A universal inequality for Neumann eigenvalues of the Laplacian on a convex domain in Euclidean space

Abstract

We obtain a new upper bound for Neumann eigenvalues of the Laplacian on a bounded convex domain in Euclidean space. As an application of the upper bound, we derive universal inequalities for Neumann eigenvalues of the Laplacian.

1 Introduction

The purpose of this article is to give a new upper bound for Neumann eigenvalues of the Laplacian on a bounded convex domain in Euclidean space and a universal inequality for Neumann eigenvalues of the Laplacian.

Let $\Omega $ be a bounded domain in Euclidean space with piecewise smooth boundary. We denote by $\lambda _k(\Omega )$ the kth positive Neumann eigenvalues of the Laplacian on $\Omega $ . For a finite sequence $\{A_{\alpha }\}_{\alpha =0}^k$ of Borel subsets of $\Omega $ , we set

$$ \begin{align*} \mathcal{D}(\{A_{\alpha}\}):=\min_{\alpha \neq \beta} d(A_{\alpha},A_{\beta}), \end{align*} $$

where $d(A_{\alpha },A_{\beta }):=\inf \{d(x,y) \mid x\in A_{\alpha },y\in A_{\beta }\}$ and d is the Euclidean distance function.

Throughout this paper, we write $\alpha \lesssim \beta $ if $\alpha \leq c\beta $ for some universal concrete constant $c>0$ (which means c does not depend on any parameters such as dimension and k, etc.).

One of the main theorems in this paper is as follows.

Theorem 1.1 Let $\Omega $ be a bounded convex domain in $\mathbb {R}^n$ with piecewise smooth boundary, and let $\{A_{\alpha }\}_{\alpha =0}^k$ be a sequence of Borel subsets of $\Omega $ . Then we have

(1.1) $$ \begin{align} \lambda_k(\Omega)\lesssim \frac{n^2}{(\mathcal{D}(\{A_{\alpha}\})\log (k+1))^2}\max_{\alpha=0,\ldots,k} \Big(\log \frac{\operatorname{vol}(\Omega)}{\operatorname{vol}(A_{\alpha})}\Big)^2. \end{align} $$

Remark 1.1 The above theorem also holds for Neumann eigenvalues of the Laplacian on bounded convex domains in a manifold of nonnegative Ricci curvature. The proof only uses Lemma 3.1, which follows from the Bishop–Gromov inequality.

In [3, 4], Chung, Grigory’an, and Yau obtained

$$ \begin{align*} \lambda_k(\Omega)\lesssim \frac{1}{\mathcal{D}(\{A_{\alpha}\})^2}\max_{\alpha=0,\ldots,k} \Big(\log \frac{\operatorname{vol}(\Omega)}{\operatorname{vol}(A_{\alpha})}\Big)^2 \end{align*} $$

for a bounded (not necessarily convex) domain $\Omega $ and its Borel subsets $\{A_{\alpha }\}$ (see also [9, 10]). Compared to their inequality, the inequality (1.1) is better for large k if we fix n. Their inequality is better for large n if we fix k. Theorem 1.1 also gives an answer to Question 5.1 in [6] up to $n^2$ factor.

As an application of Theorem 1.1, we obtain the following universal inequality for Neumann eigenvalues of the Laplacian.

Theorem 1.2 Let $\Omega $ be a bounded convex domain in $\mathbb {R}^n$ with piecewise smooth boundary. Then we have

(1.2) $$ \begin{align} \lambda_{k+1}(\Omega)\lesssim n^4 \lambda_k(\Omega). \end{align} $$

Related with (1.2), the author conjectured in [5, 7] that

$$ \begin{align*} \lambda_{k+1}(\Omega)\lesssim \lambda_k(\Omega) \end{align*} $$

holds under the same assumption of Theorem 1.2. In [6, equation (1.3)], the author proved that

$$ \begin{align*} \lambda_{k+1}(\Omega)\lesssim (n\log k)^2\lambda_k(\Omega) \end{align*} $$

for a bounded convex domain $\Omega $ . The inequality (1.2) avoids the dependence of k for the upper bound of the ratios $\lambda _{k+1}(\Omega )/\lambda _k(\Omega )$ and gives a better inequality if $\log k \geq n$ . In [5, 7], the author proved a dimension-free universal inequality $\lambda _k(\Omega )\lesssim c^k \lambda _1(\Omega )$ for a bounded convex domain in $\mathbb {R}^n$ and for some universal constant $c>1$ . In [13, Theorem 1.5], Liu showed an optimal universal inequality $\lambda _k(\Omega )\lesssim k^2\lambda _1(\Omega )$ under the same assumption. Thus, $n^2$ factor is not needed for small k (e.g., $k=2,3$ ) in (1.2). As mentioned in [6, equation (1.5)] combining Milman’s result [14] with Cheng and Li’s result [2], one can obtain $\lambda _k(\Omega )\gtrsim k^{2/n}\lambda _1(\Omega )$ under the same assumption. Together with Liu’s inequality, this shows

$$ \begin{align*} \lambda_{k+1}(\Omega)\lesssim k^{2-2/n}\lambda_k(\Omega). \end{align*} $$

The inequality (1.2) is better than this inequality for large k if we fix n. This inequality is better for large n if we fix k.

2 Preliminaries

We collect several results to use in the proof of our theorems.

Proposition 2.1 [1, Theorem 8.2.1]

Let $\Omega $ be a bounded domain in a Euclidean space with piecewise smooth boundary, and let $\{\Omega _{\alpha }\}_{\alpha =0}^{l}$ be a finite partition of $\Omega $ by subdomains in the sense that $\operatorname {vol} (\Omega _\alpha \cap \Omega _{\beta })=0$ for each different $\alpha ,\beta $ . Then we have

$$ \begin{align*} \lambda_{l+1}(\Omega)\geq \min_{\alpha}\lambda_1(\Omega_{\alpha}).\\[-18pt] \end{align*} $$

Refer to [11, Appendix ${C}_{+}$ ] for a weak form of the above proposition.

Theorem 2.2 [15, equation $(1.2)$ ]

Let $\Omega $ be a bounded convex domain in a Euclidean space with piecewise smooth boundary. Then we have

$$ \begin{align*} \lambda_1(\Omega)\geq \frac{\pi^2}{\operatorname{Diam} (\Omega)^2}. \end{align*} $$

Combining Proposition 2.1 with Theorem 2.2 in order to give a “good” lower bound for Neumann eigenvalues of the Laplacian, it is enough to provide a “good” finite convex partition of the domain.

For an upper bound of Neumann eigenvalues, we mention the following theorem.

Theorem 2.3 [12, Theorem 1.1]

Let $\Omega $ be a bounded convex domain in $\mathbb {R}^n$ with piecewise smooth boundary. For any natural number k, we have

$$ \begin{align*} \lambda_k(\Omega)\lesssim \frac{n^2k^2}{\operatorname{Diam} (\Omega)^2}. \end{align*} $$

In order to construct a “good” partition, we recall a Voronoi partition of a metric space. Let X be a metric space, and let $\{x_{\alpha }\}_{\alpha \in I}$ be a subset of X. For each $\alpha \in I$ , we define the Voronoi cell $C_{\alpha }$ associated with the point $x_{\alpha }$ as

$$ \begin{align*} C_{\alpha}:= \{x\in X \mid d(x,x_{\alpha})\leq d(x,x_{\beta}) \text{ for all }\beta\neq \alpha \}. \end{align*} $$

If X is a bounded convex domain $\Omega $ in a Euclidean space, then $\{C_{\alpha }\}_{\alpha \in I}$ is a convex partition of $\Omega $ (the boundaries $\partial C_{\alpha }$ may overlap each other). Observe also that if the balls $\{ B(x_{\alpha },r)\}_{\alpha \in I}$ of radius r cover $\Omega $ , then $C_{\alpha } \subseteq B(x_{\alpha },r)$ , and thus $\operatorname {Diam} (C_{\alpha } )\leq 2r$ for any $\alpha \in I$ .

3 Proof of Theorems 1.1 and 1.2

We use the following key lemma to prove Theorem 1.1.

Lemma 3.1 [8, Lemma 3.1]

Let $\Omega $ be a bounded convex domain in $\mathbb {R}^n$ with a piecewise smooth boundary. Given $r>0$ , suppose that $\{x_{\alpha }\}_{\alpha =0}^{l}$ is r-separated points in $\Omega $ , i.e., $d(x_{\alpha },x_{\beta })\geq r$ for distinct $\alpha $ , $\beta $ . Then we have

$$ \begin{align*} r\lesssim \frac{n}{\sqrt{\lambda_l(\Omega)}}. \end{align*} $$

Proof of Theorem 1.1

Suppose that there is a sequence $\{ A_{\alpha }\}_{\alpha =0}^{k}$ of Borel subsets such that

$$ \begin{align*} \lambda_k(\Omega)\geq \frac{c n^2}{(\mathcal{D}(\{A_{\alpha}\})\log (k+1))^2}\max_{\alpha=0,\ldots,k} \Big(\log \frac{\operatorname{vol} (\Omega)}{\operatorname{vol}(A_{\alpha})}\Big)^2 \end{align*} $$

for sufficiently large $c>0$ . Since $(k+1)\operatorname {vol}(A_{\alpha })\leq \operatorname {vol}(\Omega )$ for some $\alpha $ , we have

(3.1) $$ \begin{align} \mathcal{D}(\{A_{\alpha}\})\geq \frac{c n}{\sqrt{\lambda_k(\Omega)}}=:r_0. \end{align} $$

For each $\alpha $ , we fix a point $x_{\alpha }\in A_{\alpha }$ . The sequence $\{x_{\alpha }\}_{\alpha =0}^{k}$ is then $r_0$ -separated in $\Omega $ by (3.1). By virtue of Lemma 3.1, we get

$$ \begin{align*} \frac{c n}{\sqrt{\lambda_k(\Omega)}}= r_0 \lesssim \frac{n}{\sqrt{\lambda_k(\Omega)}}. \end{align*} $$

For sufficiently large c, this is a contradiction. This completes the proof of the theorem.

We can reduce the number of $\{A_{\alpha }\}$ in Theorem 1.1 as follows.

Lemma 3.2 Let $\Omega $ be a convex domain in $\mathbb {R}^n$ , and let $\{A_{\alpha }\}_{\alpha =0}^{k-1}$ be a sequence of Borel subsets of $\Omega $ . Then we have

$$ \begin{align*} \lambda_k(\Omega)\lesssim \frac{n^2}{(\mathcal{D}(\{A_{\alpha}\})\log (k+1))^2}\max_{\alpha=0,\ldots,k-1} \Big(\log \frac{\operatorname{vol}(\Omega)}{\operatorname{vol}(A_{\alpha})}\Big)^2. \end{align*} $$

The above lemma follows from Theorem 1.1 and [7, Theorem 3.4].

To prove Theorem 1.2, let us recall the Bishop–Gromov inequality in Riemannian geometry. See [8, Lemma 3.4] for the proof in the case of convex domains in $\mathbb {R}^n$ .

Lemma 3.3 (Bishop–Gromov inequality)

Let $\Omega $ be a convex domain in $\mathbb {R}^n$ . Then, for any $x\in \Omega $ and any $R>r>0$ , we have

$$ \begin{align*} \frac{ \operatorname{vol} (B(x,r)\cap \Omega)}{\operatorname{vol} (B(x,R)\cap \Omega)}\geq \Big(\frac{r}{R}\Big)^n. \end{align*} $$

In the proof of Theorem 1.2, we make use of a similar argument as in [6, Theorem 1.3].

Proof of Theorem 1.2

Let $R:=cn^2/\sqrt {\lambda _{k+1}(\Omega )}$ , where c is a positive number specified later. Suppose that $\Omega $ includes $k+1\ R$ -separated net $\{x_{\alpha }\}_{\alpha =0}^{k}$ in $\Omega $ . By Theorem 2.3, we have $\operatorname {Diam} (\Omega ) \leq c'n(k+1)/\sqrt {\lambda _{k+1}(\Omega )}$ for some universal constant $c'>0$ . Applying the Bishop–Gromov inequality, we have

$$ \begin{align*} \frac{\operatorname{vol}(B(x_{\alpha},R)\cap \Omega)}{\operatorname{vol}(\Omega)}\geq \frac{R^n}{(\operatorname{Diam} \Omega)^n}\geq \Big(\frac{c}{c'(k+1)}\Big)^n\geq \frac{1}{(k+1)^n} \end{align*} $$

for $c>c'$ . By Lemma 3.2, we obtain

$$ \begin{align*} \lambda_{k+1}(\Omega)\lesssim \frac{n^2(\log (k+1)^n)^2}{(\mathcal{D}(\{B(x_{\alpha},R)\cap \Omega\})\log (k+2))^2}\lesssim \frac{n^4}{R^2}=\frac{1}{c}\lambda_{k+1}(\Omega). \end{align*} $$

For sufficiently large c, this is a contradiction.

Let $x_0, x_1,x_2,\ldots ,x_l$ be maximal R-separated points in $\Omega $ , where $l\leq k-1$ . By the maximality, we have $\Omega \subseteq \bigcup _{\alpha =0}^{l} B(x_{\alpha },R)$ . If $\{ \Omega _{\alpha } \}_{\alpha =0}^{l}$ is the Voronoi partition of $\Omega $ associated with $\{x_{\alpha }\}$ , then we have $\operatorname {Diam} (\Omega _{\alpha })\leq 2R$ . Theorem 2.2 thus yields $\lambda _1(\Omega _{\alpha })\gtrsim 1/R^2$ for each $\alpha $ . According to Proposition 2.1, we obtain

$$ \begin{align*} \lambda_k(\Omega)\geq \min_{\alpha}\lambda_1(\Omega_{\alpha})\gtrsim \frac{1}{R^2}\gtrsim \frac{\lambda_{k+1}(\Omega)}{n^4}. \end{align*} $$

This completes the proof of the theorem.