
p
i =1
n +1
i=1
2p |dψi |
2
p
= (n + 1)1− 2 |dψ|p ,
(2.5)
i=1 p
where the inequality follows from the concavity of x → x 2 . Replacing (2.4) and (2.5) in (2.1) we obtain p
λ1,p (M , g ) ≤ (n + 1)1− 2
|dψ|p νg . M
Proof of Theorem 2.1. Let φ : (M , g ) → (S n , can) be a conformal immersion. From Lemma 2.7 we have that there exists
γ ∈ G(n) such that
p λ1,p (M , g ) ≤ (n + 1)| 2 −1|
|d(γ ◦ φ)|p νg . M
Since g ∈ M (M ), Hölder’s inequality implies
|d(γ ◦ φ)| νg ≤ p
M
|d(γ ◦ φ)| νg m
mp
.
M
On the other hand since γ ◦ φ : (M , g ) → (S n , can) is a conformal immersion, (γ ◦ φ)∗ can =
m
|d(γ ◦φ)|2 m
g and we have
m
|d(γ ◦ φ)|m νg = m 2 Vol(M , (γ ◦ φ)∗ can) ≤ m 2 sup Vol(M , (γ ◦ φ)∗ can). γ ∈G(n)
M
Combining the inequalities above we obtain: p 2
p λ1,p (M , g ) ≤ m (n + 1)| 2 −1|
mp
sup Vol(M , (γ ◦ φ) can) ∗
γ ∈G(n)
.
Taking the infimum over all φ ∈ In (M , [g ]) we obtain the desired inequality.
Proof of Corollary 2.4. Li and Yau [5] have noted that the n-conformal volume is bounded above by a constant depending only on the genus of the surface. Their constant has been improved by El-Soufi and Ilias [14]. If M is orientable we have Vnc
(M , [g ]) ≤ 4π
genus(M ) + 3
2
for n ≥ 2.
If M is non orientable, Vnc
(M , [g ]) ≤ 12π
genus(M ) + 3 2
for n ≥ 4.
p p p p Theorem 2.1 implies now the desired result with kp = 3| 2 −1| (8π ) 2 when M is orientable and kp = 5| 2 −1| (24π ) 2 when M is non orientable.
3. The case p > m For the sake of self-containedness we include here the variational characterizations for the first eigenvalues for the Dirichlet and the Neumann problems for ∆p . Let Ω be a domain in M and consider the Dirichlet problem:
∆p u = λ |u|p−2 u in Ω u = 0 on ∂ Ω .
The infimum λD1,p (Ω , g ) of the set of eigenvalues for this problem is itself a positive eigenvalue with the variational characterization
|du|p νg 1,p Ω λ (Ω , g ) = inf u ∈ W0 (Ω ) \ {0} . |u|p νg Ω D 1,p
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A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95
Consider now the Neumann problem on Ω :
∆p f = |f |p−2 f in Ω df (η) = 0 on ∂ Ω ,
where η denotes the exterior unit normal vector field to ∂ Ω . Here too, the infimum λN1,p (Ω , g ) of the set of nonzero eigenvalues is a positive eigenvalue with the variational characterization
|df |p νg p−2 f ∈ W 1,p (Ω , g ) \ {0}, λ (Ω , g ) := inf Ω | f | f ν = 0 . g |f |p νg Ω Ω N 1 ,p
We consider first the case of (S m , [can]): Theorem 3.1. For any p > m, S m carries Riemannian metrics of volume one, conformal to the standard metric can, with λ1,p arbitrarily large. Proof of Theorem 3.1. Let r ∈ [0, π], denote the geodesic distance on (S m , can) w.r.t. a point x0 ∈ S m . Let ε > 0 and define a radial function fε : S m → R by 4p
fε (r ) = ε m(p−m) · χ[0, π −ε]∪[ π +ε,π ] (r ) + χ( π −ε, π +ε) (r ). 2
2
2
(3.1)
2
Let
m−p 2 p | du | f ν m ε m can 1,p m p−2 2 ν λ1,p (ε) = inf Rε (u) := S u ∈ W ( S ) \ { 0 }, | u | u | f | = 0 . ε can m pf 2 ν Sm | u | ε m can S
We will show first that p
lim sup λ1,p (ε) · ε m = ∞.
(3.2)
ε→0
Classical density arguments imply that there exists uε ∈ W 1,p (S m ) \ {0} with Rε (uε ). Let u¯ ε : S m → R be a radial function defined by u¯ ε (r ) = p
1 V
S m−1
Sm
m
|uε |p−2 uε fε 2 νcan = 0 such that λ1,p (ε) =
|uε (r , ·)|p νcan ,
(3.3)
where V = Vol(S m−1 , can). Differentiating w.r.t. r we obtain p
pu¯ pε−1 u¯ ′ε =
V
S m−1
|uε |p−2 uε
∂ uε νcan . ∂r
By Hölder’s inequality we obtain u¯ pε−1 |¯u′ε | ≤
p−p 1 1p ∂ uε p ∂ uε p νcan ≤ 1 |uε |p−1 | u | ν · ν ε can ∂ r can . V S m−1 ∂r V S m−1 S m−1 1
It follows that
|¯u′ε |p ≤
∂ uε p νcan ≤ 1 |duε |p νcan . V S m−1 ∂ r V S m−1
1
On the other hand
π m m |¯uε |p fε 2 νcan = V · |¯uε |p fε 2 sin r m−1 dr 0 Sm m π p = |uε | νcan fε 2 sin r m−1 dr 0 S m−1 m = |uε |p fε 2 νcan , Sm
where the second equality follows from (3.3). Similarly (3.4) implies
m−p 2
Sm
|¯u′ε |p fε
νcan ≤
m−p 2
Sm
|duε |p fε
νcan .
(3.4)
A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95
93
In particular, we obtain that u¯ ε ∈ W 1,p (S m ) and
Sm
λ1,p (ε) = Rε (uε ) ≥
m−p 2
|¯u′ε |p fε
νcan
m 2
|¯uε | fε νcan Sm m−p m−p S m |¯u′ε |p fε 2 νcan S m |¯u′ε |p fε 2 νcan , − ≥ min + , m m m |¯uε |p fε 2 νcan |¯uε |p fε 2 νcan S Sm p
+
−
where S+ , S− denote the hemispheres centered at x0 , respectively −x0 . Without loss of generality we may assume that m
m
m S+
λ1,p (ε) ≥
m−p 2
|¯u′ε |p fε
m S+
νcan
m 2
|¯uε |p fε νcan
.
(3.5) π − ε] π 2 π and vε on ( − ε, ]
Let wε ∈ W
1,p
u¯ ε
(S+ ), wε = m
u¯ ε (
π 2
on [0,
− ε)
2
= u¯ ε − wε . Then vε = 0 on [0, π2 − ε] and wε′ = 0 on ( π2 − ε, π2 ).
2
Since vε′ and wε′ have disjoint supports, we have |¯u′ε |p = |vε′ |p + |wε′ |p . On the other hand |¯uε |p = |vε + wε |p ≤ 2p−1 (|vε |p + |wε |p ). Then (3.5) and (3.1) imply
m S+
λ1,p (ε) ≥ 21−p
m−p 2
(|vε′ |p + |wε′ |p )fε
νcan
m 2
(|vε |p + |wε |p )fε νcan − 2p ′ p ′ p m |vε | νcan + ε m m |wε | νcan S+ 1−p S+ =2 . m p p 2 m |vε | νcan + m |wε | fε νcan S S m S+
+
+
Quite to multiply u¯ ε by a constant we may assume
λ1,p (ε) ≥ 21−p
m
S+
• Case 1: lim supε→0
2p
m S+
|vε′ |p νcan + ε − m
m S+
|vε |p νcan +
m
m S+
|wε |p fε 2 νcan = 1 and the inequality above becomes
|wε′ |p νcan .
m
S+
(3.6)
|wε′ |p νcan > 0. p
p
Inequality (3.6) implies that λ1,p (ε)ε m ≥ 21−p ε − m
m S+
|wε′ |p νcan , and therefore (3.2) is verified.
|wε′ |p νcan = 0. Then we may find a sequence εn → 0 such that wεn → c strongly in Lp (M ), where c is a constant. In particular since p > m m m, {fεn } is uniformly bounded and we have limn→∞ S m fεn2 νcan = 0. It follows that limn→∞ S m |wεn |p fεn2 νcan = limn→∞ + + m m m 2 2 p p p p |wεn |p fε 2 νcan ≥ 12 m (|wεn | −|c | )fεn νcan +|c | limn→∞ m fεn νcan = 0. Hence for εn small enough, m |vεn | νcan = 1 − Sm S S S • Case 2: limε→0
m S+
+
+
+
+
and (3.6) implies p 1− 2
λ1,p (εn ) ≥ 2
− 2p
≥2
− 2p
m S+
sin
|vε′ n |p νcan ≥ 2 π 2
− εn
m−1
m S+
|vε′ n |p νcan
m S+
|vεn |p νcan
π
2
|vε′ n |p dr
2
|vεn |p dr
π −ε n 2 π π −ε n 2
− 2p
=2
π
2
|vε′ n |p sin r m−1 dr
2
|vεn |p sin r m−1 dr
π −ε n 2 π π −ε n 2
.
(3.7)
1,p Let v¯ εn ∈ W0 (−εn , εn ) be an even function such that v¯ εn (s) = vεn (s + π2 − εn ) for 0 ≤ s ≤ εn . We have then
π
2
π −ε n 2 π 2
π −ε n 2
|vε′ n |p dr p dr
|vεn |
εn 0
|¯vε′ n |p dr
0
|¯vεn |p dr
= εn
εn −ε = εn n
|¯vε′ n |p dr
vεn |p dr −εn |¯
≥ λD1,p (−εn , εn ) = εn−p λD1,p (−1, 1).
(3.8)
94
A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95 p
Inequalities (3.7), (3.8) imply λ1,p (εn ) ≥ 2− 2 [sin( π2 − εn )]m−1 εn λD1,p (−1, 1) and it is straightforward that (3.2) is verified in this case too. Fix now ε > 0 and let f˜ε ∈ C ∞ (S m ), radial with respect to x0 and such that f˜ε ≤ fε , f˜ε (r ) = fε (r ) = 1 on [ π2 − 2ε , π2 + 2ε ] −p
and f˜ε (π − r ) = f˜ (r ). Then
Vol(S m , f˜ε can) =
m
f˜ε 2 νcan =
Sm
π +ε 2 2
>V
π −ε 2 2
π
m
2
f˜ε 2 sin r m−1 dr νcan
− π2
S m−1
sin r m−1 dr
π m−1 > εV sin −ε ,
where V = Vol(S m−1 , can).
2
(3.9)
˜− We will compare now λ1,p (S m , f˜ε can) and λ1,p (ε). Let u˜ ε be an eigenfunction for λ1,p (S m , f˜ε can) and denote by u˜ + ε ,u ε the positive, respectively, the negative part of u˜ ε . Then [15] m−p 2
p ˜ |du˜ + ε | fε
Sm
λ1,p (S , f˜ε can) = m
Sm
m−p 2
p ˜ |du˜ − ε | fε
νcan
. p ˜ |˜u− ε | fε νcan m ˜− = t u˜ + uε,t0 |p−2 u˜ ε,t0 fε 2 νcan = 0 and the equation above implies ε +u ε . Then there is t0 such that S m |˜
Sm
Let t ∈ R and u˜ ε,t
νcan
m−p 2
Sm
Sm
=
p ˜ |˜u+ ε | fε νcan
|du˜ ε,t0 |p f˜ε
λ1,p (S , f˜ε can) = m
m 2
m 2
Sm
νcan
m 2
Sm
≥
|˜uε,t0 |p f˜ε νcan
m−p 2
|du˜ ε,t0 |p fε
Sm
νcan
m 2
|˜uε,t0 |p fε νcan
≥ λ1,p (ε),
(3.10)
where the first inequality follows from the fact that f˜ε ≤ fε and the second from the variational characterization for λ1,p (ε). Inequalities (3.9), (3.10) and (3.2) yield p
p
p
lim sup λ1,p (S m , f˜ε can)Vol(S m , f˜ε can) m ≥ V m · lim sup λ1,p (ε) · ε m = ∞. ε→0
ε→0
2 −m
Finally, let hε = Vol(S m , f˜ε can) Vol(S m , hε can) = 1
f˜ε . We have then lim sup λ1,p (S m , hε can) = ∞.
and
ε→0
We will extend the construction from (S m , [can]) to (M , [g ]) by means of the first eigenvalue for the Neumann problem for ∆p on a domain Ω in M. Theorem 3.2. Let (M , g ) be a compact Riemannian manifold of dimension m. Then for any p > m, [g ] contains Riemannian metrics of volume one with λ1,p arbitrarily large. Proof of Theorem 3.2. Let r denote the geodesic distance on (S m , can) w.r.t. a point x0 . Let f ∈ C ∞ (S m ) be a function radial m denote the hemisphere centered at x0 . Let v w.r.t. x0 , such that f (r ) = f (π − r ) and Vol(S m , fcan) = 1. As before, let S+
be an eigenfunction for λ
N 1 ,p
2
m S+
(S+ , fcan) and let w ∈ W m
1,p
(S ), w(r ) = m
v(r )
if 0 ≤ r ≤
v(π − r )
if
m
π 2
π
2
. Then
m
Sm
|w|p−2 wf 2 νcan =
|v|p−2 v f 2 νcan = 0 and the variational characterization for λ1,p (S m , fcan) implies λ1,p (S , fcan) ≤ m
Sm
|dw|p f
Sm
|w|
m−p 2 m
pf 2
νcan
νcan
m S+
=
|dv|p f m
S+
m−p 2 m
pf 2
|v|
νcan
νcan
= λN1,p (S+m , f can).
(3.11)
m Let Ω be a domain in M such that there exists a diffeomorphism Φ : Ω → S+ . We may assume Ω is included in the open
region of a local chart of M. In this chart we have νg = det(gij )dx1 ∧ dx2 ∧ · · · ∧ dxm and νΦ ∗ can = dx2 ∧ · · · ∧ dxm . There exist positive constants c1 , c2 such that
c1
det(gij ) ≤
det((Φ ∗ can)ij ) ≤ c2
We will compare now λ m and (S+ , fcan) we have
N 1 ,p
det(gij ) on Ω .
det((Φ ∗ can)ij )dx1 ∧
(3.12)
(S+ , f can) and λ (Ω , (f ◦ Φ )g ). Note first that since Φ is an isometry between (Ω , (f ◦ Φ )Φ ∗ can) m
N 1 ,p
λN1,p (S+m , fcan) = λN1,p (Ω , (f ◦ Φ )Φ ∗ can).
(3.13)
A.-M. Matei / Nonlinear Analysis 80 (2013) 88–95
95
Let u be an eigenfunction for λN1,p (Ω , (f ◦ Φ )g ) and denote by u+ , u− the positive, respectively, the negative part of u. Then
|dus |p (f ◦ Φ )
Ω
λN1,p (Ω , (f ◦ Φ )g ) =
Ω
c1
≥
c2
m−p 2 m 2
νg
|us | (f ◦ Φ ) νg p
≥
c1
c2
m
there is s ∈ R such that the function us = su+ + u− verifies
Ω
|us |p−2 us (f ◦ Φ ) 2 νΦ ∗ can = 0. Furthermore
|dus |p (f ◦ Φ )
Ω
Ω
m−p 2
νΦ ∗ can
m 2
|us | (f ◦ Φ ) νΦ ∗ can p
λN1,p (Ω , (f ◦ Φ )Φ ∗ can),
(3.14)
where the first inequality follows from (3.12) and the second from the variational characterization of λN1,p (Ω , (f ◦ Φ )Φ ∗ can). From (3.11), (3.13) and (3.14) we obtain c1
λN1,p (Ω , (f ◦ Φ )g ) ≥
c2
λ1,p (S m , fcan).
(3.15)
Let now δ > 0; there is an extension f ◦ Φ of f ◦ Φ on the entire manifold M such that the metric g˜ = f ◦ Φ g verifies [16]: N ˜ λ1,p (M , g ) > λ1,p (Ω , (f ◦ Φ )g ) − δ . Inequality (3.15) implies
λ1,p (M , g˜ ) >
c1 c2
λ1,p (S m , fcan) − δ.
(3.16)
On the other hand Vol(M , g˜ ) > Vol(Ω , (f ◦ Φ )g ) ≥
=
1 c2
m Vol(S+ , fcan) =
1 c2 1
2c2
Vol(Ω , (f ◦ Φ )Φ ∗ can) Vol(S m , fcan) =
1 2c2
.
(3.17) p
p
Let K > 0; from the proof of Theorem 3.1 we may assume that f is chosen such that λ1,p (S m , fcan) > 2 m +1 c1−1 c2m
p
2 −m
Finally, let h = Vol(M , g˜ )
K . For
δ < K , inequalities (3.16) and (3.17) imply p c1 ≥ λ1,p (S m , fcan) − δ (2c2 )− m > K .
δ small enough such that (2c2 ) λ1,p (M , g˜ )Vol(M , g˜ ) m
+1
p −m
c2
g˜ . Then h ∈ [g ], Vol(M , h) = 1 and λ1,p (M , h) > K .
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