Let $\Omega$ be a bounded domain with smooth boundary $\partial \Omega $ in an n-dimensional Riemannian manifold (M,g). The p-Laplacian is deflned by

${\Delta _p}:W_0^{1,P}(\Omega ) \mapsto {W^{ - 1,q}}(\Omega ), \\ u \mapsto {\Delta _p}u = div({\left| {\nabla u} \right|^{p - 2}}\nabla u),$

where $W_0^{1,P}(\Omega )$is the Sobolev space given by the closure of $C_0^\infty (\Omega )$ with norm

$\left\| u \right\|_{1,p}^p = {\int_\Omega {\left| u \right|} ^p}dv + \int_\Omega {{{\left| {\nabla u} \right|}^p}} dV,$

and $W_0^{1,P}(\Omega )$ is the dual space of$W_0^{1,P}(\Omega )$ and 1 $1 ﹤ p,q ﹤ \infty ,\frac{1}{p} + \frac{1}{q} = 1$As a generalization of the usual Laplacian, the p-Laplacian is widely used in many subjects, especially ${\Delta _p}$models the non-Newtonian fluids in physics. It describes dilatant fluids when p > 2 and pseudoplastics when p ﹤2 , whereas p =2 corresponds to Newtonian fluids. The operator${\Delta _p}$ with p≠ 2 also appears in many other applications, such as reaction-difiusion problems, flow through porous media, nonlinear elasticity, etc., see [14] for more details.

Let $(M, g, d\mu)$ be a weighted manifold, that is, a Riemannian manifold $(M,g)$ endowed with a weighted volume form $d\mu=e^{-\varphi}dV$, where $\varphi\in C^{\infty}(M)$ and $dV$ is the volume element induced by the metric $g$. With respect to the weighted measure, the weighted $p$-Laplacian is defined as follows

$\Delta_{p,\varphi}u= \mathrm{div}(|\nabla u|^{p-2}\nabla u)-|\nabla u|^{(p-2)}\langle\nabla u, \nabla \varphi\rangle =\Delta_{p}u-|\nabla u|^{(p-2)}\langle\nabla u, \nabla \varphi\rangle.$

We are interested in the following nonlinear eigenvalue problem (the Dirichlet eigenvalue problem)

$ \begin{cases} \Delta_{p,\varphi}u+\lambda|u|^{p-2}u=0 \ \mathrm{in}\ \Omega,\\ u|_{\partial\Omega}=0. \end{cases}$

(1.1)

We recall that the first eigenvalue for the weighted $p$-Laplacian has the following variational characterisation

$\lambda_{p,\varphi}(\Omega) =\min\limits_{\substack{u\in W_{0}^{1,p}(\Omega)\\u \not\equiv 0}} \frac{\displaystyle\int_{\Omega}|\nabla u|^{p}e^{-\varphi} \mathrm{d}V}{\displaystyle\int_{\Omega}|u|^{p}e^{-\varphi}\mathrm{d}V}.$

(1.2)

The problems of partial differential equations involving weighted $p$-Laplacian have been studied by many mathematicians, see [1, 18], etc.. For more researches on eigenvalue problems, we refer the readers to [6, 15,16] etc..

For the following nonlinear eigenvalue problem

$ \begin{cases} \Delta_{p}f+\lambda|f|^{p-2}f=0 \ \mathrm{in}\ \Omega,\\ f|_{\partial\Omega}=0. \end{cases} $

The first eigenvalue associated with a Riemannian metric $g$ on a manifold $M$ has been extensively studied in recent mathematical literature, such as[7- 11]，etc.. In [8]and [9], Kawohl-Fridman and Lefton-Wei used the coarea formula and the Cavalieri principle to estimate the lower bound of the first eigenvalue about this problem by the Cheeger constant

$\lambda_{1,p}(\Omega)\geq\left(\frac{h(\Omega)}{p}\right)^{p},$

where $h(\Omega)=\mathfrak{J}_{\infty}(\Omega)$ is the Cheeger constant of domain $\Omega$. This wonderful result inspires us to estimate the lower bounds of the first eigenvalues for the weighted $p$-Laplacian operator eigenvalue problems by the Cheeger constant.

In this paper, we use the coarea formula, the Cavalieri principle and the Federer-Fleming Theorem to investigate the first eigenvalues of problem $(1.1)$. We obtain the lower bounds estimations of the first eigenvalues for the weighted $p$-Laplace operator eigenvalue problems by the Cheeger constant and isoperimetric constant.

In this section, our main goal is to estimate the lower bounds of the first eigenvalues for the weighted $p$-Laplacian eigenvalue problems on weighted manifolds. First, we recall some preliminary knowledge of the isoperimetric constant, Cavalieri's Principle and the coarea formula for later use.

Definition 2.1 Let $M$ be an $n$-dimensional Riemannian manifold with $n\geq2$. For each $\nu>1$, the ${\rm \nu-isoperimetric\ constant}\ of\ M,\ \mathfrak{J}_{\nu}(M)$, is defined to be the infimum

$\mathfrak{J}_{\nu}(M)=\inf\limits_{\Omega}\frac{A(\partial\Omega)}{V(\Omega)^{1-\frac{1}{\nu}}},$

where $\Omega$ varies over open submanifolds of $M$ possessing compact closure and $C^{\infty}$ boundary. If $\nu=\infty$, $\mathfrak{J}_{\infty}(M)$ is called the Cheeger constant, that is

$\mathfrak{J}_{\infty}(M)=\inf\limits_{\Omega}\frac{A(\partial\Omega)}{V(\Omega)}.$

Remark 2.2 As stated in [3], the fact that $\mathfrak{J}_{\nu}(M)>0$ is only possible for $n\leq \nu \leq \infty.$ Indeed, let $\nu<n$, and consider a small geodesic ball $B(x;\epsilon)$, with center $x\in M$ and radius $\epsilon>0$, for the isoperimetric quotient of $B(x;\epsilon)$,

$\lim\limits_{\epsilon\rightarrow 0}\frac{A(\partial\Omega)}{V(\Omega)^{1-\frac{1}{\nu}}}\sim\lim\limits_{\epsilon\rightarrow 0} {\rm const.} \epsilon^{\frac{n}{\nu}-1}=0.$

So it seems at first glance that one only has a discussion of isoperimetric constants for $\nu\geq n$=dim$M$.

Definition 2.3 Let $M$ be an $n$-dimensional Riemannian manifold, $n\geq2$. For each $\nu>1$, the Sobolev constant of $M$, $\mathfrak{S}_{\nu}(M)$, is defined to be the infimum

$\mathfrak{S}_{\nu}(M)=\inf\limits_{f}\frac{\|\nabla f\|_{1}}{\|f\|_{\frac{\nu}{\nu-1}}},$

where $f\in C_{0}^{\infty}(M)$.

The isoperimetric constant and the Sobolev constant have the following famous relationship:

Lemma 2.4 (The Federer-Fleming Theorem) The isoperimetric and Sobolev constants are equal, that is,

$\mathfrak{J}_{\nu}(M)=\mathfrak{S}_{\nu}(M).$

(2.1)

The detailed proof of the Federer-Fleming theorem can be found in [3，4] and [12]. This elegant result was first proven in [4] by Federer and Fleming, and in [12] independently by Maz'ya in $1960$.

Lemma 2.5 (see[3] The coarea Formula) Let $M$ be a $C^{n}$ Riemannian manifold, and let $\Phi: M\rightarrow \mathbb{R}$ be a $C^{n}$ function. Then for any measurable function $u: M\rightarrow \mathbb{R}$ that is everywhere nonnegative or is in $L^{1}(M)$, one has

$\int_{M}u|\nabla\Phi|\mathrm{d}V=\int_{R}\mathrm{d}V_{1}(y)\int_{\Phi^{-1}[y]}(u|_{\Phi^{-1}[y]})\mathrm{d}A.$

Lemma 2.6 (see [3] Cavalieri's Principle) Let $\nu$ be a measure on Borel sets in $[0, \infty]$, $\phi$ its indefinite integral, given by

$\phi(t)=\nu ([0, t))﹤+\infty,\ \forall\ t>0,$

$(\Omega, \Sigma, \mu)$ a measure space, and $u$ a nonnegative $\Sigma$-measurable function on $\Omega$. Then

$\int_{\Omega}\phi(u(x))\mathrm{d}\mu(x)=\int_{0}^{\infty}\mu({u>t})\mathrm{d}\nu(t)$

or equivalently

$\int_{\Omega}\mathrm{d}\mu(x)\int_{0}^{u(x)}\mathrm{d}\nu(t)=\int_{0}^{\infty}\mathrm{d}\nu(t)\int_{\Omega}I_{\{u>t\}}\mathrm{d}\mu.$

Using the coarea formula and the Cavalieri principle, we can get the following lower bound estimation of the first eigenvalue for the weighted $p$-Laplacian on weighted Riemannian manifold by the Cheeger constant.

Theorem 2.7 Let $\Omega$ be a connected domain with smooth boundary $\partial\Omega$ in an $n$-dimensional weighted Riemannian manifold $(M, g, d\mu)$. Assume $\lambda_{p,\varphi}(\Omega)$ is the first eigenvalue of problem $(1.1)$ for $\varphi\in C^{\infty}(\Omega)$. Then

$\lambda^{\frac{1}{p}}_{p,\varphi}(\Omega)\geq\frac{1}{p}\left(h(\Omega)-C_{\varphi}\right),$

(2.2)

where $C_{\varphi}=\max\limits_{x\in\Omega}|\nabla \varphi|$ and $h(\Omega)=\mathfrak{J}_{\infty}(\Omega)$ are the the Cheeger constant of domain $\Omega$.

Proof For any $u\in C^{\infty}_{0}(\Omega)$, set

$\Omega(t)=\{x\in\Omega:|u|^{p}e^{-\varphi}>t\}$

and

$\ \mathrm{V}(t)=\mathrm{V}(\Omega(t)),\ \mathrm{A}(t)=\mathrm{A}(\partial\Omega(t)).$

It follows from the Hölder inequality that

$\int_\Omega | \nabla ({u^p}{e^{ - \varphi }})|{\rm{d}}V \le p\int_\Omega | u{|^{p - 1}}|\nabla u|{e^{ - \varphi }}{\rm{d}}V + \int_\Omega | u{|^p}|\nabla \varphi |{e^{ - \varphi }}{\rm{d}}V \\ \le p{\{ \int_\Omega | u{|^p}{e^{ - \varphi }}{\rm{d}}V\} ^{\frac{{p - 1}}{p}}}{\{ \int_\Omega | \nabla u{|^p}{e^{ - \varphi }}{\rm{d}}V\} ^{\frac{1}{p}}} + \int_\Omega | u{|^p}|\nabla \varphi |{e^{ - \varphi }}{\rm{d}}V.$

(2.3)

From the coarea formula, the Cavalieri principle and the definition of Cheeger constant, we can get

$\int_\Omega | \nabla ({u^p}{e^{ - \varphi }})|{\rm{d}}V = \int_0^\infty {\rm{A}} (t){\rm{d}}t \\ = \int_0^\infty {\frac{{{\rm{A}}(t)}}{{{\rm{V}}(t)}}} {\rm{V}}(t){\rm{d}}t \ge h(\Omega )\int_0^\infty {\rm{V}} (t){\rm{d}}t \\ = h(\Omega )\int_0^\infty {\rm{d}} t\int_\Omega {{I_{\{ |u{|^p}{e^{ - \varphi }} > t\} }}} {\rm{d}}V \\ = h(\Omega )\int_\Omega {\rm{d}} V\int_0^{|u{|^p}{e^{ - \varphi }}} {\rm{d}} t \\ = h(\Omega )\int_\Omega | u{|^p}{e^{ - \varphi }}{\rm{d}}V,$

since $C^{\infty}_{0}(\Omega)$ is dense in $ W_{0}^{1,p}(\Omega)$, the above relation holds also for any $u\in W_{0}^{1,p}(\Omega)$, which together with $(2.3)$ implies

$\frac{\{\displaystyle\int_{\Omega}|\nabla u|^{p}e^{-\varphi} \mathrm{d}V\}^{\frac{1}{p}}}{\{\displaystyle\int_{\Omega}| u|^{p}e^{-\varphi}\mathrm{d}V\}^{\frac{1}{p}}} \geq\frac{1}{p}\left(h(\Omega)-C_{\varphi}\right),$

this inequality and $(1.2)$ imply

$\lambda^{\frac{1}{p}}_{p,\varphi}(\Omega)=\min\limits_{\substack{u\in W_{0}^{1,p}(\Omega)\\u \not\equiv 0}}\frac{\{\displaystyle\int_{\Omega}|\nabla u|^{p}e^{-\varphi} \mathrm{d}V\}^{\frac{1}{p}}}{\{\displaystyle\int_{\Omega}| u|^{p}e^{-\varphi}\mathrm{d}V\}^{\frac{1}{p}}} \geq\frac{1}{p}\left(h(\Omega)-C_{\varphi}\right),$

which completes the proof.

Using the Federer-Fleming theorem, we can get the following lower bound estimation of the first eigenvalue by the isoperimetric constant.

Theorem 2.8 Assume that $\Omega$ satisfies the conditions of Theorem $2.7$, and the isoperimetric constant $\mathfrak{J}_{\nu}(\Omega)$ is positive for some $\nu>1$. Then

$\lambda^{\frac{1}{p}}_{p, \varphi}(\Omega) \geq\frac{1}{p}\left(\frac{\mathfrak{J}_{\nu}(\Omega)}{V(\Omega)^{\frac{1}{\nu}}}-C_{\varphi}\right).$

(2.4)

Proof For any $u\in W^{1,p}_{0}(\Omega)$, let $f(u)=|u|^{p-1}ue^{-\varphi}$, then, we first have by the Hölder inequality that

$\int_{\Omega}|f|\mathrm{d}V\leq \left\{\int_{\Omega}|f|^\frac{\nu}{\nu-1}\mathrm{d}V\right\}^\frac{\nu-1}{\nu}\left\{\int_{\Omega}1\mathrm{d}V\right\}^\frac{1}{\nu} =\left\{\int_{\Omega}|f|^\frac{\nu}{\nu-1}\mathrm{d}V\right\}^\frac{\nu-1}{\nu}V(\Omega)^\frac{1}{\nu}.$

(2.5)

According to the Federer-Fleming theorem $(2.1)$ and the definition of the sobolev constant, we deduce

$\mathfrak{J}_{\nu}(\Omega)\left\{\int_{\Omega}|f|^\frac{\nu}{\nu-1}\mathrm{d}V\right\}^\frac{\nu-1}{\nu}\leq\int_{\Omega}|\nabla f|\mathrm{d}V,$

which together with $(2.5)$ gives us

$\int_{\Omega}|f|\mathrm{d}V\leq\frac{V(\Omega)^{\frac{1}{\nu}}}{\mathfrak{J}_{\nu}(\Omega)}\int_{\Omega}|\nabla f|\mathrm{d}V.$

(2.6)

Again, by the Hölder inequality, we have

$\int_\Omega | \nabla f|{\rm{d}}V = p\int_\Omega | u{|^{p - 1}}|\nabla u|{e^{ - \varphi }}{\rm{d}}V + \int_\Omega | u{|^p}|\nabla \varphi |{e^{ - \varphi }}{\rm{d}}V \\ \le {\rm{ }}p{\left\{ {\int_\Omega | u{|^p}{e^{ - \varphi }}{\rm{d}}V} \right\}^{\frac{{p - 1}}{p}}}{\left\{ {\int_\Omega | \nabla u{|^p}{e^{ - \varphi }}{\rm{d}}V} \right\}^{\frac{1}{p}}} + \int_\Omega | u{|^p}|\nabla \varphi |{e^{ - \varphi }}{\rm{d}}V.$

(2.7)

The combination of $(2.6)$ and $(2.7)$ can yield

$\int_\Omega | u{|^p}{e^{ - \varphi }}{\rm{d}}V \\ \le {\rm{ }}\frac{{V{{(\Omega )}^{\frac{1}{\nu }}}}}{{{_\nu }(\Omega )}}\left\{ {p{{\left\{ {\int_\Omega | u{|^p}{e^{ - \varphi }}{\rm{d}}V} \right\}}^{\frac{{p - 1}}{p}}}{{\left\{ {\int_\Omega | \nabla u{|^p}{e^{ - \varphi }}{\rm{d}}V} \right\}}^{\frac{1}{p}}} + \int_\Omega | u{|^p}|\nabla \varphi |{e^{ - \varphi }}{\rm{d}}V} \right\},$

this inequality implies

$\frac{\{\displaystyle\int_{\Omega}|\nabla u|^{p}e^{-\varphi} \mathrm{d}V\}^{\frac{1}{p}}}{\{\displaystyle\int_{\Omega}| u|^{p}e^{-\varphi}\mathrm{d}V\}^{\frac{1}{p}}}\geq\frac{1}{p}\left(\frac{\mathfrak{J}_{\nu}(\Omega)}{V(\Omega)^{\frac{1}{\nu}}}-C_{\varphi}\right),$

from this inequality and $(1.2)$, it is obvious that

$\lambda^{\frac{1}{p}}_{p,\varphi}(\Omega)=\min\limits_{\substack{u\in W_{0}^{1,p}(\Omega)\\u \not\equiv 0}}\frac{\{\displaystyle\int_{\Omega}|\nabla u|^{p}e^{-\varphi} \mathrm{d}V\}^{\frac{1}{p}}}{\{\displaystyle\int_{\Omega}| u|^{p}e^{-\varphi}\mathrm{d}V\}^{\frac{1}{p}}}\geq\frac{1}{p}\left(\frac{\mathfrak{J}_{\nu}(\Omega)}{V(\Omega)^{\frac{1}{\nu}}}-C_{\varphi}\right),$

which completes the proof.

Remark 2.9 It is obvious that, if we take $\nu=\infty$, then from $(2.4)$ we have

$\lambda^{\frac{1}{p}}_{p,\varphi}(\Omega)=\min\limits_{\substack{u\in W_{0}^{1,p}(\Omega)\\ u \not\equiv 0}}\frac{\{\displaystyle\int_{\Omega}|\nabla u|^{p}e^{-\varphi} \mathrm{d}V\}^{\frac{1}{p}}}{\{\displaystyle\int_{\Omega}| u|^{p}e^{-\varphi}\mathrm{d}V\}^{\frac{1}{p}}} \\ \geq \lim\limits_{\nu\rightarrow\infty}\frac{1}{p}\left(\frac{\mathfrak{J}_{\nu}(\Omega)}{V(\Omega)^{\frac{1}{\nu}}} -C_{\varphi}\right) =\frac{1}{p}\left(h(\Omega)-C_{\varphi}\right).$

Corollary 2.10 Let $\Omega$ be a connected domain with smooth boundary $\partial\Omega$ in the Euclidean space $\mathbb{R}^{n}$. Then

$\lambda^{\frac{1}{p}}_{p,\varphi}(\Omega)\geq \frac{n}{p}\left(\left(\frac{\omega_{n}}{V(\Omega)}\right)^{\frac{1}{n}}-nC_{\varphi}\right),$

(2.8)

where $\omega_{n}$ denotes the volume of the unit ball in $R^{n}$.

Proof It is well known that

$\mathfrak{J}_{n}(\Omega)=n\omega_{n}^{\frac{1}{n}}$

for any domain $\Omega\subseteq\mathbb{R}^{n}$, where $\omega_{n}$ denotes the volume of the unit ball in $\mathbb{R}^{n}$. From this fact and $(2.4)$, we can get

$\lambda^{\frac{1}{p}}_{p,\varphi}(\Omega)\geq \frac{n}{p}\left(\left(\frac{\omega_{n}}{V(\Omega)}\right)^{\frac{1}{n}}-nC_{\varphi}\right),$

which completes the proof.

Example 1 If $\Omega=B_{n}(R)$ is a ball in $\mathbb{R}^{n}$ with radius R, then the volume of $\Omega$ is $V(\Omega)=\omega_{n}R^{n}$, and we can get

$\lambda^{\frac{1}{p}}_{p,\varphi}(\Omega)\geq\frac{n}{p}\left(\frac{1}{R}-nC_{\varphi}\right)$

directly by $(2.8)$. Since any ball is trivial Cheeger set (see [2]), by simply calculation, we can obtain

$h(\Omega)=\frac{A(\partial\Omega)}{V(\Omega)}=\frac{n}{R}$

from inequality (2.2), thus, we can get the same inequality as above.

Example 2 Let $S^{n}$ be a unit sphere with sectional curvature $1$, and $\Omega\subseteq S^{n}$ (small enough) be a relatively compact domain with smooth boundary $\partial\Omega$. Then the Ricci curvature of $S^{n}$ is $n-1$. From [17,Theorem 1.4] , we know that for any connected domain $\Omega\subset S^{n},\ n=2,3,4,5$,

$\frac{A(\partial\Omega)}{V(\Omega)^{1-\frac{1}{n}}}\geq n\omega_{n}^{\frac{1}{n}}\left(1-\tau V(\Omega)^{\frac{2}{n}}\right)^{\frac{1}{n}},$

where $\tau=\frac{n(n-1)}{2(n+2)\omega_{n}^{\frac{2}{n}}}$. According to Definition $2.1$, we derive

$\mathfrak{J}_{n}(\Omega)\geq n\omega_{n}^{\frac{1}{n}}\left(1-\tau V(\Omega)^{\frac{2}{n}}\right)^{\frac{1}{n}}.$

Then from $(2.4)$, we have

$\lambda^{\frac{1}{p}}_{p,\varphi}(\Omega)\geq\frac{1}{p}\left(\frac{n\omega_{n}^{\frac{1}{n}}\left(1-\tau V(\Omega)^{\frac{2}{n}}\right)^{\frac{1}{n}}}{V(\Omega)^{\frac{1}{n}}}-C_{\varphi}\right).$