New York Journal of Mathematics

New York J. Math. 14 (2008) 393–401.

Convexity of the first eigenfunction of the drifting Laplacian operator and its

applications

Li Ma and Baiyu Liu

Abstract. In this short note, we prove the convexity of the first eigenfunction of the drifting Laplacian operator with zero Dirichlet boundary value provided a suitable assumption to the drifting term is added. After giving a gradient estimate, we then use the convexity of the first eigenfunction to get a lower bound of the difference of the first and second eigenvalues of the drifting Laplacian.

Contents

1. Introduction 393 2. Convexity of the first eigenfunction 396 3. Gradient estimate for the quotient of eigenfunctions 396 4. Lower bound for the eigenvalue difference 399 References 400

1. Introduction

Given a smooth bounded convex domain Ω ⊂ ℝ^{n} with smooth boundary and h a smooth function on the closure of Ω. We consider the following Dirichlet eigenvalue problem on the domain Ω ⊂ ℝ^{n}.

(1)  \begin{cases}−Δ_{h}u = λu, & in Ω \\ u = 0, & on ∂Ω, \end{cases}

Received January 20, 2008. Mathematics Subject Classification. 35C44. Key words and phrases. Convexity, first eigenfunction, drifting Laplacian, warped

product.

This research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 20060003002.

393^{ISSN 1076-9803/08}

where Δ_{h} = Δ−∇h·∇ is the drifting Laplacian (also called the h-Laplacian since Δ_{h}u = e^{h}div(e^{−h}∇u)) and Ω is a convex bounded regular domain in R^{n}. Here we assume Δu = u′′ on the real line R. People may refer to papers of J. Lott [L03] and G. Perelman [P02] for more geometric background of the h-Laplacian. We point out that the problem (1) is closely related to eigenvalue problems on warped product space (see [CM04], [Ma06] and [MZ07]). In fact, associated to the function h, we let

f(x) = \frac{1}{4}(2Δh − |∇h|^{2}).

Then f(x) is the scalar curvature of the metric g_{0} + e^{−h}dθ^{2} on the product Ω × S^{1} with g_{0} being the Euclidean metric in the domain Ω. We mention that the modified scalar curvature of a metric g and dilation function h, as introduced by Perelman in [P02, Remark 1.3], is R^{m} = R + 4f, where R is the scalar curvature of g and f = f(x) was defined above. Just as in the case when h = 0, we need to show the convexity of the first eigenfunction of the h-Laplacian; however, the known results developed by others can not be used directly when h is nontrivial. We find a trick to overcome this difficulty. Once we obtain the required convexity for the first eigenfunction, we can proceed as in [SWYY85] to find the lower bound of the gap between the first eigenvalue and the second eigenvalue of the h-Laplacian.

Let us state these results precisely. Denote the first and second eigenvalues of (1) by λ_{1} and λ_{2}, respectively, and denote the corresponding first and second eigenfunctions by f_{1} and f_{2}.

Following the works of Caffarelli and Spruck [CS82] and Korevaar [K83], who were in turn independently generalizing results of Brascamp and Lieb [BL76], we first show:

Theorem 1. Assume that h is a smooth concave function on the closure of the domain Ω. Assume that

f = \frac{1}{4}(2Δh − |∇h|^{2})

is concave on \overline{Ω}. Let u_{1} = f_{1} be the first eigenfunction to the first eigenvalue problem

−Δu + ∇h · ∇u = λ_{1}u, u > 0 in Ω

with the Dirichlet boundary condition u = 0 on ∂Ω. Let v = − log u_{1}. Then v is convex in Ω.

Note that there are many h’s satisfying the assumptions in Theorem 1. In fact, it is clear that the linear function h(x) = ∑_{j} a_{j}x_{j}, where all a_{j} are real constants, and the more geometrically natural quadratic function

h(x) = − ∑_{j} λ_{j}x_{j}^{2},

where λ_{j} > 0 are positive constants, satisfy the assumptions in Theorem 1. The special case when h(x) = −|x|^{2} corresponds to the metric

g = dx^{2} + exp(|x|^{2})dθ^{2}

with its Riemannian measure dv = exp(|x|^{2})dxdθ. Note that when h(r) = 2 log r (r > 0), the assumptions in Theorem 1 are also satisfied and the metric g = dr^{2} + e^{−2 log r}dθ^{2} = dr^{2} + r^{−2}dθ^{2} is the dual metric to the flat metric dr^{2} + r^{2}dθ^{2}.

Another nontrivial example is h(x) = −e^{−x} in the real line.

Let u = \frac{f_{2}}{f_{1}} and λ = λ_{2} − λ_{1}.

Theorem 2. Let Ω be a smooth, bounded, convex domain in ℝ^{n}, h and f be as in Theorem 1. Then we have

|∇u|^{2} + λ(μ − u)^{2} ≤ sup_{Ω} λ(μ − u)^{2},

where μ is a constant no less than sup_{Ω} u.

Using this gradient estimate, we can give a lower bound for the difference of eigenvalues λ.

Theorem 3. Assume the same conditions as in Theorem 1. Then we have

λ_{2} − λ_{1} ≥ \frac{π^{2}}{4d^{2}} ,

where d is the diameter of Ω .

We remark that when h = constant, Theorem 3 was obtained by I. M. Singer, Bun Wong, Shing-Tung Yau and Stephen S.-T. Yau in [SWYY85]. Note that, some of our results may be extended to more general divergenceform elliptic operators. For example, let Δ_{g} denote the Laplace–Beltrami operator associated to a Riemannian metric g on the closure of the domain Ω. Then Theorem 1 should apply with Δ replaced by Δ_{g}. Theorem 2 should apply as long as the Ricci curvature of g is nonnegative, in which case the well-known Bochner formula for Δ_{g}|∇u|^{2} preserves the necessary inequality in (2) below. The trouble is that it is not easy to formulate the convexity property of the domain. It is not easy to extend our results to the case where a more general divergence-form elliptic operator

Lu = ∑_{ij} ∂_{i}(a_{ij}(x)∂_{j}u)

is used in place of the Laplacian operator. The difficult part is to find a suitable corresponding Bochner formula (see (2) below). We leave these as questions. However, using a trick from the recent work [Y03] of S.-T. Yau, we can extend part of our results here to more general elliptic operators (see [ML08]).

The plan of the paper is as follows: we prove Theorem 1 in Section 2. Theorem 2 will be proved in Section 3, and Theorem 3 will be proved in Section 4.

2. Convexity of the first eigenfunction

We shall follow the method introduced by Korevaar [K83], see also [CS82] and [Ka85] for improvements. Their results can not be used directly. So we need a transformation to overcome this difficulty.

Proof of Theorem 1. By direct computation, we have

Δv = ∇h · ∇v + λ_{1} + |∇v|^{2}.

Let

w = v + \frac{h}{2}.

Then we have

Δw = |∇w|^{2} + λ_{1} + f(x),

where f(x) = \frac{1}{4}(2Δh − |∇h|^{2}) as before. Then by the result of Korevaar (see Theorem 1.3 in [K83] and Lemma 3.2 in [CS82]), we know that w is convex, and then we know that v is convex. □

3. Gradient estimate for the quotient of eigenfunctions

We first set two lemmata to get a gradient estimate for u.

Lemma 1. Let Ω ⊂ ℝ^{n} be smooth, bounded and convex domain. Then u = \frac{f_{2}}{f_{1}} is smooth up to the boundary ∂Ω.

Proof. For ∀p ∈ ∂Ω, choose local coordinates {x_{1}, x_{2}, …, x_{n}} on a sufficiently small neighborhood U such that U ∩ ∂Ω = U ∩ {x_{1} = 0}. Since

\begin{cases}f_{1} = 0 & on ∂Ω \\ f_{1} > 0 & in Ω, \end{cases}

by Hopf’s lemma we have \frac{∂f_{1}}{∂x_{1}} < 0 on ∂Ω. Using the Malgrange preparation theorem, we have f_{1} = g_{1}x_{1} in \overline{Ω} ∩ U, where g_{1} satisfies g_{1} ≠ 0 and smooth on \overline{Ω} ∩ U.

For f_{2} = 0 on ∂Ω, we can write f_{2} = g_{2}x_{1}, where g_{2} is smooth on \overline{Ω} ∩ U. Thus, u = \frac{f_{2}}{f_{1}} = \frac{g_{2}}{g_{1}} is smooth on \overline{Ω} ∩ U. □

Lemma 2.

Δ_{h}u = −λu − 2∇u · ∇ log f_{1}.

Proof. By direct computation, we have

Δu = \frac{Δf_{2}}{f_{1}} − 2\frac{∇f_{1} · ∇f_{2}}{f_{1}^{2}} − \frac{f_{2}}{f_{1}^{2}} Δf_{1} + 2\frac{f_{2}}{f_{1}^{3}} |∇f_{1}|^{2}

= \frac{1}{f_{1}^{2}} (−λ_{2}f_{1}f_{2} + λ_{1}f_{1}f_{2}) + \frac{1}{f_{1}^{2}} (f_{1}∇h · ∇f_{2} − f_{2}∇h · ∇f_{1})

− 2\frac{∇f_{1} · ∇f_{2}}{f_{1}^{2}} + 2f_{2} \frac{|∇f_{1}|^{2}}{f_{1}^{3}}

= −λu + \frac{∇h · ∇f_{2}}{f_{1}} − \frac{f_{2}}{f_{1}^{2}} ∇h · ∇f_{1} − 2\frac{∇f_{1} · ∇f_{2}}{f_{1}^{2}} + 2f_{2} \frac{|∇f_{1}|^{2}}{f_{1}^{3}} .

Because

∇u · ∇ log f_{1} = \frac{∇f_{1} · ∇f_{2}}{f_{1}^{2}} − f_{2} \frac{|∇f_{1}|^{2}}{f_{1}^{3}}

and

∇h · ∇u = \frac{∇h · ∇f_{2}}{f_{1}} − \frac{f_{2}}{f_{1}^{2}} ∇h · ∇f_{1},

we obtain

Δu = −λu + ∇h · ∇u − 2∇u · ∇ log f_{1},

which proves the lemma. □

Proof of Theorem 2. Define a function G by

G = |∇u|^{2} + λ(μ − u)^{2}.

Since u is smooth by Lemma 1, so is G.

Now, we compute

G_{i} = 2 ∑_{j=1}^{n} u_{j}u_{ji} − 2λ(μ − u)u_{i},

(2) ΔG = 2|D^{2}u|^{2} + 2∇u · ∇Δu − 2λ(μ − u)Δu + 2λ|∇u|^{2}

= 2|D^{2}u|^{2} + 2∇u · ∇(−λu + ∇h · ∇u − 2∇u · ∇ log f_{1})

− 2λ(μ − u)(−λu + ∇h · ∇u − 2∇u · ∇ log f_{1}) + 2λ|∇u|^{2}

= 2|D^{2}u|^{2} − 2λ|∇u|^{2} + 2∇u · ∇(∇h · ∇u)
− 4∇u · ∇(∇u · ∇ log f_{1}) + 2λ^{2}(μ − u)u
− 2λ(μ − u)∇h · ∇u + 4λ(μ − u)∇u · ∇ log f_{1}
+ 2λ|∇u|^{2}.

Let G(p) = max_{x∈\overline{Ω}} G(x). All we need to prove is |∇u|(p) = 0.

Suppose |∇u|(p) ≠ 0.

If p ∈ Ω, then we can choose coordinates such that

u_{1}(p) ≠ 0, u_{i}(p) = 0, for 2 ≤ i ≤ n.

Applying the maximum principle, we have

(3)  ∇G(p) = 0, ΔG(p) ≤ 0.

We now have

0 = G_{i}(p) = 2(u_{1}u_{1i} − λ(μ − u)u_{i})(p),

and

u_{11}(p) = λ(μ − u(p)), u_{1i}(p) = 0, for 2 ≤ i ≤ n.

Therefore, at the point p,

(4)  (∇u · ∇ log f_{1})_{1} = u_{11}(∇ log f_{1})_{1} + u_{1}(∇ log f_{1})_{11}
= λ(μ − u)(∇ log f_{1})_{1} + u_{1}(∇ log f_{1})_{11},

and

(5)  (∇h · ∇u)_{1} = h_{11}u_{1} + h_{1}u_{11}
= u_{1}h_{11} + h_{1}λ(μ − u).

A computation that uses (2), (3), (4), (5) yields

(6)  0 ≥ ΔG(p)
= 2|D^{2}u| + 2λ^{2}(μ − u)u
+ 2u_{1}(∇h · ∇u)_{1} − 2λ(μ − u)u_{1}h_{1}
+ 4λ(μ − u)u_{1}(∇ log f_{1})_{1} − 4u_{1}(∇u · ∇ log f_{1})_{1}
= 2(u_{11}^{2} + λ^{2}(μ − u)u) + 2u_{1}^{2}(h_{11} − 2(log f_{1})_{11})
≥ 2((λ(μ − u))^{2} + λ^{2}(μ − u)u)
= 2λ^{2}μ(μ − u),

where we have used the fact that h − 2 log f_{1} is convex (see the proof of Theorem 1). For μ ≥ sup_{Ω} u > 0, (6) gives rise to a contradiction.

Else, i.e., p ∈ ∂Ω, we can choose an orthonormal frame {l_{1}, …, l_{n}} around p such that \frac{∂}{∂x_{1}} = l_{1}|_{∂Ω} is the outward normal vector. By computation, we have

(7)  \frac{∂G}{∂x_{1}} (p) = 2u_{1}u_{11}(p) + 2 ∑_{j=2}^{n} u_{j}u_{j1}(p) − 2λ(μ − u)u_{1}(p) ≥ 0.

Consider

(8)  Δu = −λu + ∇h · ∇u − 2∇u · ∇ log f_{1},

where Δu, u, ∇h · ∇u are smooth up to the boundary and achieves finite values on ∂Ω. Therefore, ∇u · ∇ log f_{1} = \frac{u_{1}(f_{1})_{1}}{f_{1}} + \frac{∑_{i=2}^{n} u_{i}(f_{1})_{i}}{f_{1}} also attains finite values on ∂Ω. For f_{1} = 0, on ∂Ω, we have (f_{1})_{i} = 0, 2 ≤ i ≤ n. Then, on ∂Ω,

0 = f_{1}(∇u · ∇ log f_{1}) = u_{1}(f_{1})_{1} + ∑_{i=2}^{n} u_{i}(f_{1})_{i} = u_{1}(f_{1})_{1}.

Therefore, u_{1}(f_{1})_{1} = 0 on ∂Ω. Since (f_{1})_{1} ≠ 0 on ∂Ω by Hopf’s lemma and f_{1} = 0 on ∂Ω, we get u_{1} = 0 on ∂Ω. (7) is simplified to

\frac{∂G}{∂x_{1}} (p) = 2 ∑_{j=2}^{n} u_{j}u_{j1}(p) ≥ 0.

The definition of Hessian gives us that

u_{j1} = l_{j}l_{1}u − (∇_{l_{j}}l_{1})u.

Since u_{1} = 0 on ∂Ω, we have

u_{j1} = −(∇_{l_{j}}l_{1})u.

Using the second fundamental form (I_{ij}) of the hypersurface ∂Ω in ℝ^{n}, we have

u_{i1} = − ∑_{j=2}^{n} I_{ij}u_{j}, i ≠ 1.

Then we obtain

\frac{∂G}{∂x_{1}} (p) = −2 ∑_{i,j=2}^{n} u_{j}I_{ij}u_{j}(p) ≥ 0.

From the convexity of ∂Ω and u_{1}(p) = 0, we get |∇u|(p) = 0. □

Remark. Since we only need the convexity of h − 2 log f_{1}, we can use it to replace the assumption that h is concave from the assumptions of Theorem 2 and Theorem 3.

4. Lower bound for the eigenvalue difference

We follow the method used in [SWYY85] to prove Theorem 3.

Proof of Theorem 3. Choosing μ = sup_{Ω} u in Theorem 2, we have

(9)  \sqrt{λ} ≥ \frac{|∇u|}{\sqrt{(sup u − inf u)^{2} − (sup u − u)^{2}}} .

Let A = sup u − inf u and W = sup u − u, (9) can be written as

(10)  \sqrt{λ} ≥ \frac{|∇W |}{\sqrt{A^{2} − W^{2}}} .

400 Li Ma and Baiyu Liu

Let u(q_{1}) = sup_{Ω} u, u(q_{2}) = inf_{Ω} u. For the convexity of Ω, q_{1} and q_{2} can be connected by a line σ ⊂ \overline{Ω}. Integrate (10) along σ from q_{1} to q_{2}, we obtain

∫_{0}^{A} \frac{|dW |}{\sqrt{A^{2} − W^{2}}} ≥ \sqrt{λ}ds.

By calculation, we have

\frac{π}{2} ≤ \sqrt{λ}l(σ) ≤ \sqrt{λ}d,

where l(σ) is the length of σ, d is the diameter of Ω.

Then we get λ_{2} − λ_{1} = λ ≥ \frac{π^{2}}{4d^{2}} . □

Acknowledgements. We thank the unknown referee very much for helpful suggestions which improved the original manuscript.

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Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China lma@math.tsinghua.edu.cn liuby05@mails.tsinghua.edu.cn

This paper is available via http://nyjm.albany.edu/j/2008/14-18.html.