On the Lower Bound of the Principal Eigenvalue of a Nonlinear Operator

We prove sharp lower bound estimates for the first nonzero eigenvalue of the non-linear elliptic diffusion operator $L_p$ on a smooth metric measure space, without boundary or with a convex boundary and Neumann boundary condition, satisfying $BE(\kappa,N)$ for $\kappa\neq 0$. Our results extends the work of Koerber[5] for case $\kappa=0$ and Naber-Valtorta[10] for the $p$-Laplacian.


Introduction
Let M be a compact manifold. The Laplacian operator on M plays a key role in studying the geometry of M , and one of the key quantity related to the Laplacian is its first nonzero eigenvalue λ 1 , also called the principal eigenvalue. There have been a lot of works on the estimate of λ 1 for the Neumann boundary value problem: In [11] Payne and Weinberger showed λ 1 ≥ π 2 /D 2 for the Laplacian on the convex subset of R n with diameter D. Later in [4] Cheeger gave a lower bound of λ 1 in terms of the isoperimetric constant on compact Riemannian manifolds. In [7], given that M has nonnegative Ricci curvature, P. Li and Yau proved the lower bound π 2 /4D 2 by using gradient estimate. Later Zhong and Yang [14] used a barrier argument to prove the sharp lower bound π 2 /D 2 for compact Riemannian manifold with nonnegative Ricci curvature. Afterwards Kroger in [6] used a gradient comparison technique to recover the result of Zhong and Yang, and furthermore he was able to deal with a negative Ricci lower bound case. It was in Bakry and Emery's work [2] that the situation is generalized into a manifold with weighted volume measure, and the Laplacian is replaced by a general elliptic diffusion operator L. By defining curvature-dimension condition, we can make sense of Ricci lower bound in the senario of smooth measure spaces. Later Bakry and Qian [3] used gradient comparison technique similar to Kroger to prove the sharp lower bound π 2 /D 2 for λ 1 (L) assuming M to be BE(0, N ) for some N ≥ 1. Later Andrews and Ni [1] recovered this result with a simple modulus of continuity method.
In recent years there is much attention to the nonlinear operator called p−Laplacian ∆ p .
In [12] he showed the sharp estimate λ 1 ≥ (p − 1) π p p D p for Riemannian manifolds with Ricci lower bound 0 and p > 1, where π p is the half period of p−sine function which will be defined later. The method used is a gradient comparison via Bochner formula for p-Laplacian, and a fine ODE analysis of the one dimensional model solution. Later Naber and Valtorta [10] extended the result to the case Ric ≥ κ(n − 1) for κ < 0. The key improvement is the better understanding of the one dimensional model equation in the κ < 0 case, which is considerably more complicated than non-negative case. Very recently Li-Wang in [8] and [9] used a modulus of continuity method to get the gradient comparison, in the case of drifted p−Laplacian, which fits in the setting of a Bakry-Emery manifold with weight e −f , thus opening the possibility of studying the non-linear version of L operator with drifted terms in metric measure spaces Satisfying BE(κ, N ). For κ = 0 case, [5] showed that (p − 1) π p p D p is the sharp lower bound.
In this paper we follow the approaches of [3] and [10] to study the non-linear operator L p on a compact manifold with possibly convex boundary(to be defined later). We extend the Theorem 1.1 of [5] to the κ = 0 case, more precisely: Theorem 1.1. Let M be compact and connected and L be an elliptic diffusion operator with invariant measure m. Assume that L satisfies BE(κ, N ) where κ = 0. Let D be diameter defined by the intrinsic distance metric on M . Let u be an eigenfunction associated with λ satisfying Neumann boundary condition if ∂M = ∅, where λ is the first nonzero eigenvalue of L p . Then denoting w (p−1) := |w| p−2 w, we have (1) When κ > 0, assuming further that D ≤ π/ √ κ, we have a sharp comparison: where λ D is the first nonzero eigenvalue of the following Neumann eigenvalue prob- (2) When κ < 0, we have a sharp comparison: where λ D is the first nonzero eigenvalue of the following Neumann eigenvalue problem on [−D/2, D/2]: The rest of this paper is devoted to the proof of Theorem 1.1. The basic structure is the following. In section 2 we introduce the setting and definitions related to the linear elliptic diffusion operator L. In section 3 we define the non-linear operator L p and its Neumann eigenvalue problem. In section 4 we use the Bochner formula to derive a useful estimate (Prop 4.2) which will be used in proving the gradient comparison theorem in section 5. In section 6 we study the associated three one-dimensional model equations and combined with section 7, we get maximum comparison between our model solutions and the solution to the Neumann eigenvalue problem. Finally combining the gradient, maximum and diameter comparison we prove the theorem in section 8.

The geometry of elliptic diffusion operators
In this section we give some definitions which will be used later in our proof. First we introduce the elliptic diffusion operator, which is a natural generalization of a second order linear differential operator on a Riemannian manifold.
and Γ(f, f ) ≥ 0 with equality if and only if df = 0. Here Γ is defined as Definition 2.2. We say that a locally finite Borel measure m is L-invariant if there is a generalized function ν such that holds for all smooth f, g. ν is called the outward normal function and is defined to be a set of and the Γ 2 -operator by Definition 2.5. We can define the N -Ricci curvature as Let κ ∈ R and N ∈ [1, ∞], we say that L satisfies BE(κ, N ) condition if and only if for any f ∈ C ∞ (M ).
If M has a boundary, the geometry of ∂M also plays an important role in the eigenvalue estimate. We define the convexity of ∂M as follows.
Definition 2.6. Let ν be the outward normal direction, and U ⊂ M be an open set, φ,η ∈ C ∞ (U ) such that Γ(ν, η) and Γ(ν, φ) = 0 on U ∩ ∂M . We define the second fundamental form on ∂M by If for any φ as above with Γ(φ) > 0 on U ∩ ∂M we have Then we say ∂M is convex. If we have strict inequality then ∂M is called strictly convex.

The generalized p-Laplacian and its eigenvalue problem
Now we are going to work on the eigenvalue problem of the non-linear operator L p derived from the previously defined L. The generalized p-Laplacian is defined by We also define which is the linearization of L p . Now we define the eigenvalue of L p . If λ ∈ R and u ∈ C 2 (M ) satisfies the Neumann boundary problem: Then we call λ an eignevalue, and u an eigenfunction of L p , however, we may not always find a classical solution. To define the eigenfunction in a weak sense, we first use the invariance of m to deduce the following integration-by-parts formula: Lemma 3.1. Let φ ∈ C ∞ (M ) and u ∈ C 2 (M ) and Γ(u) > 0 on supp(φ). Then we have So we define the eigenvalue and eigenfunction by Definition 3.1. We say that λ is an eigenvalue of L p if there is a u ∈ W 1,p (M ) such that for any φ ∈ C ∞ (M ) the following identity holds: We have the following result concerning the regularity of principal eigenfunctions. Then the principal eigenfunction is in C 1,α (M ) for some α > 0, and u is smooth near points x ∈ M such that Γ(u)(x) = 0 and u(x) = 0; for p < 2, u is C 3,α , and for p > 2, u is C 2,α near x where Γ(u)(x) = 0 and u(x) = 0.

Bochner formula
In this section, we will derive the Bochner formula and an estimate which is helpful to prove the gradient estimate in the next section.
Proposition 4.1 (Bochner formula). Let u ∈ C 1,α (M ) be a first eigenfunction of L p , and x ∈ M be a point such that Γ(u)(x) = and u(x) = 0. Then at x we have the following formula: Proof. c.f. [5], Lemma 3.1.
Then for any n ≥ N , we have for n ∈ (1, ∞), Proof. Following [5] Lemma 3.3, we can scale u on both sides so that Γ(u)(x) = 1. We can assume n = N since B(κ, N ) implies B(κ, n) for n ≥ N . When n = 1, by the curvaturedimension inequality and Lu = trH u = A u , we get . Then by standard computations, together with the assumption Γ(u) = 1, we have Then we get Since B(φ(u), φ(u)) ≥ 0 for any φ, we have non-positive discriminant

Gradient Comparison Theorem and Its Applications
In this section we prove the gradient comparison theorem of the eigenfunction with the solution to the one-dimensional model.
such that w is strictly increasing on [a, b] and the range of u is contained the range of w.
Then for all x ∈ M , Proof. By scaling u so that min(u) = −1, we can assume that the range of u is contained in the range of w. By chain rule of Γ what we need to show, equivalently, is for all x ∈ M . Since T κ depends smoothly on κ, we will first prove that for anyκ < κ, the gradient comparison holds when T κ is replaced by Tκ in (5.1), and then we can takeκ → κ. This will give us a room to use proof by contradiction.
Now we denote Z c0 as Z, φ c0 as φ when there is no confusion. When x 0 is in the interior of M , this clearly implies the following equations: If x 0 ∈ ∂M , since Γ(u,ν) = 0 by the Neumann boundary condition, we have that Γ(Z, u) = 0 at x 0 . Since Z achieves maximum at x 0 and ∂M is convex, we have This implies that the second derivative of Z along the normal direction is nonpositive. On the other hand, the second derivatives along tangential directions are nonpositive, hence the ellipticity of L u p implies that L u p (Z)(x 0 ) ≤ 0. Hence we comfirmed the three equations above for all x ∈ M .
and by differentiating the ODE satisfied by w we have By the ODE evaluated at w −1 (u(x 0 )), we have Plugging the above equation into the third term of (5.5), we have We can assume thatκ has the same sign as κ since we can pickκ sufficiently close to κ. Now we have T ′ κ = T 2 κ /(n − 1) +κ to rewrite the second term and finally get 1 By Proposition 4.1 and 4.2, together with the fact that L p u = −λu (p−1) we have Hence in both cases, we have 1 which is a contradiction with the second derivative test. Therefore we conclude that Z 1 ≤ 0 on M , which implies our gradient comparison result.
Remark 5.2. When 1 < p < 2 we know that u ∈ C 2,α near x 0 , hence the Bochner formula can not be directly applied to x 0 . In this case notice that u does not vanish identically in a neighborhood of x 0 , we can choose x ′ → x 0 with u ′ (x ′ ) = 0. As we apply the Bochner formula at x ′ , The first term Γ(u) −1) , u) since u is a eigenfunction. Now this diverging term will cancel with −λ(p−1)u p−2 Γ(u) p 2 in the expression of 1 p L u p (φ(u)), which makes it possible to define 1 p L u p Γ(u) p 2 − φ(u) (x 0 ) to be the limit of 1 p L u p (φ(u))(x ′ ) as x ′ → x 0 . Therefore the previous proof still works when 1 < p < 2.

One-dimensional Models
In this section we will study the one-dimensional comparison model ODEs and discuss some fine properties of their solutions. Let n ≥ 2, p > 1 be fixed.
6.1. The Model ODE. Let a ∈ R, p > 1 be fixed, we will consider the following form of initial value problem: where T is defined over a subset of R, to be specified according to the cases κ > 0 or κ < 0.
To study this ODE we first define the p-sin and p-cos functions.
Definition 6.1. For every p ∈ (1, ∞), let π p be defined by: and we extend it to a periodic function on R. Let cos p (t) = d dt sin p (t), and we have the following identity which resembles the case of usual sin and cos: | sin p (t)| p + | cos p (t)| p = 1.
6.4. Fine analysis of the model equation (6.1). The central question we need to address here is the existence of solution to (6.1) whose range matches the range of u. Due to the normalization that min w = min u = −1, we will consider the maximum of w. For this purpose we introduce some notations. For a ∈ R, let w i,a be the solution to the equation (6.1) with T = T i , and b(i, a) be the first critical point of w i,a after a. If w ′ i,a (t) > 0 for t > a, then we say b(i, a) = ∞. Also let δ i,a = b(i, a) − a and m(i, a) = w i, a (b(i, a)). We shall prove the following statement in the current and next section: Theorem 6.3. Under the same setting as Theorem 1.1, let u be an eigenfunction of L p operator, normalized so that min u = −1 and max u ≤ 1. Then we have the following existence results: (i) (κ > 0) There is some a ∈ I 0 and a solution w 0,a such that m(0, a) = max u.
To prove Theorem 6.3, first we establish the following existence and uniqueness of a solution to the model equation.
Proof of Proposition 6.1. In the case a ∈ I i for i = 0, 1, 2, 3 we obtain the existence and uniqueness result from the fact that T i is a Lipschitz continuous function starting with a. Hence we need to confirm the boundary cases only. When (1) κ > 0 and a = −π/(2 √ κt) and (2) κ < 0 and a = 0 for model T 1 , we can use fixed-point theorem argument to prove the existence and uniqueness of the solution by slightly modifying the proof in [13], section 3.
Then we shall look at the case κ > 0. In order to find a such that w 0,a matches the maximum of u, we use the continuous dependence of m(0, a) on a. We need to show that Proposition 6.2. Fix α > 0, n ≥ 1 and κ > 0. Then there always exists a uniqueā ∈ I 0 such that the solution w 3,−ā is odd, and in particular, the maximum of w restricted to [−ā,ā] is 1.
The proof of Proposition 6.2 requires certain weaker estimate of λ. We define the first Neumann eigenvalue of the equation (6.1) on I 0 to be First we claim that Lemma 6.1. If λ ≤ λ 0 , κ > 0, then equation (6.1) admits a odd solution w such that w ′ (t) > 0 for all t ∈ I 0 .
Proof. Consider the initial value problem starting with 0: This problem admits a solution w up to π/(2 √ κ), the singularity of T 0 , which can be extended to an odd solution on I 0 . Now we claim that w ′ (t) > 0 for all t ∈ I 0 . Suppose for some t 0 ∈ (0, π/(2 √ κ), w ′ (t 0 ) = 0. This is an eigenfunction corresponding to λ, hence by the monotonicity of first eigenvalue, we get λ > λ 0 , which contradicts with our assumption that λ ≤ λ 0 . Therefore, we have w ′ (t) > 0 on I 0 .
Proof. Suppose that on the contrary, λ ≤ λ 0 , then by Lemma 6.1, we get an odd function w such that w ′ > 0 on I 0 . Suppose the first eigenfuntion u is scaled so that −1 = min u ≤ max u ≤ 1. When w is bounded, we can scale w so that max w = w(π/(2 √ κ)) = max u. Pick x, y ∈ M such that u(x) = − max u and u(y) = max u. Then by the gradient comparison theorem, which is a contradiction. When w is unbounded, we can choose c k ց 0 and t k ր π/(2 √ κ) such that c k w(t k ) = max u. Again using the gradient comparison theorem, we can prove D ≥ 2t k → π/ √ κ, which again gives a contradiction. Hence λ > λ 0 .
Proof of Proposition 6.2. Using the Prüfer Transformation, we consider the following initial value problem: Since λ > λ 0 , equation 6.5 has a solution φ such that φ(â) = π p /2 for someâ < π/(2 √ κ). This implies that e achieves maximum atâ from the equation satisfied by e. Therefore we conclude that w = e sin φ/α achieves maximum atâ, and w can be extended to an odd function on [−â,â] such that w ′ (−â) = 0, i.e. w is a solution to the model equation (6.1).
identical under a translation. However, by the T 0 model equation, this can only happen when a = a ′ . Therefore m(0, a) is invertible, and it is monotonic. To see the continuity of m(0, a), note that when a > −π/(2 √ κ), the continuous dependence of the solution on the initial value problem is automatic. When a = −π/(2 √ κ), Lemma 6.3 shows that m(0, a) is continuous. Now let us turn to the case κ < 0, which is more delicate. We will get a similar result as Proposition 6.2: Proposition 6.4 ([10] Proposition 6.1). Fix α > 0, n ≥ 1 and κ < 0. Then there always exists a uniqueā > 0 such that the solution w 3,−ā is odd, and in particular, the maximum of w restricted to [−ā,ā] is 1.
By studying the equation of φ one can show that there is a critical valueᾱ at which the oscillatory behavior of w changes. For the modeal T 3 , we have Lemma 6.4 ([10] Proposition 6.4). There exists a limiting valueᾱ > 0 such that for α >ᾱ we have δ(3, a) < ∞ for every a ∈ R. For α <ᾱ, we have for sufficiently large a we have When α =ᾱ, we have lim a→∞ δ(3, a) = ∞.
Both cases α <ᾱ and α ≥ᾱ need to be considered in proving the case (2) of Theorem 6.3. When α <ᾱ we can always use model T 3 to produce the whole range comparison solutions w, i.e. 0 < max w ≤ 1, and when α ≥ᾱ we have restriction on the maximum value that u can achieve. More precisely we have: Combining the Proposition 6.4 and Lemmas above, we know that in the case κ < 0, if m(1, 0) ≤ max u ≤ 1, there is always a model solution w to T 1 , T 2 or T 3 such that max w = max u.
6.5. Diameter Comparison. In order to get the eigenvalue comparison with one-dimensional moder of the same diameter bound, we still need to understand how λ D varies with the diameter. Again we will follow [10]. Definition 6.4. We define the minimum diameter of the one-dimesional model associated with λ to beδ i (λ) = min{δ(i, a)|i = 0, 1, 2, 3, a ∈ I i } The following propositions deals with the lower bound ofδ i (λ) for i = 0, 1, 2, 3: Proposition 6.5. For i = 0 and any a ∈ I 0 , we have δ(0, a) > 2â, whereâ < π/(2 √ κ) is such that w 0,−â is odd.
Proof. The proof here is based on the symmetry and convexity of the model T 0 . See [10] Proposition 8.4 for the proof.
For the case κ < 0, we cite the following results from [10]: Model 3 needs a little bit careful attention. For this one we notice first that there is always a > 0 with an odd solution for initial data at −ā. Namely w 3,ā is odd function with min −1 and max 1. This is a critical situation which minimizes the diameter D given λ: and if a = −ā, the inequality is strict.

Maximum of Eigenfunctions
In this section we are going to compare the maximum of the eigenfunctions u and the model functions w. First, we define a measure on the interval [a, b(a)] which is essentially the pullback of the volume measure on M by w −1 • u. By the ODE satisfied by w, w ′′ is positive before w hits its first zero.
First we have a theorem which can be seen as a comparison between the model function and the eigenfunction. This result is equivalent to the following statement: To prove the maximum comparison we study the volume of a small ball around the minimum of u. By the gradient comparison we have the following: Proof. To keep notations short, let w = w 1,0 . Let ǫ be small such that −1 + ǫ < −2 −p+1 . Then we have u (p−1) < − 1 2 when u < −1 + ǫ. Let t 0 be the first zero of w, then by Theorem 6.1 we have E(t) ≤ E(t 0 ). Therefore by Theorem 6.2 we get Since ǫ can be arbitrarily small, we have the claim holds for r sufficiently small. (1) If κ > 0, and w 0,−π/(2 √ κ) is the corresponding eigenfunction. Then max u ≥ m(0, −π/ (2 √ κ)).
Proof. Let m denotes m(0, −π/(2 √ κ)) or m(1, 0) in either cases, and suppose that max u < m. Since m is the least possible value among max w for all model solutions w, by continuous dependence of the solution of model equation on n, we can find n ′ > n so that max u is still less that the maximum of the correspoding model equation. Since BE(κ, n ′ ) is still satisfied, we have by Theorem 6.3, that m(B x0 (r)) ≤ cr n ′ for r sufficiently small. However by Bishop-Gromov volume comparison we have m(B x0 (r)) ≥ Cr N . This is a contradiction since n ′ > n ≥ N . Proof. We scale u so that min u = −1 and max u ≤ 1. By Proposition 6.3 we can find a model function w i,a such that max u = max w i,a . By the gradient comparison theorem, Γ(w −1 i,a • u) ≤ 1. Let x and y on M be points where u attains maximum and minimum, then we have a (m(i, a)) − w −1 i,a (−1) = δ(i, a, λ) ≥ δ(i,ā) Therefore by the monotonicity of eigenvalue of the model equation, we have that To check the sharpness of this result when κ < 0, we have the following examples: let M i = [−D/2, D/2] × i −1 τ3 S n−1 be a warped product where S n−1 is the standard unit sphere, and τ 3 (t) = cosh( √ −κt). If we consider L being the classical Laplacian on M , then standard computation shows that M i has Ric ≥ −(n − 1)κ and geodeiscially convex boundary. Hence it also satisfy the BE(κ, N ) condition. If we take u(t, x) = w(t) where w is the solution to our one-dimensional model equation with λ = λ D . Since the diameter of M i tends to d as i → ∞, we see that the first eigenvalue on M i converges to λ d , which shows the sharpness of our lower bound when κ < 0. For κ > 0, the round sphere S n−1 ( π √ κ ) serves as a model for sharp lower bound of λ.