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

\[\mu(\varphi_n) \leq 2n-2 \quad \text{for all } n \geq 2,\]

which implies

\[\limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq 2.\]

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

\[|\Omega|^\frac{p}{N} \lambda_1(\Omega) \geq |B_1|^\frac{p}{N} \lambda_1(B_1),\]

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

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\[|\Omega| \lambda_n(\Omega)^\frac{N}{p} \geq \mu(\varphi_n) |B_1| \lambda_1(B_1)^\frac{N}{p}.\]

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

\[m \approx \frac{|\Omega|}{h^N}.\]

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

\[\lambda_n(\Omega) \leq \lambda_1(Q_{h_n}),\]

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

\[\lambda_1(Q_h) = \lambda_1(Q_1) h^{-p}.\]

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.

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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

\[\lambda_1(Q_1) \leq \widetilde{\pi}_p^p N \quad \text{for} \quad p<2\]

and

\[\lambda_1(Q_1) \leq \widetilde{\pi}_p^p N^\frac{p}{2} \quad \text{for} \quad p>2,\]

where

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

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

\[\lambda_1(B_1) \geq N \left(\frac{p}{p-1}\right)^{p-1} \quad \text{for} \quad p<2,\]

see [⁶] or [⁷]; and

\[\lambda_1(B_1) \geq Np \quad \text{for} \quad p>2,\]

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.

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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

\[m \approx \frac{|\Omega|}{2 \sqrt{3} r^2}.\]

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

\[\lambda_n(\Omega) \leq \lambda_1(H_{r_n}) = \lambda_1(H_1) r_n^{-p} \approx \lambda_1(H_1) \left(\frac{2\sqrt{3} n}{|\Omega|}\right)^\frac{p}{2} \quad \text{as } n \to \infty,\]

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

\[\limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{2\sqrt{3}}{\pi} = 1.1026\dots\]

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.

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

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Last modified: 18-Feb-20

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