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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In this post, we would like to discuss some combinatorial aspects of the $p$-Laplacian. Namely, let $\int_\Omega |\nabla u|^p \, dx$ be the $p$-Dirichlet energy, where $u \in W^{1,p}(\Omega)$ and $p>1$. Its first variation is given by

\[D^1 \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1) = p \int_\Omega |\nabla u|^{p-2} (\nabla u, \nabla \xi_1) \, dx,\]

where $\xi_1 \in W^{1,p}(\Omega)$.

The second variation (if exists) is also easy to compute:

\begin{align} \notag D^2 \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \xi_2) &= p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_2) \, dx \newline \notag &+ p \int_\Omega |\nabla u|^{p-2} (\nabla \xi_1, \nabla \xi_2) \, dx, \end{align}

where $\xi_2 \in W^{1,p}(\Omega)$.

Let us make some effort to calculate the third variation:

\begin{align} \notag D^3 \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \xi_2, \xi_3) &= p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_2) (\nabla u, \nabla \xi_3) \, dx \newline \notag &+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_1) (\nabla \xi_2, \nabla \xi_3) \, dx \newline \notag &+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_2) (\nabla \xi_1, \nabla \xi_3) \, dx \newline \notag &+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_3) (\nabla \xi_1, \nabla \xi_2) \, dx, \end{align}

where $\xi_3 \in W^{1,p}(\Omega)$.

We already start seeing some structure. So, let us now try to derive a general formula for the $n$-th variation of the energy functional. Our main result is the following one.

Proposition. Let $u \in W^{1,p}(\Omega)$. If for a natural $n \geq 1$ there exists $n$-th variation of the $p$-Direchlet energy of $u$ in direction $(\xi_1,\dots,\xi_n) \in (W^{1,p}(\Omega))^n$, then

\begin{align} \notag &D^n \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \dots, \xi_n) \newline \notag &= \int_\Omega \left( \sum\limits_{i=0}^{\lfloor \frac{n}{2} \rfloor} |\nabla u|^{p-2(n-i)} \prod\limits_{j=0}^{n-i-1} (p-2j) \left[\sum\limits_{\sigma \in B(n,n-2i)} \prod\limits_{k=1}^{n-2i} (\nabla u, \nabla \xi_{\sigma(k)}) \left(\sum\limits_{\omega \in P(n,\sigma)} \prod\limits_{l=1}^{i} (\nabla \xi_{\omega(l,1)}, \nabla \xi_{\omega(l,2)}) \right) \right] \right) dx, \end{align}

where

• $B(n,n-2i)$ is the set of all possible $(n-2i)$-combinations of $\{1,2,\dots,n\}$ such that the ordering inside each $\sigma \in B(n,n-2i)$ is immaterial. Evidently, the cardinality of $B(n,n-2i)$ is ${n \choose n-2i}$. In particular, if $i=0$, then $card(B(n,n-2i)) = 1$.

• $P(n,\sigma)$ is the set of all possible partitions of the set $\{1,2,\dots,n\} \setminus \sigma$ into pairs such that the ordering of pairs and inside a pair is immaterial. Note that $card(\sigma)=n-2i$, and hence the number of pairs in each $\omega \in P(\sigma)$ is $i$. It is not hard to see that the cardinality of $P(\sigma)$ is $\frac{(2i)!}{2^i i!}$. We represent $\omega$ as a $i \times 2$-matrix $(\omega(s,t))_{s=1..i,~t=1,2}$. For instance, if $n=6$ and $\sigma = \{1,2\}$, then

\[P(\sigma) = \left\{ \begin{pmatrix} 3 & 5 \newline 4 & 6 \end{pmatrix}, \begin{pmatrix} 3 & 4 \newline 6 & 5 \end{pmatrix}, \begin{pmatrix} 3 & 4 \newline 4 & 6 \end{pmatrix} \right\}.\]

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Lemma. Let $\Omega \subset \mathbb{R}^N$ be a bounded open set, $N \geq 1$. Let $1 \leq q \leq p \leq \gamma$, and $\gamma \leq p^*$ if $N>p$ and $\gamma<\infty$ if $N < p$. Then there exists $C=C(\Omega,q,p,\gamma)$ such that for any $u \in W_0^{1,p}(\Omega)$ the following inequality is satisfied:

\[\left(\int_\Omega |u|^q \, dx \right)^{\gamma-p} \left(\int_\Omega |u|^\gamma \, dx \right)^{p-q} \leq C \left(\int_\Omega |\nabla u|^p \, dx \right)^{\gamma-q}.\]

In fact, this inequality easily follows from the general Sobolev inequality just by applying the latter one to each term on the lhs of the former one. (I’m almost sure there could be some investigation of an inequality of this type in the literature, but I was not able to find it.)

There are a couple of things about this inequality:

1) It is just beautiful that the exponents are permuted)

2) If one considers the optimization problem for the constant $C$, then it is possible to see that a corresponding minimizer of

\[\inf_{u \in W_0^{1,p}(\Omega) \setminus \{0\}} \frac{\left(\int_\Omega |\nabla u|^p \, dx \right)^{\gamma-q}}{\left(\int_\Omega |u|^q \, dx \right)^{\gamma-p} \left(\int_\Omega |u|^\gamma \, dx \right)^{p-q}}\]

exists and satisfies the convex-concave equation

\[-\Delta_p u = \lambda |u|^{q-2} u + |u|^{\gamma-2} u.\]

In fact, such a minimization problem was explicitly considered in connection with the above equation in [¹].

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Consider the standard divisor function $\sigma_\alpha(n)$ defined as

\[\sigma_\alpha(n) = \Sigma_{d | n} d^\alpha.\]

Assume that $\alpha > 1$. It was proved at least by Gronwall in [¹] (yeah, this is the same Gronwall who obtained Gronwall’s inequality) that $\sigma_\alpha(n)$ satisfies

\[\sigma_\alpha(n) < \zeta(\alpha) n^\alpha\]

for all $n \in \mathbb{N}$, and this upper bound is achieved. Here $\zeta(\alpha)$ is the Riemann Zeta function.

Recently, thinking with Falko Baustian on one problem, we were partially interested in the asymptotic behaviour of $\sigma_\alpha(n)$ for odd numbers $n$. The natural question is whether we can improve the upper bound for odd $n$’s?

Proposition. Let $\alpha>1$, and let $n \in \mathbb{N}$ be odd. Then

\[\sigma_\alpha(n) < \left(1-\frac{1}{2^\alpha}\right) \zeta(\alpha) n^\alpha,\]

and this upper bound is achieved.

Proof. We will argue sketchely and along the same lines as in [¹], pp. 114-116. First, we decompose an arbitrary $n$ into prime factors as

\[n = p_{\lambda_1}^{\nu_1} \cdot \ldots \cdot p_{\lambda_k}^{\nu_k},\]

where $p_i$ is a prime, $\lambda_i < \lambda_{i+1}$, and $\nu_i > 0$. Note that since $n$ is odd, we have $p_i \neq 2$.

As in formula (5) of [¹], we have

\begin{equation}\label{eq:sigma} \sigma_\alpha(n) = n^\alpha \Pi_{i=1}^k \frac{1-\frac{1}{p_{\lambda_i}^{\alpha(\nu_i+1)}}}{1-\frac{1}{p_{\lambda_i}^\alpha}}. \end{equation}

Since the numerator and denominator in \eqref{eq:sigma} are less than one and since $n$ is odd, we can estimate $\sigma_\alpha(n)$ as

\[\sigma_\alpha(n) < n^\alpha \Pi_{i=1}^k \frac{1}{1-\frac{1}{p_{\lambda_i}^\alpha}} < n^\alpha \Pi_{i=2}^\infty \frac{1}{1-\frac{1}{p_{i}^\alpha}} = n^\alpha \left(1-\frac{1}{2^\alpha}\right) \Pi_{i=1}^\infty \frac{1}{1-\frac{1}{p_{i}^\alpha}} = n^\alpha \left(1-\frac{1}{2^\alpha}\right) \zeta(\alpha),\]

where $p_i$ are consequtive primes $p_1 = 2$, $p_2=3$, etc. Therefore, the desired upper bound is obtained.

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Consider a standard 3D cylinder $C_{R,a} := B_R \times [0,a]$, where $B_R \subset \mathbb{R}^2$ is a disk of radius $R>0$, and $a>0$ is the height of the cylinder. It is not hard to see (we will discuss it below) that the spectrum of the Laplace operator on $C_{R,a}$ under zero Dirichlet conditions consists of eigenvalues

\[\lambda_{m,n,k}(R,a) = \frac{j_{m,n}^2}{R^2} + \frac{k^2 \pi^2}{a^2}, \quad m \in \mathbb{N} \cup \{0\}, ~ n,k \in \mathbb{N},\]

where $j_{m,n}$ is the $n$-th positive zero of the Bessel function $J_{m}$ of order $m$. That is, the first term corresponds to the spectrum of $B_R$, and the second term - to the spectrum of $[0,a]$.

Recalling the properties of the spectrum of $B_R$, we know that the multiplicity of any $\lambda_{m,n,k}(R,a)$ can be, at least, one or two. Let us give the following more detailed result whose proof follows directly from the expression of $\lambda_{m,n,k}(R,a)$.

Lemma. Let $R,a>0$. If there exist $m_i,n_i,k_i$, $i=1,2$, such that $\lambda_{m_1,n_1,k_1}(R,a) = \lambda_{m_2,n_2,k_2}(R,a)$, then

\[a = \pi R \left(\frac{k_1^2-k_2^2}{j_{m_2,n_2}^2-j_{m_1,n_1}^2}\right)^{1/2}.\]

In particular, if $R>0$ is fixed, then the set of $a$’s for which the multiplicity of some $\lambda_{m,n,k}(R,a)$ exceeds two is at most countable. The same holds true if we fix $a>0$ and vary $R>0$. Moreover, clearly, analogous assertions can be stated for higher-dimensional cylinders $B_R \times [0,a]$, where a ball $B_R \subset \mathbb{R}^N$.

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Yesterday, during a discussion with P. Drabek, I have learned about one nice fact (see Lemma below) on the multiplicity of eigenvalues for the following one-dimensional (nonlinear) eigenvalue problem for the weighted $p$-Laplacian:

\[\left\{ \begin{aligned} -(a(x)|u'|^{p-2} u')' &= \lambda b(x) |u|^{p-2}u \quad {\rm in}\ (0,1), \\\ u(0) &= u(1) = 0, \end{aligned} \right. \tag{1}\label{1}\]

where $p \geq 2$, $a,b \in C^1(a,b)$, $a,b > 0$ on $(0,1)$. Denote by $\{\lambda_k\}$ a set of eigenvalues of \eqref{1}. (In fact, $\{\lambda_k\}$ is countable, see below.)

Unlike the linear case $p=2$, in the general nonlinear case $p \neq 2$ it is necessary to specify what one means under the multiplicity of an eigenvalue $\lambda_k$. Naturally, there are two types of the multiplicity:

• Algebraic multiplicity, say $m$, means that there exist $m$ eigenvalues $\lambda_{l}, \dots, \lambda_{l+m-1}$ with $l \leq k \leq l+m-1$ such that

\[\lambda_{l-1} < \lambda_{l} = \dots = \lambda_{k} = \dots = \lambda_{l+m-1} < \lambda_{l+m}.\]

• Geometric multiplicity, say $m$, means that there exist $m$ distinct eigenfunctions associated with $\lambda_k$. In fact, it is not entirely clear what it should be meant under “distinct”. Definitely, each eigenfunction has to be defined uniquely up to scaling. Moreover, in the linear case $p=2$, the eigenfunctions have to be linearly independent. However, in the case $p \neq 2$, it seems to be unnecessary to demand the linear independency from eigenfuctions. That is, if $u_1$ and $u_2$ are linearly independent and both associated with $\lambda_k$, and, by some chance, $c_1 u_1 + c_2 u_2$ is again an eigenfunction associated with $\lambda_k$ for some $c_1,c_2 \neq 0$, then all three eigenfunctions $u_1$, $u_2$, $c_1 u_1 + c_2 u_2$ can be considered as distinct.

Just by definition, the algebraic multiplicity implies the geometric one. The natural (and seems to be open) question is the converse assertion: Does the geometric multiplicity imply the algebraic one? Surely, this question is worth for a higher-dimensional problem, too.

In this direction, the following interesting result was proved by J. Nečas (see Appendix V, Theorme 1.1, p. 233 in [¹]):

Lemma. All eingevalues of \eqref{1} form a countable set $0 < \lambda_1 < \lambda_2 < \ldots$ such that $\lim\limits_{k\to\infty} \lambda_k = \infty$. All (normed) eigenfunctions are isolated. Moreover, each $\lambda_k$ has only a finite number of (normed) eigenfunctions.

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Recently, I accidentally stumbled upon the old article [¹] which claimed that the Bessel functions $J_{\nu}$ and $J_{\mu}$ have no common positive zero for any positive real orders $\nu, \mu \in \mathbb{R}^+$.

Recall that the same assertion for $\nu, \mu \in \mathbb{N}$ holds true and usually referred as Bourget’s hypothesis, see p.485 in [²].

However, it is quite easy to see that Bourget’s hypothesis fails, in general, if $\nu, \mu \in \mathbb{R}^+$. Thus, the result of [¹] is already incorrect. The authors of [¹] provided, as a main tool, the following integral criterion for Bourget’s hypothesis:

$J_{\nu}(\lambda) = J_{\mu}(\lambda) = 0$ for some $\lambda, \nu, \mu \in \mathbb{R}^+$ if and only if $\int_0^\lambda \frac{1}{t} J_{\nu}(t) J_{\mu}(t) \, dt = 0$.

This result would be an interesting observation if it were not false. In fact, it is ok in one direction. However, if the integral vanishes, it doesn’t imply the vanishing of $J_\nu$ and $J_\mu$. This can be easily shown numerically by considering, say, $\nu=1$, $\mu=3.2$, and looking for the value of the integral and a common zero (which doesn’t exist) in the interval $(14,15)$.

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P.S. Already in 1964 the authors speak about Bourget’s hypothesis as an “old fashion problem”. I feel a bit of shame that, after almost 60 years from [¹], I’m still interested in this very outdated problem, especially in connection with multiplicity of Dirichlet eigenvalues in annuli, see the previous post :-)

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Recently, I’ve submitted a preprint to arXiv, where I studied some properties of zeros of the cross-product of Bessel functions

\begin{equation} \label{eq:cp} J_\nu(R z) Y_\nu(z) - J_\nu(z) Y_\nu(R z), \quad \nu\geq 0, ~R>1. \end{equation}

This research originated from the previous post, where I studied the explicit expression (and value) of the Pleijel constant for the disk; see also this preprint. Recall that the expression for $Pl(B)$ crucially relies on the result of Elbert & Laforgia [¹] (see also [²]) on the asymptotic $j_{kx,k}$ as $k\to \infty$ for the zero $j_{kx,k}$ of the Bessel function $J_{kx}$. More precisely, Elbert & Laforgia [¹] shown that

\[\lim_{k \to \infty} \frac{j_{kx,x}}{k} = \iota(x), \quad x > -1,\]

where $\iota(x)$ is a unique solution of the intial value problem

\[\frac{dy}{dx} = \frac{\arccos\left(\frac{x}{y}\right)}{\sqrt{1 - \left(\frac{x}{y}\right)^2}}, \qquad y(0) = \pi.\]

In fact, $\iota(x)$ admits a closed-form representation in terms of a solution of a transcendental equation (2.1) in [¹]; see also Section 1.5 in [²].

However, if one would like to obtain an explicit expression for the Pleijel constant of annuli in a similar fashion as for the disk, then a similar asymptotic relation as of Elbert & Laforgia but for zeros of \eqref{eq:cp} is needed. (Let me recall that eigenvalues of the Dirichlet Laplace operator on the annulus $A_{R} = \{x \in \mathbb{R}^2:~ 1 < |x| < R\}$ are given by $\lambda_{\nu,k} = a_{\nu,k}^2$, where $a_{\nu,k}$ is the $k$-th positive zero of \eqref{eq:cp}, $\nu \in \mathbb{N}$.) Unfortunately, I was not able to find any appropriate result for $a_{\nu,k}$ in the literature. Moreover, it really seems that the theory of zeros of \eqref{eq:cp} is much much less developed than the theory of zeros of $J_{\nu}$, and only few bounds and asymptotic results are known. (But, from eigenbusiness point of view, zeros of \eqref{eq:cp} are no less important than zeros of $J_\nu$.)

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