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
$$ \begin{align} \lambda_k(\Omega)\lesssim \frac{n^2}{(\mathcal{D}(\{A_{\alpha}\})\log (k+1))^2}\max_{\alpha=0,\ldots,k} \Big(\log \frac{\operatorname{vol}(\Omega)}{\operatorname{vol}(A_{\alpha})}\Big)^2. \end{align} $$
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 [Reference Chung, Grigor’yan and Yau3, Reference Chung, Grigor’yan and Yau4], 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 [Reference Funano and Sakurai9, Reference Gozlan and Herry10]). 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 [Reference Funano6] 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
$$ \begin{align} \lambda_{k+1}(\Omega)\lesssim n^4 \lambda_k(\Omega). \end{align} $$
Related with (1.2), the author conjectured in [Reference Funano5, Reference Funano7] that
$$ \begin{align*} \lambda_{k+1}(\Omega)\lesssim \lambda_k(\Omega) \end{align*} $$
holds under the same assumption of Theorem 1.2. In [Reference Funano6, 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 [Reference Funano5, Reference Funano7], 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 [Reference Liu13, 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 [Reference Funano6, equation (1.5)] combining Milman’s result [Reference Milman14] with Cheng and Li’s result [Reference Cheng and Li2], 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 [Reference Buser1, 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 [Reference Gromov11, Appendix
${C}_{+}$
] for a weak form of the above proposition.
Theorem 2.2 [Reference Payne and Weinberger15, 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 [Reference Kröger12, 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 [Reference Funano8, 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
$$ \begin{align} \mathcal{D}(\{A_{\alpha}\})\geq \frac{c n}{\sqrt{\lambda_k(\Omega)}}=:r_0. \end{align} $$
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 [Reference Funano7, Theorem 3.4].
To prove Theorem 1.2, let us recall the Bishop–Gromov inequality in Riemannian geometry. See [Reference Funano8, 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 [Reference Funano6, 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.
Acknowledgment
The author would like to express many thanks to the anonymous referees for their helpful and useful comments.
