Some sharp bounds for Steklov eigenvalues

TL;DR

This paper establishes sharp lower bounds for Steklov eigenvalues on star-shaped domains in hypersurfaces of revolution and paraboloids, extending classical results to more general geometries.

Contribution

It provides new sharp lower bounds for Steklov eigenvalues on star-shaped domains in curved spaces, generalizing previous Euclidean results.

Findings

Derived sharp lower bounds for Steklov eigenvalues.

Extended bounds to domains in hypersurfaces of revolution.

Applied bounds to domains in paraboloids.

Abstract

This work is an extension of a result given by Kuttler and Sigillito (SIAM Rev $10$:$368-370$, $1968$) on a star-shaped bounded domain in $\mathbb{R}^2$. Let $\Omega$ be a star-shaped bounded domain in a hypersurface of revolution, having smooth boundary. In this article, we obtain a sharp lower bound for all Steklov eigenvalues on $\Omega$ in terms of the Steklov eigenvalues of the largest geodesic ball contained in $\Omega$ with the same center as $\Omega$. We also obtain similar bounds for all Steklov eigenvalues on star-shaped bounded domain in paraboloid, $P = \left\lbrace (x, y, z) \in \mathbb{R}^{3} : z = x^2 + y^2\right\rbrace$.

Equations

$$\displaystyle\begin{array}[]{rcll}\Delta\varphi&=&0&\mbox{ in }\Omega,\\ \frac{\partial\varphi}{\partial\nu}&=&\mu\varphi&\mbox{ on }\partial\Omega,\end{array}$$

$${\mu }_l(\Omega )=\underset{E}{\sup }\underset{0\ne \phi \in E^{\perp }}{\inf }\frac{{\int }_{\Omega }\Vert \nabla \phi {\Vert }^{2}dv}{{\int }_{\partial \Omega }{\phi }^{2}ds},$$

$${\mu }_2(\Omega )\ge \frac{{R_m}^{n-1}}{{R_M}^{n+1}}{h}_m.$$

$$\displaystyle\mu_{2k+1}(\Omega)\geq\mu_{2k}(\Omega)\geq\frac{k\left[1-{2}\bigg{/}\left(1+\sqrt{1+4\,\min\left({R(\theta)}/{R^{\prime}(\theta)}\right)^{2}}\right)\right]}{\max\sqrt{R^{2}(\theta)+{R^{\prime}}^{2}(\theta)}},$$

$${\mu }_2(\Omega )\ge \frac{(R_m{)}^{n-2}}{(R_M{)}^{n-1}}\frac{\left\{2+a-\sqrt{a^2+4a}\right\}}{2\sqrt{a+1}}.$$

$${\mu }_2(\Omega )$$

$$\partial \Omega$$

$$\Omega \backslash \left\{p\right\}$$

$$\text{ for }0\le a\le 1,0\le ar\le r\text{ and }\frac{h(r)}{r}\le \frac{h(ar)}{ar}$$

$$\text{ for }a\ge 1,ar\ge r\text{ and }\frac{h(r)}{r}\ge \frac{h(ar)}{ar}.$$

$${\mu }_l(\Omega )$$

$${\int }_{\Omega }\Vert \nabla f{\Vert }^{2}dv={\int }_{U_p\Omega }{\int }_0^{R_u}\left[{\left(\frac{\partial f}{\partial r}\right)}^2+\frac{1}{h^2(r)}\Vert \overline{\nabla }f{\Vert }^2\right]h^{n-1}(r)drdu.$$

$${\int }_{\Omega }\Vert \nabla f{\Vert }^{2}dv$$

$$+\Vert \overline{\nabla }f{\Vert }^2-\frac{2\rho }{R_u}\frac{\partial f}{\partial \rho }⟨\overline{\nabla }f,\overline{\nabla }{R}_u⟩\}]h^{n-1}\left(\frac{\rho R_u}{R_m}\right)\left(\frac{R_u}{R_m}\right)d\rho du.$$

$$-\frac{2\rho }{R_m{h}^2\left(\frac{\rho R_u}{R_m}\right)}\left(\frac{\partial f}{\partial \rho }\right)⟨\overline{\nabla }f,\overline{\nabla }{R}_u⟩$$

$$-\frac{{\beta }^2{R}_u}{R_m{h}^2\left(\frac{\rho R_u}{R_m}\right)}\Vert \overline{\nabla }f{\Vert }^2.$$

$${\int }_{\Omega }\Vert \nabla f{\Vert }^{2}dv$$

$$+\frac{R_u\left(1-{\beta }^2\right)}{R_m{h}^2\left(\frac{\rho R_u}{R_m}\right)}\Vert \overline{\nabla }f{\Vert }^2]h^{n-1}\left(\frac{\rho R_u}{R_m}\right)d\rho du.$$

$$\begin{array}{r}\frac{\rho }{R_m}h(R_u)\le h\left(\frac{\rho R_u}{R_m}\right)\le \frac{R_u}{R_m}h(\rho ),\\ 0\le h^{n-1}(\rho )\le h^{n-1}\left(\frac{\rho R_u}{R_m}\right).\end{array}$$

$${\int }_{\Omega }\Vert \nabla f{\Vert }^{2}dv$$

$$+\frac{R_u\left(1-{\beta }^2\right)}{R_m}{\left(\frac{R_m}{R_{u}h(\rho )}\right)}^2\Vert \overline{\nabla }f{\Vert }^2]h^{n-1}\left(\rho \right)d\rho du$$

$$\ge \left(\frac{R_m}{R_M}\right){\int }_{U_p\Omega }{\int }_0^{R_m}[\left\{1-\left(\frac{1}{{\beta }^2}-1\right)a\right\}{\left(\frac{\partial f}{\partial \rho }\right)}^2$$

$$+\frac{\left(1-{\beta }^2\right)}{h^2(\rho )}\Vert \overline{\nabla }f{\Vert }^2]h^{n-1}\left(\rho \right)d\rho du.$$

$$1-\left(\frac{1}{{\beta }^2}-1\right)a=1-{\beta }^2=\frac{(2+a)-\sqrt{a^2+4a}}{2}>0.$$

$${\int }_{\Omega }\Vert \nabla f{\Vert }^{2}dv$$

$$+\frac{1}{h^2(\rho )}\Vert \overline{\nabla }f{\Vert }^2]h^{n-1}\left(\rho \right)d\rho du$$

$$=\left(\frac{R_m}{R_M}\right)\left(\frac{(2+a)-\sqrt{a^2+4a}}{2}\right){\int }_{B(R_m)}\Vert \nabla f{\Vert }^{2}dv.$$

$${\int }_{\partial \Omega }{f}^{2}ds={\int }_{U_p\Omega }{f}^2\sec (\theta )h^{n-1}\left(R_u\right)du.$$

$${\int }_{\partial \Omega }{f}^{2}ds$$

$$\frac{{\int }_{\Omega }\Vert \nabla f{\Vert }^{2}dv}{{\int }_{\partial \Omega }{f}^{2}ds}$$

$${\int }_{\partial B(R_m)}\phi {\varphi }_i{h}^{n-2}(R_u)\sqrt{h^2(R_u)+\Vert \overline{\nabla }{R}_u{\Vert }^2}ds=0.$$

$${\int }_{\partial \Omega }\phi {\varphi }_{i}ds={\int }_{U_p\Omega }\phi {\varphi }_i\frac{\sqrt{h^2(R_u)+\Vert \overline{\nabla }{R}_u{\Vert }^2}}{h(R_u)}{h}^{n-1}\left(R_u\right)du.$$

$${\int }_{\partial \Omega }\phi {\varphi }_{i}ds$$

$$=0.$$

$${\mu }_l(\Omega )$$

$$\ge \left(\frac{R_m}{R_M}\right)\left(\frac{(2+a)-\sqrt{a^2+4a}}{2\sqrt{1+a}}\right)\frac{h^{n-1}\left(R_m\right)}{h^{n-1}\left(R_M\right)}$$

Full text

1. Introduction

Let $\Omega$ be a bounded domain in a compact connected Riemannian manifold with smooth boundary $\partial\Omega$. The Steklov eigenvalue problem is to find all real numbers $\mu$ for which there exists a nontrivial function $\varphi\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega})$ such that

[TABLE]

where $\nu$ is the outward unit normal to the boundary $\partial\Omega$. This problem was introduced by Steklov [13] for bounded domains in the plane in $1902$. Its importance lies in the fact that the set of eigenvalues of the Steklov problem is same as the set of eigenvalues of the well-known Dirichlet-Neumann map. This map associates to each function defined on $\partial\Omega$, the normal derivative of its harmonic extension on $\Omega$. The eigenvalues of the Steklov problem are discrete and form an increasing sequence $0=\mu_{1}<\mu_{2}\leq\mu_{3}\leq\cdots\nearrow\infty$. The variational characterization of $\mu_{l}$, $1\leq l<\infty$ is given by

[TABLE]

where $E$ is a set of ${l-1}$ functions $\phi_{1},\phi_{2},\ldots,\phi_{l-1}$ such that $\phi_{i}\in H^{1}(\Omega)$, $1\leq i\leq{l-1}$ and $E^{\perp}=\left\{\varphi\in H^{1}(\Omega):\int_{\partial\Omega}\varphi\phi_{i}ds=0,1\leq i\leq{l-1}\right\}$. For background on this problem, see [9].

There are several results which estimate first nonzero eigenvalue of the Steklov eigenvalue problem [1, 2, 5, 6]. The first upper bound for $\mu_{2}$ was given by Weinstock [15] in $1954$. He proved that among all simply connected planar domains with analytic boundary of fixed perimeter, the circle maximizes $\mu_{2}$. Later F. Brock [3] obtained a sharp upper bound for $\mu_{2}$ by fixing the volume of the domain. He proved that for a bounded Lipschitz domain $\Omega\subset\mathbb{R}^{n}$, $\mu_{2}(\Omega)\,\left(\text{vol}(\Omega)\right)^{\frac{1}{n}}\leq{\omega_{n}}^{\frac{1}{n}}$, where $\omega_{n}$ is the volume of the unit ball in $\mathbb{R}^{n}$ and equality holds if and only if $\Omega$ is a ball. In several recent papers, bounds for all eigenvalues of the Steklov problem have been studied [4, 8, 11, 16]. In particular, sharp upper bounds for some specific functions of the Steklov eigenvalues have been derived in [8]. Weyl-type bounds have also been obtained for Steklov eigenvalues in [11, 16].

Let $\Omega\subset\mathbb{R}^{n}$ be a star-shaped domain with smooth boundary $\partial\Omega$. Let $p$ be a center of $\Omega$. Let $R_{m}=\min\left\{d(p,x)|x\in\partial\Omega\right\}$, $R_{M}=\max\left\{d(p,x)|x\in\partial\Omega\right\}$ and $h_{m}=\min\left\{\langle x,\nu\rangle|x\in\partial\Omega\right\}$, where $\nu$ is the outward unit normal to $\partial\Omega$. With these notations, Bramble and Payne [2] proved that

[TABLE]

Equality holds when $\Omega$ is a ball.

Kuttler and Sigillito [10] proved the following lower bound for a star-shaped bounded domain in $\mathbb{R}^{2}$.

Theorem 1.1** ([10]).**

Let $\Omega$ be a star-shaped bounded domain in $\mathbb{R}^{2}$ with smooth boundary and centered at the origin. Then, for $1\leq k<\infty$,

[TABLE]

where $R(\theta)=\max\left\{|x|:x\in\Omega,x=|x|e^{i\theta}\right\}$ and equality holds for a disc.

Following the idea of Kuttler and Sigillito [10], Garcia and Montano [7] and the first author [12] obtained a similar bound for the first nonzero Steklov eigenvalue on a star-shaped domain in $\mathbb{R}^{n}$ and $\mathbb{S}^{n}$, respectively. Let $\Omega$ be a star-shaped bounded domain with smooth boundary $\partial\Omega$ centered at a point $p$ and $\nu$ be the outward unit normal to $\partial\Omega$. For any point $q\in\partial\Omega$, let $0\leq\theta(q)\leq\alpha<\frac{\pi}{2}$, where $\cos(\theta(q))=\langle\nu(q),\partial_{r}(q)\rangle$. Let $a=\tan^{2}\alpha$.

Theorem 1.2** ([7]).**

Let $\Omega\subset\mathbb{R}^{n}$. Then with the above notations, the first nonzero eigenvalue of the Steklov problem $\mu_{2}(\Omega)$ satisfies

[TABLE]

Theorem 1.3** ([12]).**

Let $\Omega$ be a star-shaped bounded domain in $\mathbb{S}^{n}$ such that $\Omega\subset\mathbb{S}^{n}\backslash\left\{-p\right\}$. Then the first nonzero Steklov eigenvalue $\mu_{2}(\Omega)$ satisfies

[TABLE]

Here $R_{m}$ and $R_{M}$ are defined as above.

In Theorem 2.2, we obtain a lower bound similar to [12], for all Steklov eigenvalues on a star-shaped domain $\Omega$ in hypersurface of revolution centered at pole. In Theorem 3.1, we prove a result for a star-shaped domain in a paraboloid in $\mathbb{R}^{3}$ analogous to the above. The main tool used to prove these results is the construction of suitable test function for the variational characterization of the corresponding eigenvalues.

2. Eigenvalues on hypersurface of revolution

Let $M$ be a hypersurface of revolution with metric $g=dr^{2}+h^{2}(r)g_{\mathbb{S}^{n-1}}$, where $g_{\mathbb{S}^{n-1}}$ is the usual metric on $\mathbb{S}^{n-1}$ and $r\in\left[0,L\right]$ for some $L\in\mathbb{R}^{+}$. Moreover, We assume that $h$ satisfies $h(0)=0$, $h^{\prime}(0)=1$. Let $\Omega$ be a star-shaped bounded domain in $M$ with respect to the pole $p$ of $M$. Let $\partial\Omega$ be the smooth boundary of $\Omega$ with outward unit normal $\nu$. Since $\Omega$ is star-shaped with respect to the point $p$ and have smooth boundary, then for every point $q\in\partial\Omega,$ there exists a unique unit vector $u\in T_{p}M$ and $R_{u}>0$ such that $q=\exp_{p}(R_{u}\,u)$. Observe that in geodesic polar coordinates, $\Omega$ and $\partial\Omega$ can be written as

[TABLE]

Define $R_{m}=\min{R_{u}},\,R_{M}=\max{R_{u}}$.

Let $\partial_{r}$ be the radial vector field starting at $p$, the center of $\Omega$ and $\nu$ be the unit outward normal to $\partial\Omega$. Since $\Omega$ is a star-shaped bounded domain, for any point $q\in\partial\Omega$, $\cos(\theta(q))=\langle\nu(q),\partial_{r}(q)\rangle>0$. Therefore $\theta(q)<\frac{\pi}{2}$ for all $q\in\partial\Omega$. By compactness of $\partial\Omega$, there exists a constant $\alpha$ such that $0\leq\theta(q)\leq\alpha<\frac{\pi}{2}$ for all $q\in\partial\Omega$. Recall that for any point $q\in\partial\Omega$, $\tan^{2}(\theta(q))=\frac{\|\overline{\nabla}R_{u}\|^{2}}{h^{2}(R_{u})}$. Additionally, assume that $h$ also satisfies the following conditions

1. (a)

$\frac{h(r)}{r}$ is a decreasing function of $r$ on $\left[0,R_{M}\right]$, 2. (b)

$h(r)$ is an increasing function of $r$ on $\left[0,R_{M}\right]$.

Lemma 2.1**.**

Let $h(r)$ be a function defined on $[0,R]$ such that $\frac{h(r)}{r}$ is a decreasing function. Then $h(r)$ satisfies the following properties:

1. (a)

If $0\leq a\leq 1$, then $h(ar)\geq ah(r)$. 2. (b)

If $a\geq 1$, then $h(ar)\leq ah(r)$.

Proof.

Since $\frac{h(r)}{r}$ is a decreasing function of $r$,

[TABLE]

Which gives the desired results. ∎

The following theorem gives a sharp lower bound for all Steklov eigenvalues on a star-shaped domain in $M$.

Theorem 2.2**.**

Let $\Omega\subset M$, $\nu$, $\alpha$, $R_{m}$ and $R_{M}$ be as the above. Let $a=\tan^{2}(\alpha)$. Then $\mu_{l}(\Omega)$, $1\leq l<\infty$ satisfies the following inequality.

[TABLE]

where $B\left(R_{m}\right)\subset M$ is the geodesic ball of radius $R_{m}$ centered at $p$. Further, if $\Omega$ is a geodesic ball, then equality occurs. Conversely, if equality holds for some $l$, then $\Omega$ is a geodesic ball of radius $R_{m}$.

Proof.

For a continuously differential real valued function $f$ defined on $\overline{\Omega}$, we first find a lower bound for $\int_{\Omega}{\|\nabla f\|^{2}}\,dv$ and then an upper bound for $\int_{\partial\Omega}{f^{2}}\,ds$ to find a lower bound for $\frac{\int_{\Omega}{\|\nabla f\|^{2}}\,dv}{\int_{\partial\Omega}{f^{2}}\,ds}$.

Let $f$ be a continuously differential real valued function defined on $\overline{\Omega}$. Then for $q\in\Omega$, $\|\nabla f\|^{2}=\left(\frac{\partial f}{\partial r}\right)^{2}+\frac{1}{h^{2}(r)}\|\overline{\nabla}f\|^{2}$. Therefore

[TABLE]

Let $u^{\prime}=u$, $\rho=\frac{r\,R_{m}}{R_{u}}$. Then $\overline{\nabla}f=\overline{\nabla}_{u^{\prime}}f-\frac{\rho}{R_{u^{\prime}}}\frac{\partial f}{\partial\rho}\,\overline{\nabla}_{u^{\prime}}R_{u^{\prime}}$. By abuse of notations, we denote $u^{\prime}$ by $u$ and $\overline{\nabla}_{u^{\prime}}$ by $\overline{\nabla}$. Then the above integral can be written as

[TABLE]

Next we estimate $\langle\overline{\nabla}f,\overline{\nabla}R_{u}\rangle$. For any function $\beta^{2}$ on $\overline{\Omega}$, Cauchy-Schwarz inequality gives

[TABLE]

Thus

[TABLE]

Note that $0\leq\frac{\rho}{R_{m}}\leq 1\leq\frac{R_{u}}{R_{m}}$ and $0\leq\rho\leq\frac{\rho\,R_{u}}{R_{m}}\leq R_{u}$. Hence

[TABLE]

We assume $\beta^{2}<1$ and by substituting above inequalities in (6), we get

[TABLE]

By solving the equation $1-\left(\frac{1}{\beta^{2}}-1\right)a=1-\beta^{2}$ for $\beta^{2}$ we see that

[TABLE]

From this it follows that

[TABLE]

Now we find an upper bound for $\int_{\partial\Omega}{f^{2}}\,ds$.

Recall that the Riemannian volume element on $\partial\Omega$, denoted $ds$, is given by $ds=\sec(\theta)\,h^{n-1}\left(R_{u}\right)\,du$ (see [14]). Then

[TABLE]

By using the fact that $h^{n-1}(R_{m})\leq h^{n-1}(R_{u})\leq h^{n-1}(R_{M})$ and substituting $r=\frac{\rho\,R_{u}}{R_{m}}$, this integral becomes

[TABLE]

By inequalities (8) and (9), we have

[TABLE]

We now construct some specific test functions for the variational characterization of $\mu_{l}(\Omega)$.

We choose the functions $\phi_{i}$, $1\leq i<\infty$ such that $\phi_{i}h^{n-2}(R_{u})\sqrt{h^{2}(R_{u})+\|\overline{\nabla}R_{u}\|^{2}}$ is the $i$th Steklov eigenfunction of $B(R_{m})$. Let $\varphi$ be an arbitrary function which satisfies

[TABLE]

Note that

[TABLE]

By substituting $r=\frac{\rho\,R_{u}}{R_{m}}$, the above integral becomes

[TABLE]

Fix $E=\left\{\phi_{1},\phi_{2},\ldots,\phi_{l-1}\right\}$ in (4). Then it follows from (4) that

[TABLE]

Since $\phi_{i}h^{n-2}(R_{u})\sqrt{h^{2}(R_{u})+\|\overline{\nabla}R_{u}\|^{2}}$ is the $i$th Steklov eigenfunction of $B(R_{m})$, we have

[TABLE]

By substituting the above value in (11), we get (5). If $\Omega$ is a geodesic ball, then $R_{m}=R_{M}$ and $a=0$, hence equality holds in (5). Next if equality holds in (5) for some $l$, then equality holds in (7) and $R_{u}=R_{m}$. Hence $\Omega$ is a geodesic ball. ∎

Remark 2.3**.**

In [7] and [12], authors obtained a lower bound for the first nonzero Steklov eigenvalue on a star-shaped bounded domain in $\mathbb{R}^{n}$ and $\mathbb{S}^{n}$, respectively. Using the above idea, a similar bound can be obtained for all nonzero Steklov eigenvalues on a star-shaped bounded domain in $\mathbb{R}^{n}$ and $\mathbb{S}^{n}$.

3. Eigenvalues on a paraboloid in $\mathbb{R}^{3}$

In this section, we state and prove the result for a star-shaped bounded domain in a paraboloid $P=\left\{(x,y,z)\in\mathbb{R}^{3}:z=x^{2}+y^{2}\right\}$. We first fix some notations which will be used to state the main result of this section.

We use the parametrization $\left(r\cos\theta,r\sin\theta,r^{2}\right)$ for paraboloid $P$, where $\theta\in[0,2\pi)$ and $r\geq 0$. Then the line element $ds^{2}$ and the area element $dA$ on $P$ is given by $ds^{2}=\left(1+4r^{2}\right)dr^{2}+r^{2}\,d\theta^{2}$ and $dA=r\sqrt{1+4r^{2}}\,dr\,d\theta$, respectively. Let $\Omega\subset P$ be a star-shaped bounded domain with respect to the origin and have smooth boundary $\partial\Omega$. Then there exists a function $R:[0,2\pi)\longrightarrow\mathbb{R}^{+}$ such that

[TABLE]

Hereafter, we denote $R(\theta)$ by $R_{\theta}$. Let $R_{m}=\text{min}\left\{R_{\theta}:\theta\in[0,2\pi)\right\}$ and $R_{M}=\text{max}\left\{R_{\theta}:\theta\in[0,2\pi)\right\}$. Define $B(R_{m})=\left\{(R_{m},\theta):\theta\in[0,2\pi)\right\}$. Let $\nu$ be the outward unit normal to $\partial\Omega$. Let $a=\max\left\{\left(1+4R_{\theta}^{2}\right)\left(\frac{R_{\theta}^{\prime}}{R_{\theta}}\right)^{2}:\theta\in[0,2\pi)\right\}$. With these notations, we prove the following theorem.

Theorem 3.1**.**

Let $\Omega$, $\nu$, $a$, $R_{m}$ and $R_{M}$ be as the above. Then $\mu_{l}(\Omega)$, $1\leq l<\infty$ satisfies

[TABLE]

Furthermore, if equality holds for some $l$ then $\Omega$ is a geodesic ball of radius $R_{m}$ and if $\Omega$ is a geodesic ball then equality holds in (12).

Proof.

Let $f$ be a continuously differentiable real valued function defined on $\overline{\Omega}$. We first obtain a lower bound for $\int_{\Omega}{\|\nabla f\|^{2}}\,dA$.

[TABLE]

Let $\phi=\theta$, $\rho=\frac{r\,R_{m}}{R_{\theta}}$. Since $\rho=\frac{r\,R_{m}}{R_{\theta}}\leq r$, we have $\sqrt{1+4r^{2}}\geq\sqrt{1+4\rho^{2}}$ and $\frac{r}{\sqrt{1+4r^{2}}}\geq\frac{\rho}{\sqrt{1+4\rho^{2}}}$. Thus the above integral can be written as

[TABLE]

For any function $\beta^{2}$ on $\overline{\Omega}$, Cauchy-Schwarz inequality gives

[TABLE]

As a consequence, we have

[TABLE]

Note that $\left(1+4\rho^{2}\right)\left(\frac{R_{\phi}^{\prime}}{R_{\phi}}\right)^{2}\leq\left(1+4R_{\phi}^{2}\right)\left(\frac{R_{\phi}^{\prime}}{R_{\phi}}\right)^{2}\leq a$ and $\frac{R_{m}}{R_{\phi}}\geq\frac{R_{m}}{R_{M}}$. Let’s assume $\beta^{2}<1$, then the above integral becomes

[TABLE]

Solving the equation $1-\left(\frac{1}{\beta^{2}}-1\right)a=1-\beta^{2}$ for $\beta^{2}$, we obtain

[TABLE]

By substituting these values, we have

[TABLE]

Now we give a lower bound for $\int_{\partial\Omega}{f^{2}}\,ds$.

[TABLE]

By substituting $\phi=\theta$, $\rho=\frac{r\,R_{m}}{R_{\theta}}$ and using the fact that $R_{\theta}\leq R_{M}$, we get

[TABLE]

Hence for a continuously differentiable real valued function $f$ defined on $\overline{\Omega}$, it follows from (13) and (14) that

[TABLE]

Now using the same argument as in Theorem 2.2, we get the desired result. ∎

Acknowledgment

The authors would like to thank Dr. Prosenjit Roy for various useful discussions. Some part of this work was done when the first author was working under project PDA/IITK/MATH/96062 at IIT Kanpur.