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:

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:

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:

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

# General domains

$\bullet$ Let us start with the following implication, which is nothing else than the domain monotonicity property, see e.g. [1]:

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

$$\label{eq:1} \lambda_1(B_r) \geq \lambda_1(\Omega) \geq \lambda_1(B_R).$$

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

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

$\bullet$ 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

$$\label{eq:FK} \lambda_1(\Omega) \geq \lambda_1(B),$$

or, in a rescaled form,

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

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].

$\bullet$ 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

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

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

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:

Then it was proved in [4] that there exists $C = C(N,p) > 0$, independent of $\Omega$, such that

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

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

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

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].

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

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

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

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:

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

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]:

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

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

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

for all $p>1$, see [10]. If $\Omega \subset \mathbb{R}^2$ is of connectivity $k \geq 2$, then

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

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.

$\bullet$ 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])

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

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

$\bullet$ 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

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

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

It was shown in [12] that

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

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:

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

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

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

$\bullet$ 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:

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

where $h(\Omega)$ is the Cheeger constant of $\Omega$ defined by (see e.g. [15] for the precise definition)

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]):

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

Moreover, if $p \to \infty$, then it is known that (see [17]):

$$\label{eq:asympJut} \lim_{p \to \infty} \lambda_1(\Omega) = \frac{1}{r_\Omega}.$$

$\bullet$ 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

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

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

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

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 $p>1$.

# 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

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

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

Another estimate is

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

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

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

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

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

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

# Special domains. Spherical shell

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

It was proved by Kajikiya [9] that

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

On the other hand, using \eqref{eq:LW} and the explicit value of $h(A(a,b))$ (see e.g. [22]), one gets

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

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

# 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

and

where

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

# Bibliography

1. Allegretto, W., & Huang, X. Y. (1998). A Picone’s identity for the $p$-Laplacian and applications. Nonlinear Analysis: Theory, Methods & Applications, 32(7), 819-830.  2

2. Brothers, J. E., & Ziemer, W. P. (1988). Minimal rearrangements of Sobolev functions. J. Reine Angew. Math, 384(1), 988.

3. Bhattacharya, T. (2001). Some observations on the first eigenvalue of the $p$-Laplacian and its connections with asymmetry. Electron. J. Differential Equations, 35(2001), 1-15.  2

4. Fusco, N., Maggi, F., & Pratelli, A. (2009). Stability estimates for certain Faber-Krahn, isocapacitary and Cheeger inequalities. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie IV, 8(1), 51.  2

5. Fusco, N., & Zhang, Y. R. Y. (2017). A quantitative form of Faber–Krahn inequality. Calculus of Variations and Partial Differential Equations, 56(5), 138.

6. Hansen, W., & Nadirashvili, N. (1994). Isoperimetric inequalities in potential theory. Potential Analysis, 3(1), 1-14.  2

7. Bourgain, J. (2013). On Pleijel’s nodal domain theorem. arXiv preprint arXiv:1308.4422.

8. Poliquin, G. (2015). Principal frequency of the $p$-Laplacian and the inradius of Euclidean domains. Journal of Topology and Analysis, 7(03), 505-511.  2

9. Kajikiya, R. (2015). A priori estimate for the first eigenvalue of the $p$-Laplacian. Differential and Integral Equations, 28(9/10), 1011-1028.  2

10. Poliquin, G. (2014). Bounds on the Principal Frequency of the $p$-Laplacian. Contemp. Math, 630, 349-366.  2 3

11. Gilbarg, D., & Trudinger, N. S. (1998). Elliptic partial differential equations of second order. Springer.

12. Bueno, H., Ercole, G., & Zumpano, A. (2009). Positive solutions for the $p$-Laplacian and bounds for its first eigenvalue. Advanced Nonlinear Studies, 9(2), 313-338.  2 3

13. Brasco, L. (2014). On torsional rigidity and principal frequencies: an invitation to the Kohler− Jobin rearrangement technique. ESAIM: Control, Optimisation and Calculus of Variations, 20(2), 315-338.

14. Lefton, L., & Wei, D. (1997). Numerical approximation of the first eigenpair of the $p$-Laplacian using finite elements and the penalty method. Numerical Functional Analysis and Optimization, 18(3-4), 389-399.

15. Leonardi, G. P. (2015). An overview on the Cheeger problem. In New trends in shape optimization (pp. 117-139). Springer.

16. Kawohl, B., & Fridman, V. (2003). Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant. Comment. Math. Univ. Carolin, 44(4), 659-667.

17. Juutinen, P., Lindqvist, P., & Manfredi, J. J. (1999). The $\infty$-eigenvalue problem. Archive for rational mechanics and analysis, 148(2), 89-105.

18. Lindqvist, P. (1993). On non-linear Rayleigh quotients. Potential Analysis, 2(3), 199-218.

19. Benedikt, J., & Drábek, P. (2012). Estimates of the principal eigenvalue of the $p$-Laplacian. Journal of Mathematical Analysis and Applications, 393(1), 311-315.

20. Benedikt, J., & Drábek, P. (2013). Asymptotics for the principal eigenvalue of the $p$-Laplacian on the ball as p approaches 1. Nonlinear Analysis: Theory, Methods & Applications, 93, 23-29.

21. Huang, Y. (1997). On the eigenvalues of the $p$-Laplacian with varying $p$. Proceedings of the American Mathematical Society, 125(11), 3347-3354.  2

22. Bueno, H., & Ercole, G. (2015). On the $p$-torsion functions of an annulus. Asymptotic Analysis, 92(3-4), 235-247.

23. Bonder, J. F., & Pinasco, J. P. (2008). Estimates for eigenvalues of quasilinear elliptic systems. Part II. Journal of Differential Equations, 245(4), 875-891.