The aim of this post is to collect in one place various lower bounds for the first eigenvalue of the $p$-Laplacian. Although the linear case $p=2$ is well-developed, the situation in the general nonlinear case $p>1$ is less known.

Let us consider the nonlinear eigenvalue problem for the $p$-Laplacian:

\[\left\{ \begin{aligned} -\text{div}(|\nabla u|^{p-2} \nabla u) &= \lambda |u|^{p-2} u &&\text{in } \Omega,\\ u &= 0 &&\text{on } \partial \Omega. \end{aligned} \right.\]

The nonlinear operator on the left-hand side is usually denoted as $\Delta_p$. Here $1 < p < \infty$, $\lambda \in \mathbb{R}$, and $\Omega$ is a bounded domain in $\mathbb{R}^N$, $N \geq 1$. This problem has to be understood in a weak sense, with the working space $W_0^{1,p}(\Omega)$.

The first eigenvalue of the $-\Delta_p$ on $\Omega$ can be defined as follows:

\[\lambda_1(\Omega) = \inf_{u \in W_0^{1,p}(\Omega) \setminus \{0\}} \frac{\int_\Omega |\nabla u|^p \, dx}{\int_\Omega |u|^p \, dx}.\]

It is very well known that $\lambda_1(\Omega) > 0$, it is simple and isolated. Moreover, $\lambda_1(\Omega)$ changes under the scaling of $\Omega$ as follows:

\[\lambda_1(s\Omega) = \lambda_1(\Omega) s^{p},\]

where $s\Omega := \{ x \in \mathbb{R}^N:~ s x \in \Omega \}$.

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General domains

$\bullet$ Let us start with the following implication, which is nothing else than the domain monotonicity property, see e.g. [_¹]:

\[\Omega_1 \subset \Omega_2 \quad \implies \quad \lambda_1(\Omega_1) \geq \lambda_1(\Omega_2).\]

As a corollary, we see that if $B_r \subset \Omega \subset B_R$, where $B_r$ and $B_R$ are open balls with radii $r$ and $R$, respectively, then

\begin{equation}\label{eq:1} \lambda_1(B_r) \geq \lambda_1(\Omega) \geq \lambda_1(B_R). \end{equation}

Equivalently, noting that $\lambda_1(B_s) = \lambda_1(B_1) s^{-p}$, we see that \eqref{eq:1} can be written in the form

\begin{equation}\label{eq:2} \lambda_1(B_1) r^{-p} \geq \lambda_1(\Omega) \geq \lambda_1(B_1) R^{-p}. \end{equation}

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$\bullet$ Inequality \eqref{eq:1} can be specified in the following way. Let $B$ be a ball such that $|\Omega| = |B|$, i.e., the volumes are equal. The famous Faber-Krahn inequality (or one can also say Pólya–Szegő inequality) states that

\begin{equation}\label{eq:FK} \lambda_1(\Omega) \geq \lambda_1(B), \end{equation}

or, in a rescaled form,

\begin{equation}\label{eq:FK2} |\Omega|^\frac{p}{N} \lambda_1(\Omega) \geq |B_1|^\frac{p}{N} \lambda_1(B_1). \end{equation}

The Faber-Krahn inequality easily follows form the Schwarz symmetrization. Moreover, here the equality holds if and only if $\Omega = B$. The last fact is already a much more subtle matter, see [_²].

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$\bullet$ The Faber-Krahn inequality \eqref{eq:FK} can be refined. In the planar case $N=2$, this was done by Bhattacharya [_³] in the following way. Define the asymmetry of a domain $\Omega$ as

\[\alpha(\Omega) = \inf_x \frac{|\Omega \setminus B(x)|}{|\Omega|},\]

where $B(x)$ is a ball of the same volume as $\Omega$ and centered at a point $x$. Then, it was proved in [_³] that there exists a constant $C>0$, independent of $\Omega$, such that

\begin{equation}\label{eq:Bhat1} \lambda_1(\Omega) \geq \lambda_1(B)(1 + C \alpha^3). \end{equation}

In the higher-dimensional case $N \geq 2$, a similar result was obtained by Fusco, Maggi, Pratelli [_⁴]. Namely, consider the Fraenkel asymmetry of $\Omega$ defined analogously to $\alpha(\Omega)$ as follows:

\[A(\Omega) = \inf_x \frac{|\Omega \Delta B(x)|}{|\Omega|}.\]

Then it was proved in [_⁴] that there exists $C = C(N,p) > 0$, independent of $\Omega$, such that

\begin{equation}\label{eq:FMP} \lambda_1(\Omega) \geq \lambda_1(B)(1 + C A(\Omega)^{2+p}). \end{equation}

Let us remark that \eqref{eq:FMP} holds for any open set $\Omega \subset \mathbb{R}^N$ with finite measure, i.e., $\Omega$ is not necessarily bounded.

A sharp version of \eqref{eq:FMP} is given by Fusco, Zhang [_⁵], where the authors have shown that

\begin{equation}\label{eq:FZ} \lambda_1(\Omega) \geq \lambda_1(B)(1 + C A(\Omega)^{2}). \end{equation}

Let us also remark, that a small disadvantage of \eqref{eq:Bhat1}, \eqref{eq:FMP} and \eqref{eq:FZ} is that the constant $C$ is not quantified there. However, in the case $p=2$ and $N=2$ some explicit lower bounds are known, see [_⁶] and [_⁷]. See also this post, where we treated the case $p \geq 2$ and $N = 2$ by the same arguments as in [_⁶].

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$\bullet$ To show another variation of inequality \eqref{eq:1}, let us denote by

\[r_\Omega := \sup \{ r:~ \exists B_r \subset \Omega\}\]

the inradius of $\Omega$. Although the upper bound for $\lambda_1(\Omega)$ through $r_\Omega$ is a trivial consequence of \eqref{eq:1}, it is quite surprising that $\lambda_1(\Omega)$ also has a lower estimate through $r_\Omega$ for some $p$ and $N$. In particular, it was shown by Poliquin [_⁸] that if $p>N$, then there exists a constant $C = C(N,p) > 0$ (and $C_1 = C_1(N,p) > 0$) such that

\begin{equation}\label{eq:Pol1} \lambda_1(\Omega) \geq C \, \lambda_1(B_{r_\Omega}) \equiv \frac{C_1}{r_\Omega^p}. \end{equation}

Moreover, if $\partial \Omega$ is connected, then \eqref{eq:Pol1} holds for all $p > N-1$.

In fact, the validity of \eqref{eq:Pol1} in the linear case $p=2$ with $N=2$ and its violation for $p=2$ and $N \geq 3$ were known much earlier, see [_⁸] for the related references.

Let us remark that in the case when $\Omega$ is convex, \eqref{eq:Pol1} holds for any $p>1$. Moreover, in this case, the constant $C_1$ in \eqref{eq:Pol1} can be quantified in the following way:

\begin{equation}\label{eq:Kaj} \lambda_1(\Omega) \geq \lambda_1((-r_\Omega, r_\Omega)) = \frac{p-1}{r_\Omega^p}\left(\frac{\pi}{p \sin(\pi/p)}\right)^p, \end{equation}

see Kajikiya [_⁹], where $\lambda_1((-r_\Omega, r_\Omega))$ is the first eigenvalue of the $p$-Laplacian on the one-dimensional interval $(-r_\Omega, r_\Omega)$. Or as in Poliquin [_¹⁰]:

\begin{equation}\label{eq:Pol21} \lambda_1(\Omega) \geq \left(\frac{1}{p \, r_\Omega}\right)^p. \end{equation}

Estimate \eqref{eq:Pol1} also holds for any $p>1$ in the planar case $\Omega \subset \mathbb{R}^2$. Namely, if $\Omega \subset \mathbb{R}^2$ is simply connected, then

\begin{equation}\label{eq:Pol22} \lambda_1(\Omega) \geq \left(\frac{|\Omega|+ \pi r_\Omega^2}{|\Omega| \, p \, r_\Omega}\right)^p \end{equation}

for all $p>1$, see [_¹⁰]. If $\Omega \subset \mathbb{R}^2$ is of connectivity $k \geq 2$, then

\begin{equation}\label{eq:Pol23} \lambda_1(\Omega) \geq \left(\frac{\sqrt{2}}{\sqrt{k} \, p \, r_\Omega}\right)^p \end{equation}

for all $p>1$, see again [_¹⁰]. Let us remark that proofs of \eqref{eq:Pol21}-\eqref{eq:Pol23} are based on estimates for the Cheeger constant $h(\Omega)$, see below.

On the other hand, it is not hard to see that no uniform upper bound for $\lambda_1(\Omega)$ by the outradius $R_\Omega$ is possible, just by taking a domain with a thin long spike.

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$\bullet$ Notice that, as a consequence of \eqref{eq:FK} and \eqref{eq:Kaj} or \eqref{eq:Pol21}, or just by using the Sobolev embedding theorem, one can easily get the bound of the form (see, e.g., (7.44) on p. 164 in Gilbarg, Trudinger [_¹¹])

\begin{equation}\label{eq:GT} \lambda_1(\Omega) \geq \frac{C(N,p)}{|\Omega|^{p/N}}, \end{equation}

which shows that $\lambda_1(\Omega)$ grows to infinity, provided $|\Omega| \to 0$.

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$\bullet$ Barta’s inequality for $\lambda_1(\Omega)$ reads as follows, see Section 2.2 in [_¹]: For any $v \in C^{1,\alpha}(\Omega)$ with $v>0$ and $\Delta_p v \in C$ there holds

\[\lambda_1(\Omega) \geq \inf_{x \in \Omega} \frac{-\Delta_p v(x)}{v(x)^{p-1}}.\]

One particular case was obtained by Bueno, Ercole, Zumpano [_¹²] and based on the supremum of the solution $\varphi_p$ of the torsional creep problem

\[\left\{ \begin{aligned} -\text{div}(|\nabla \varphi_p|^{p-2} \nabla \varphi_p) &= 1 &&\text{in } \Omega,\\ \varphi_p &= 0 &&\text{on } \partial \Omega. \end{aligned} \right.\]

Note that $\varphi_p$ is unique, $\varphi_p > 0$ in $\Omega$, and $\varphi \in C^1(\Omega)$.

It was shown in [_¹²] that

\begin{equation}\label{eq:BEZ} \lambda_1(\Omega) > \left(\max_{x \in \Omega}\,\varphi_p(x)\right)^{1-p}. \end{equation}

Thus, one can try to get upper bounds for $\text{max}\,\varphi_p$ to produce lower bounds for $\lambda_1(\Omega)$. In particular, in the case when $\Omega$ is a ball, \eqref{eq:BEZ} can be used to to get the inequality \eqref{eq:BD2}, see below.

Another bound for $\lambda_1(\Omega)$ through the torsional creep problem (and in the spirit of improved Faber-Krahn inequalities from above) was proved by Brasco [_¹³] and called Kohler-Jobin inequality:

\begin{equation}\label{eq:Br} \lambda_1(\Omega) \, T_p(\Omega)^{\alpha(p,N)} \geq \lambda_1(B) \, T_p(B)^{\alpha(p,N)}, \end{equation}

where $B$ is a ball such that $|B|=|\Omega|$, $T_p(\Omega)$ is torsional rigidity defined as

\[T_p(\Omega) := \max_{v \in W_0^{1,p}(\Omega)} \frac{\left(\int_\Omega |v| \, dx \right)^p}{\int_\Omega |\nabla v|^p \, dx},\]

and $\alpha(p,N) := \frac{p}{p+N(p-1)}$.

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$\bullet$ There is a famous lower bound via the geometry of $\Omega$ which is due to Cheeger in the linear case $p=2$, and consequently called Cheeger inequality. The Cheeger inequality for general $p>1$ was obtained by Lefton, Wei [_¹⁴], and reads as follows:

\begin{equation}\label{eq:LW} \lambda_1(\Omega) \geq \left(\frac{h(\Omega)}{p}\right)^p, \end{equation}

where $h(\Omega)$ is the Cheeger constant of $\Omega$ defined by (see e.g. [_¹⁵] for the precise definition)

\[h(\Omega) := \inf_{E \subset \Omega} \frac{|\partial E|}{|E|}.\]

A lot of lower bounds for $\lambda_1(\Omega)$ are based on \eqref{eq:LW}.

Let us remark that the following asymptotic holds as $p \to 1$ (see [_¹⁶]):

\begin{equation}\label{eq:asympCh} \lim_{p \to 1} \lambda_1(\Omega) = h(\Omega). \end{equation}

Moreover, if $p \to \infty$, then it is known that (see [_¹⁷]):

\begin{equation}\label{eq:asympJut} \lim_{p \to \infty} \lambda_1^{1/p}(\Omega) = \frac{1}{r_\Omega}. \end{equation}

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$\bullet$ One interesting estimate involves the varying $p$. Let us temporarily denote the first eigenvalue as $\lambda_1(p,\Omega)$ to reflect the dependence on $p$. Then it was shown by Lindqvist [_¹⁸] that if $p > s$, then

\begin{equation}\label{eq:Lind1} p \, \left(\lambda_1(p,\Omega)\right)^{1/p} \geq s \, \left(\lambda_1(s,\Omega)\right)^{1/s}. \end{equation}

In particular, taking $s = 2$, we have for all $p \geq 2$

\begin{equation}\label{eq:Lind2} \lambda_1(p,\Omega) \geq \left(\frac{2}{p}\right)^p \left(\lambda_1(2,\Omega)\right)^{p/2}. \end{equation}

Then, one can apply various lower bounds for $\lambda_1(2,\Omega)$ to get more estimates to $\lambda_1(p,\Omega)$ whenever $p > 2$. Recall that the available literature on the bounds for the linear case $\lambda_1(2,\Omega)$ is more developed than for general $p>1$.

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Special domains. Ball

$\bullet$ Let us note that the Faber-Krahn inequality and its improved versions reveal that estimates for $\lambda_1(\Omega)$ when $\Omega = B_s$, a ball with radius $s>0$, are of fundamental importance. Moreover, the precise value of the nonlinear eigenvalue $\lambda_1(\Omega)$ is not known (to our best knowledge) except for the 1D-case $\Omega \subset \mathbb{R}$, see \eqref{eq:Kaj}.

In this respect, we can show the estimate

\begin{equation}\label{eq:Ch1} \lambda_1(B_s) \geq \left(\frac{N}{s p}\right)^p. \end{equation}

which follows from \eqref{eq:LW} by noting that $h(B_s) = N/s$.

Another estimate is

\begin{equation}\label{eq:BD1} \lambda_1(B_s) \geq \frac{Np}{s^p}, \end{equation}

see Benedikt, Drabek [_¹⁹]. Note that \eqref{eq:BD1} is better than \eqref{eq:Ch1} for large $p>1$, but it could be worse for small $p>1$.

Another lower bound is

\begin{equation}\label{eq:BD2} \lambda_1(B_s) \geq \frac{N}{s^p} \left(\frac{p}{p-1}\right)^{p-1}, \end{equation}

see Bueno, Ercole, Zumpano [_¹²] or Benedikt, Drabek [_²⁰]. Note that \eqref{eq:BD2} is better than \eqref{eq:Ch1} and \eqref{eq:BD1} for small $p>1$, and better than \eqref{eq:Ch1} for large $p>1$. However, it can be worse than \eqref{eq:Ch1} for some middle values of $p>1$.

Yet another lower estimate of this fashion is due to Huang [_²¹] and can be stated as follows (with a correction of a small misprint in [_²¹], Theorem 3.2). For $p \in (1,2)$ it holds

\begin{equation}\label{eq:Hu} \lambda_1(B_s) \geq \frac{(N+p-2) \, p^{p/2}}{(p-1)^{p-1}(2-p)^{1-p/2}} \frac{1}{s^{p}}, \end{equation}

which is good for middle ranges of $p \in (1,2)$.

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Special domains. Spherical shell

$\bullet$ Let us consider the lower estimate when $\Omega$ is a spherical shell (or, equivalently, annulus) defined as

\[A(a,b) := \{ x \in \mathbb{R}^N:~ a< |x| < b \}.\]

It was proved by Kajikiya [_⁹] that

\begin{equation}\label{eq:Kaj2} \lambda_1(A(a,b)) > \left(\frac{a}{b}\right)^{N-1} \lambda_1((a,b)) \equiv \left(\frac{a}{b}\right)^{N-1} \frac{2^p(p-1)}{(b-a)^p}\left(\frac{\pi}{p \sin(\pi/p)}\right)^p. \end{equation}

On the other hand, using \eqref{eq:LW} and the explicit value of $h(A(a,b))$ (see e.g. [_²²]), one gets

\begin{equation}\label{eq:sphershell2} \lambda_1(A(a,b)) \geq \left(\frac{N(b^{N-1} + a^{N-1})}{p(b^N - a^N)}\right)^p. \end{equation}

Estimate \eqref{eq:sphershell2} is better than \eqref{eq:Kaj2} for small $p>1$, but worse for large $p>1$.

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Special domains. Hypercube

$\bullet$ Let $Q_L = (0,L) \times \dots \times (0,L)$ be a $N$-dimensional cube. It was proved in [_²³], Proposition 2.7, that

\[\lambda_1(Q_L) \geq \frac{\widetilde{\pi}_p^p N^\frac{p}{2}}{L^p} \quad \text{for} \quad p<2\]

and

\[\lambda_1(Q_L) \geq \frac{\widetilde{\pi}_p^p N}{L^p} \quad \text{for} \quad p>2,\]

where

\[\widetilde{\pi}_p = (p-1)^\frac{1}{p} \frac{2 \pi}{p \sin(\pi /p)} \equiv 2 (p-1)^\frac{1}{p} \int_0^1 \frac{ds}{(1-s^p)^\frac{1}{p}}.\]

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P.S. Probably, this post will be updated from time to time with other estimates.) Comments are welcome!

Last modified: 14-Nov-21

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