On maximizing the fundamental frequency of the complement of an obstacle

TL;DR

This paper investigates how the placement and shape of an obstacle within a domain influence the maximum fundamental frequency of the domain's complement, providing bounds and geometric insights.

Contribution

It establishes bounds on the first Dirichlet eigenvalue for domains with obstacles and characterizes the location of maximizers relative to the ground state maxima.

Findings

Maximizers of the eigenvalue are close to the ground state maxima.

Large eigenvalues imply obstacle placement near maximum points.

Convex obstacles contain all ground state maxima when eigenvalues are large.

Abstract

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain satisfying a Hayman-type asymmetry condition, and let $ D $ be an arbitrary bounded domain referred to as "obstacle". We are interested in the behaviour of the first Dirichlet eigenvalue $ \lambda_1(\Omega \setminus (x+D)) $. First, we prove an upper bound on $ \lambda_1(\Omega \setminus (x+D)) $ in terms of the distance of the set $ x+D $ to the set of maximum points $ x_0 $ of the first Dirichlet ground state $ \phi_{\lambda_1} > 0 $ of $ \Omega $. In short, a direct corollary is that if \begin{equation} \mu_\Omega := \max_{x}\lambda_1(\Omega \setminus (x+D)) \end{equation} is large enough in terms of $ \lambda_1(\Omega) $, then all maximizer sets $ x+D $ of $ \mu_\Omega $ are close to each maximum point $ x_0 $ of $ \phi_{\lambda_1} $. Second, we discuss the distribution of $ \phi_{\lambda_1(\Omega)} $ and the possibility to inscribe wavelength balls at a given point in $ \Omega $. Finally, we specify our observations to convex obstacles $ D $ and show that if $ \mu_\Omega $ is sufficiently large with respect to $ \lambda_1(\Omega) $, then all maximizers $ x+D $ of $ \mu_\Omega $ contain all maximum points $ x_0 $ of $ \phi_{\lambda_1(\Omega)} $.

Equations

$${\mu }_{\Omega }:=\underset{x}{\max }{\lambda }_1(\Omega \setminus (x+D))$$

$$\frac{|B_{r_0}(x)\setminus \Omega |}{|B_{r_0}(x)|}\ge \alpha .$$

$$M:=\{x\in \Omega |{\varphi }_{{\lambda }_1}(x)=\Vert {\varphi }_{{\lambda }_1(\Omega )}{\Vert }_{L^{\infty }(\Omega )}\}$$

$${\mu }_{\Omega }:=\underset{x}{\max }{\lambda }_1(\Omega \setminus (x+D)).$$

$${\rho }_x:=\underset{y\in M}{\max }d(y,x+D),$$

$${\lambda }_1(\Omega \setminus (x+D))\le \beta ({\rho }_x){\lambda }_1(\Omega ),$$

$$\beta (\rho )=\{\begin{array}{l}{\beta }_0={\beta }_0(n,\alpha ),\rho \sqrt{{\lambda }_1(\Omega )}>r_0:=r_0(n,\alpha ),\\ \frac{c_0}{{\rho }^2{\lambda }_1(\Omega )},\rho \sqrt{{\lambda }_1(\Omega )}\le r_0,c_0=c_0(n),\end{array}$$

$$\frac{|B_{r_0}\cap \Omega |}{|B_{r_0}|}\ge 1-{ϵ}_0,$$

$${\rho }_0:=\min \left(\frac{r_0}{\sqrt{{\lambda }_1(\Omega )}},{\rho }_x\right).$$

$${\lambda }_1(\Omega \setminus (x+D))\le {\lambda }_1(B_{{\rho }_0}(x_0))=\frac{C}{{\rho }_0^2},$$

$${\rho }_{\bar{x}}\le {\beta }^{-1}(C_0).$$

$${\rho }_{\bar{x}}\lesssim \frac{1}{\sqrt{C_0{\lambda }_1(\Omega )}}.$$

$$\frac{C_1(\alpha ,n)}{\sqrt{{\lambda }_1(\Omega )}}\le \mathrm{inrad}(\Omega )\le \frac{C_2(n)}{\sqrt{{\lambda }_1(\Omega )}}.$$

$${\varphi }_{\lambda }(\tilde{x})>C,$$

$$|\Omega |=\kappa r^n,\kappa >{\omega }_n,$$

$$\begin{gathered} {\lambda }_1(\Omega \setminus B_r(\tilde{x}))\ge \frac{C}{|\Omega \setminus B_r(\tilde{x}){|}^{2/n}}=\frac{C}{(|\Omega |-{\omega }_n{r}^n{)}^{2/n}}=\frac{C}{(\kappa -{\omega }_n{)}^{2/n}{r}^2}= \\ =\frac{C}{(\kappa -{\omega }_n{)}^{2/n}(c\mathrm{inrad}(\Omega ){)}^2}\ge \frac{C{C}_2(n)}{c^2(\kappa -{\omega }_n{)}^{2/n}}{\lambda }_1(\Omega )=:{\tilde{C}}_0{\lambda }_1(\Omega ). \end{gathered}$$

$${\rho }_{\tilde{x}}\le {\beta }^{-1}({\tilde{C}}_0)=\sqrt{\frac{c_0}{{\tilde{C}}_0{\lambda }_1(\Omega )}}.$$

$$\Vert \nabla {\varphi }_{{\lambda }_1(\Omega )}{\Vert }_{L^{\infty }(\Omega )}\le \gamma (\Omega ,n)\sqrt{{\lambda }_1(\Omega )}.$$

$${\varphi }_{{\lambda }_1(\Omega )}(\tilde{x})\ge C=C(c_0,{\tilde{C}}_0,\gamma ),$$

$$C{\lambda }_1(\Omega )\le {\lambda }_1(\Omega \setminus (\bar{x}+D))\le {\lambda }_1(B^{1/2})\le \frac{C_1}{(\mathrm{inrad}(\Omega ){)}^2}=C_2{\lambda }_1(\Omega ),$$

Full text

First, we prove an upper bound on $\lambda_{1}(\Omega\setminus(x+D))$ in terms of the distance of the set $x+D$ to the set of maximum points $x_{0}$ of the first Dirichlet ground state $\phi_{\lambda_{1}}>0$ of $\Omega$. In short, a direct corollary is that if

[TABLE]

is large enough in terms of $\lambda_{1}(\Omega)$, then all maximizer sets $x+D$ of $\mu_{\Omega}$ are close to each maximum point $x_{0}$ of $\phi_{\lambda_{1}}$.

Second, we discuss the distribution of $\phi_{\lambda_{1}(\Omega)}$ and the possibility to inscribe wavelength balls at a given point in $\Omega$.

Finally, we specify our observations to convex obstacles $D$ and show that if $\mu_{\Omega}$ is sufficiently large with respect to $\lambda_{1}(\Omega)$, then all maximizers $x+D$ of $\mu_{\Omega}$ contain all maximum points $x_{0}$ of $\phi_{\lambda_{1}(\Omega)}$.

1. Introduction and background

We consider the problem of placing of an obstacle in a domain so as to maximize the fundamental frequency of the complement of the obstacle. To be more precise, let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain, and let $D$ be another bounded domain referred to as ”obstacle”. The problem is to determine the optimal translate $x+D$ so that the fundamental Dirichlet Laplacian eigenvalue $\lambda_{1}(\Omega\setminus(x+D))$ is maximized/minimized.

In case the obstacle $D$ is a ball, physical intuition suggests that for sufficiently regular domains and sufficiently small balls, $\Omega$, $\lambda_{1}(\Omega\setminus B_{r}(x))$ will be maximized when $x=x_{0}$, a point of maximum of the ground state Dirichlet eigenfunction $\phi_{\lambda_{1}}$ of $\Omega$. Heuristically, such maximum points $x_{0}$ seem to be situated deeply in $\Omega$, hence removing a ball around $x_{0}$ should be an optimal way of truncating the lowest possible frequency. Our methods give equally good results for Schrödinger operators on a large class of bounded domains sitting inside Riemannian manifolds (see the remarks at the end of Section 2).

The following well-known result of Harrell-Kröger-Kurata treats the case when $\Omega$ satisfies convexity and symmetry conditions:

Theorem 1.1** ([9]).**

*Let $\Omega$ be a convex domain in $\mathbb{R}^{n}$ and $B$ a ball contained in $\Omega$. Assume that $\Omega$ is symmetric with respect to some hyperplane $H$. Then,

(a) at the maximizing position, $B$ is centered on $H$, and

(b) at the minimizing position, $B$ touches the boundary of $\Omega$.*

The last result of Harrell-Kröger-Kurata seems to work under rather strong symmetry assumption. We also recall that the proof of Harrell-Kröger-Kurata proceeds via a moving planes method which essentially measures the derivative of $\lambda_{1}(\Omega\setminus B)$ when $B$ is shifted in a normal direction to the hyperplane (also see pp 58 of [11]).

There does not seem to be any result in the literature treating domains without symmetry or convexity properties.

In our note, we consider bounded domains $\Omega\subset\mathbb{R}^{n}$ which satisfy an asymmetry assumption in the following sense:

Definition 1.2**.**

A bounded domain $\Omega\subset\mathbb{R}^{n}$ is said to satisfy the asymmetry assumption with coefficient $\alpha$ (or $\Omega$ is $\alpha$-asymmetric) if for all $x\in\partial\Omega$, and all $r_{0}>0$,

[TABLE]

This condition seems to have been introduced in [10]. Further, the $\alpha$-asymmetry property was utilized by D. Mangoubi in order to obtain inradius bounds for Laplacian nodal domains (cf. [13]) as nodal domains are asymmetric with $\alpha=\frac{C}{\lambda^{(n-1)/2}}$.

From our perspective, the notion of asymmetry is useful as it basically rules out narrow ”spikes” (i.e. with relatively small volume) entering deeply into $\Omega$. For example, let us also observe that convex domains trivially satisfy our asymmetry assumption with coefficient $\alpha=\frac{1}{2}$.

2. The basic estimate for general obstacles

With the above in mind, we consider any bounded $\alpha$-asymmetric domain $\Omega\subset\mathbb{R}^{n}$ and a bounded obstacle domain $D$. We denote the first positive Dirichlet eigenvalue and eigenfunction of $\Omega$ by $\lambda_{1}$ and $\phi_{\lambda_{1}(\Omega)}$ respectively and let

[TABLE]

be the set of maximum points of $\phi_{\lambda_{1}(\Omega)}$.

Let us also put

[TABLE]

Finally, for a given translate $x+D$ of the obstacle let us set

[TABLE]

measuring the maximum distance from a maximum point of $\phi_{\lambda_{1}(\Omega)}$ to the translate $x+D$.

Our main estimate is the following.

Theorem 2.1**.**

Let us fix a translate $(x+D)$ and assume that $\rho_{x}>0$. Then

[TABLE]

where $\beta$ is a continuous decreasing function defined as

[TABLE]

where $\beta_{0}r_{0}=c_{0}$.

We remark that in particular if $\rho_{x}$ is of sub-wavelength order (i.e. $\lesssim\frac{1}{\sqrt{\lambda_{1}(\Omega)}}$), then $\lambda_{1}(\Omega\setminus(x+D))\lesssim\frac{1}{\rho_{x}^{2}}$. If the obstacle $D$ is convex, we can say more (see Theorem 4.1 below).

Proof of Theorem 2.1.

The proof essentially exploits the fact that there are “almost inscribed” wavelength balls centered at maximum points of $\phi_{\lambda_{1}(\Omega)}$. To make this statement precise, we recall the following theorem from [6], which works for all domains in compact Riemannian manifolds of dimension $n\geq 3$ (planar domains are known to have wavelength inradius from the work of Hayman ([10])):

Theorem 2.2**.**

Let $\dim M\geq 3,\epsilon_{0}>0$ be fixed, $\Omega$ a domain inside $M$, and $x_{0}\in\Omega$ be such that $|\varphi_{\lambda}(x_{0})|=max_{\Omega}|\varphi_{\lambda}|$, where $\varphi_{\lambda}$ is the ground state Dirichlet eigenfunction of $\Omega$. There exists $r_{0}=r_{0}(\epsilon_{0})$, such that

[TABLE]

where $B_{r_{0}}$ denotes $B\left(x_{0},\frac{r_{0}}{\sqrt{\lambda_{1}}}\right)$.

We also note that it follows from the proof that $r_{0}$ can be taken as $r_{0}=\epsilon_{0}^{\frac{n-2}{2n}}$. Moreover, let us for completeness recall that Theorem 2.2 relies on two main ingredients - namely, the Feynman-Kac formula and certain capacity estimates related to hitting probabilities of Brownian motion. We refer to [6] and [14] for more details.

Now, it is clear that under the $\alpha$-asymmetry assumption, there exists an $r_{0}:=r_{0}(\alpha,n)$, such that around each maximum point $x_{0}\in\Omega$ of $\phi_{\lambda_{1}(\Omega)}$ one can find a fully inscribed ball $B_{r_{0}/\sqrt{\lambda_{1}(\Omega)}}(x_{0})\subseteq\Omega$. By the definition of $\rho_{x}$ it follows that we can find a maximum point $x_{0}\in(\Omega\setminus(x+D))$ and an inscribed ball $B_{\rho_{0}}(x_{0})$ where

[TABLE]

As the first eigenvalue is monotonic with respect to inclusion, we see that

[TABLE]

where $C=C(n)$ is a universal constant.

Expressing the right hand side of the last inequality in terms of $\lambda_{1}(\Omega)$ we define the function $\beta(\rho)$ as above.

This concludes the proof. ∎

Here, we have considered the obstacle problem in the case of Euclidean spaces, on reasonably well-behaved domains, and for the operator $-\Delta+\lambda_{1}(\Omega)$, as that seems to be the primary case of interest. However, we also include some remarks outlining some straightforward generalizations.

** Remark 2.3****.**

It is clear that removing capacity zero sets from $\alpha$-asymmetric domains considered in Definition 1.2 will lead to the same conclusions. Indeed, in this situation we will not be dealing with fully inscribed balls as above - instead, we will have balls whose first eigenvalue is comparable to the one of an inscribed one.

** Remark 2.4****.**

Also, in the setting of curved spaces, one has absolutely similar results for $\Omega\subseteq M$, where $(M,g)$ is a smooth compact Riemannian manifold, if we allow the constants to depend on the dimension, asymmetry and the metric $g$.

** Remark 2.5****.**

Lastly, it is clear that the results of [14] allow us to extend our discussion here from operators of the form $-\Delta+\lambda_{1}(\Omega)$ to Schrödinger operators of the form $-\Delta+V$, where $V$ is bounded above. The conclusions are analogous with $\lambda_{1}(\Omega)$ replaced by $\|V\|_{L^{\infty}}$and the proofs are identical.

Now, as an immediate implication of Theorem 2.1 we have the following corollary.

Corollary 2.6**.**

Suppose that $\mu_{\Omega}=C_{0}\lambda_{1}(\Omega)$, where $C_{0}>\frac{c_{0}}{r_{0}^{2}}$ is a given fixed constant and $c_{0},r_{0}$ are the constants in Theorem 2.1. Then, for a maximizer $\bar{x}+D$ of $\mu_{\Omega}$ we have

[TABLE]

In particular, if $C_{0}$ is large,

[TABLE]

In other words the above corollary can be interpreted as follows: either $\mu_{\Omega}$ is comparable to $\lambda_{1}(\Omega)$, or the maximum points of $\phi_{\lambda_{1}(\Omega)}$ are near the maximizer sets $\bar{x}+D$ of $\mu_{\Omega}$.

We note that the localization in the Corollary above gets better when $C_{0}$ is large. By Faber-Krahn’s inequality, straightforward examples with large $C_{0}$ are domains $\Omega$ for which $|\Omega\setminus(x+D)|$ is sufficiently small for some $x$.

Particularly, for bounded convex domains in $\mathbb{R}^{n}$, by a theorem of Brascamp-Lieb, the level sets of $\phi_{\lambda_{1}(\Omega)}$ are convex. Since $\phi_{\lambda_{1}(\Omega)}$ is real analytic and it can be assumed positive on $\Omega\setminus\partial\Omega$ without loss of generality, this means that it has a unique point of maximum. So, in this setting, our result heuristically says that if removal of a ball $B_{r}$ has a “significant effect” on the vibration of $\Omega\setminus B_{r}$, then $B_{r}$ must be centered quite close to the max point of the ground state Dirichlet eigenfunction $\phi_{\lambda_{1}}$ of the domain $\Omega$, where the bound on $\rho_{x}$ gives the quantitative relation between the “effect” and the order of “closeness”. In a sense, this can be seen to be complementary to Corollary II.3 of [9].

3. Inscribed balls and distribution of $\phi_{\lambda_{1}(\Omega)}$

Further, we specify our results to the obstacle being a ball $D$. We point out a few statements related to the connection between the distribution of $\phi_{\lambda_{1}(\Omega)}$ and the possibility to inscribe a large ball at a given point $x$ in $\Omega$.

First, by Theorem 2.2 above we immediately have the following observation:

Proposition 3.1**.**

Let $\Omega$ be $\alpha$-asymmetric and let $\operatorname{inrad}(\Omega)$ denote the inner radius of $\Omega$. If $x_{0}$ is a point of maximum of $\phi_{\lambda_{1}(\Omega)}$, then there exists an inscribed ball $B_{C\operatorname{inrad}(\Omega)}(x_{0})\subseteq\Omega$, where $C=C(n,\alpha)$.

Proof of Proposition 3.1.

We observe that by the results of [13], $\alpha$-asymmetric domains $\Omega$ satisfy

[TABLE]

Now, it follows from our Theorem 2.2 (see [6]) that there exists an inscribed wavelength ball at the max point $x_{0}$, which concludes the proof. ∎

In particular, the last proposition applies for convex domains. We mention in this connection that localization results for maximum points of $\phi_{\lambda_{1}(\Omega)}$ in case $\Omega$ is a planar convex domain can be found in the work of Grieser-Jerison (see [8]).

On the other hand, it is natural to ask how large is $\phi_{\lambda_{1}(\Omega)}$ at points admitting a large inscribed ball. For reasonably nicely behaved domains, we have the following:

Corollary 3.2**.**

Let $\Omega$ be a $C^{2,\beta}$-regular $\alpha$-asymmetric domain and let $\phi_{\lambda_{1}(\Omega)}$ be normalized so that $\|\phi_{\lambda_{1}(\Omega)}\|_{L^{\infty}(\Omega)}=1$. Suppose that for $\tilde{x}\in\Omega$ there exists a maximal inscribed ball $B_{r}(\tilde{x})\subseteq\Omega$ where $r:=c\operatorname{inrad}(\Omega)$ for some $0<c\leq 1$, such that $\frac{|\Omega\setminus B_{r}(\tilde{x})|}{|\Omega|}$ is sufficiently small. Then

[TABLE]

where $C=C(|\Omega|,\partial\Omega,c,n)$.

Analogously, one can show a similar statement by demanding that $|B_{r}(\tilde{x})\cap\Omega|$ is sufficiently large in comparison to $|\Omega|$.

Proof of Corollary 3.2.

Let us first suppose that

[TABLE]

where $\omega_{n}$ is the volume of a ball of radius $1$. We use the Faber-Krahn inequality to obtain

[TABLE]

By assumption, $\tilde{C}_{0}$ is sufficiently large, i.e., in particular $\tilde{C}_{0}>\frac{c_{0}}{r^{2}_{0}}$, so we may apply Corollary 2.6 to obtain that

[TABLE]

On the other hand, the Schauder a priori estimates up to the boundary for $\phi_{\lambda_{1}(\Omega)}$ (see [7], Theorem 6.6) yield the existence of $\gamma=\gamma(\Omega,n)$, such that

[TABLE]

As by assumption $\phi_{\lambda_{1}(\Omega)}(x_{0})=1$ and $\tilde{C}_{0}$ is sufficiently large, then

[TABLE]

which concludes the claim. ∎

4. Relation between maximum points and convex obstacles

Note that Theorem 2.1 holds for arbitrary obstacles and gives a bound on the distance $\rho_{x}$ to maximum points of $\phi_{\lambda_{1}(\Omega)}$. However, it is desirable to deduce that $\rho_{x}=0$, i.e. maximizers actually contain the maximum points of $\phi_{\lambda_{1}(\Omega)}$.

From Proposition 3.1 and Theorem 2.1 we deduce the following:

Theorem 4.1**.**

Let $D$ be a convex obstacle, and $\bar{x}+D$ maximize $\lambda_{1}(\Omega\setminus(x+D))$. Then there exists a constant $C_{0}=C_{0}(\alpha,n)$ such that if $\lambda_{1}(\Omega\setminus(\bar{x}+D))\geq C\lambda_{1}(\Omega)$ for some $C\geq C_{0}$, then $\rho_{\bar{x}}=0$.

In other words, either $\mu_{\Omega}\sim\lambda_{1}(\Omega)$ or $\rho_{\bar{x}}=0$.

Proof.

To the contrary let us suppose that $\rho_{\bar{x}}=d(\bar{x}+D,x_{0})>0$ where $x_{0}$ is a maximum point of $\phi_{\lambda_{1}(\Omega)}$ and $\lambda_{1}(\Omega\setminus(\bar{x}+D))\geq C\lambda_{1}(\Omega)$ for an arbitrary large $C>0$.

We apply the statement of Proposition 3.1 and deduce that there is a wavelength inscribed ball $B$ at $x_{0}$. As $D$ is a convex domain, we can find a wavelength half-ball $B^{1/2}\subset\Omega\setminus(\bar{x}+D)$ containing $x_{0}$. By the assumption and eigenvalue monotonicity with respect to inclusion:

[TABLE]

where $C_{2}=C_{2}(n,\alpha)$. Taking $C$ sufficiently large we get a contradiction.

∎

It is clear that for explicit applications, particularly in the case of convex domains, Theorem 4.1 is dependent on a precise knowledge of the location of the maximum point of $\phi_{\lambda_{1}(\Omega)}$. Localization of the maximum point of $\phi_{\lambda_{1}(\Omega)}$ (or more generally, the “hot spot”) is a problem which is far from being settled. Here we take the space to augment Theorem 4.1 with the recent results of [2].

First we recall the definition of the “heart” of a convex body $\Omega$. The following intuitive definition appears in [4], and it is equivalent to the (more technical) definition presented in [2].

Definition 4.2**.**

Let $P$ be a hyperplane in $\mathbb{R}^{n}$ which intersects $\Omega$ so that $\Omega\setminus P$ is the union of two components located on either side of $P$. The domain $\Omega$ is said to have the interior reflection property with respect to $P$ if the reflection through $P$ of one of these subsets, denoted $\Omega_{s}$, is contained in $\Omega$, and in that case $P$ is called a hyperplane of interior reflection for $\Omega$. When $\Omega$ is convex, the heart of $\Omega$, denoted by $\heartsuit(\Omega)$, is defined as the intersection of all such $\Omega\setminus\Omega_{s}$ with respect to hyperplanes of interior reflection of $\Omega$.

The following result is contained in Proposition 4.1 of [2].

Proposition 4.3** ([2]).**

The unique maximum point $x_{0}$ of $\phi_{\lambda_{1}(\Omega)}$ is contained in $\heartsuit(\Omega)$. Furthermore, $x_{0}$ is contained in the interior of $\heartsuit(\Omega)$, if the latter is non-empty.

Acknowledgements

We thank Saskia Roos for drawing our attention to the reference [11]. We are grateful to Antoine Henrot and Kazuhiro Kurata for their comments on a draft version. We also gratefully acknowledge the Max Planck Institute for Mathematics, Bonn and the Technion, Haifa for providing ideal working conditions.