Consider the sequence $\{\lambda_n(\Omega)\}$ of eigenvalues of the Dirichelet $p$-Laplacian in a bounded domain $\Omega \subset \mathbb{R}^N$ obtained via the Lusternik–Schnirelmann min-max approach. Let $\varphi_n$ be an eigenfunction associated to $\lambda_n(\Omega)$. We are interested in the estimates for the number of nodal domains of $\varphi_n$ which we denote as $\mu(\varphi_n)$.

In the linear case $p=2$, the well-known Courant nodal domain theorem says that $\mu(\varphi_n) \leq n$ for all $n \geq 1$. Its generalization to the nonlinear case $p \neq 2$ obtained in [1] asserts that

which implies

On the other hand, in the linear case $p=2$, there is a result of Pleijel [2] on the following asymptotic refinement of the Courant nodal domain theorem:

\begin{equation}\label{eq:P} \limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{4}{j_{0,1}^2} = 0.69166\ldots, \end{equation}

see, e.g., this post for a discussion.

The aim of the present post is to generalize the result of Pleijel to the $p$-Laplacian settings. Pleijel’s approach is purely variational and consists of two main ingredients: the Faber-Krahn inequality and the Weyl law.

  1. The Faber-Krahn inequality is easily available for the $p$-Laplacian, and it can be formulated as

where $B_1$ is a unit ball in $\mathbb{R}^N$; see, e.g., the discussion here. Therefore, noting that $\lambda_n(\Omega) = \lambda_1(\Omega_i)$ for any $i=1..\mu(\varphi_n)$ where $\Omega_i$ is a nodal domain of $\varphi_n$, we get


Equivalently,

\begin{equation}\label{eq:FKP} \mu(\varphi_n) \leq \frac{|\Omega| \lambda_n(\Omega)^\frac{N}{p}}{|B_1| \lambda_1(B_1)^\frac{N}{p}}. \end{equation}

  1. The Weyl law is used to estimate $\lambda_n(\Omega)$ in \eqref{eq:FKP} in terms of $n$. Unfortunately, this law is not available for the $p$-Laplacian in the required form; see the discussion in [3] and [4]. Instead, we will obtain the simplest explicit Weyl-type upper bound for $\lambda_n(\Omega)$, and this will be enough to get a Pleijel’s type result. Let $Q_h$ stands for the $N$-dimensional cube with the side length $h$. First, if $h \to 0$, then the number $m$ of cubes $Q_h$ disjointly inscribed in $\Omega$ is given by

Second, by the variational characterization of $\lambda_n(\Omega)$, we can estimate

where $h_n$ is such that there are $n$ disjoint cubes $Q_{h_n}$ inscribed in $\Omega$. We can assume that $h_n$ is maximal.

Third, we know that

Combining the previous three facts, we get

\begin{equation}\label{eq:WP} \lambda_n(\Omega) \leq \lambda_1(Q_{h_n}) = \lambda_1(Q_1) h_n^{-p} \approx \lambda_1(Q_1) \left(\frac{n}{|\Omega|}\right)^\frac{p}{N} \quad \text{as } n \to \infty. \end{equation}

Finally, mixing \eqref{eq:FKP} and \eqref{eq:WP}, we deduce that

\begin{equation}\label{eq:Plp} \boxed{\limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{1}{|B_1|} \left(\frac{\lambda_1(Q_1)}{\lambda_1(B_1)} \right)^\frac{N}{p}.} \end{equation}

Notice that this upper bound does not depend on $\Omega$. Below, we will discuss a possible way how to improve this bound.


All we need now is to get a ‘‘good’’ upper bound for $\lambda_1(Q_1)$ and a ‘‘good’’ lower bound for $\lambda_1(B_1)$.

Let us start with an upper bound for $\lambda_1(Q_1)$. From Proposition 2.7 of [5] we know that

and

where

As lower estimates for $\lambda_1(B_1)$, we use the estimate

see [6] or [7]; and

see [8] and, in general, this post for a discussion of lower bounds.

Thus, substituting all these things into \eqref{eq:Plp}, we get

\begin{equation}\label{eq:Pp<2} \limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{\Gamma\left(\frac{N}{2}+1\right)\pi^\frac{N}{2} 2^N (p-1)^N}{p^\frac{(2p-1)N}{p} \sin(\pi /p)^N} \quad \text{for} \quad p<2 \end{equation}

and

\begin{equation}\label{eq:Pp>2} \limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{\Gamma\left(\frac{N}{2}+1\right)\pi^\frac{N}{2} 2^N N^\frac{(p-2)N}{2p} (p-1)^\frac{N}{p}}{p^\frac{(p+1)N}{p} \sin(\pi /p)^N} \quad \text{for} \quad p>2. \end{equation}

The corresponding plot is depicted below by the increasing line. We see that these upper bounds does not give us a Pleijel constant smaller than $1$ even in the dimension $N=2$, which is quite sad. Note that if $p \to 1$, then the bound \eqref{eq:Pp<2} approaches $\frac{4}{\pi}=1.2732\dots$, while if $p \to \infty$, then the bound \eqref{eq:Pp>2} approaches $\frac{8}{\pi}=2.5464\dots$, see the blue line on figure below.

$N=2$.

Let us now discuss a possible improvement of \eqref{eq:Plp} which concerns an improvement of the Weyl-type upper bound. For simplicity, let us fix $N=2$. First, we can inscribe in $\Omega$ not a square tiling, but a hexagonal tiling. If $H_r$ stands for a hexagon with the inradius $r$, and if $r \to 0$, then the number $m$ of $H_r$’s disjointly inscribed in $\Omega$ is given by

Therefore, analogously to \eqref{eq:WP} we get

and hence, from \eqref{eq:FKP},

\begin{equation}\label{eq:Plp1} \boxed{\limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{2\sqrt{3}}{|B_1|} \left(\frac{\lambda_1(H_1)}{\lambda_1(B_1)} \right)^\frac{2}{p}.} \end{equation}

Noting that $B_1 \subset H_1$, we get $\lambda_1(H_1) \leq \lambda_1(B_1)$, which yields

Moreover, if $p \to \infty$, then, by the known result from [9], $\lambda_1(H_1)^\frac{1}{p} \to 1$ and $\lambda_1(B_1)^\frac{1}{p} \to 1$, i.e., this upper estimate of the upper estimate \eqref{eq:Plp1} is sharp for $p \to \infty$. See the green line on the figure above.

Thus, unfortunately, even if $n \to \infty$, we cannot show that $\mu(\varphi_n) \leq n$ for all $p>1$ without getting a substantial improvement of the Weyl-type upper bound for $\lambda_n(\Omega)$. Such an improvement is clearly a prominent problem which needs to be studied much closer.


The PDF-version of this post can be found here.


Last modified: 18-Feb-20


Bibliography

  1. Drábek, P., & Robinson, S. B. (2002). On the generalization of the Courant nodal domain theorem. Journal of Differential Equations, 181(1), 58-71. 

  2. Pleijel, Å. (1956). Remarks on Courant’s nodal line theorem. Communications on pure and applied mathematics, 9(3), 543-550. 

  3. Friedlander, L. (1989). Asymptotic behaviour of the eigenvalues of the $p$-laplacian. Communications in Partial Differential Equations, 14(8-9), 1059-1069. 

  4. Azorero, J. G., & Peral Alonso, I. (1988). Comportement asymptotique des valeurs propres du $p$-laplacien. Comptes rendus de l’Académie des sciences. Série 1, Mathématique, 307(2), 75-78. 

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

  6. 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. 

  7. 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. 

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

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