Let $\Omega \subset \mathbb{R}^2$ be a bounded domain with the smooth boundary $\partial \Omega$. Consider the eigenvalues $\lambda_k$ of the Laplace operator on $\Omega$ under zero Dirichlet boundary conditions. Let us denote by $m(\lambda_k)$ the multiplicity of $\lambda_k$.

It was proved by Hoffmann-Ostenhof, Michor and Nadirashvili ¹ that $m(\lambda_k) \leq 2k-3$ for any $k \geq 3$. Here, I would like to provide the following Pleijel-type remark about asymptotic behavior of $m(\lambda_k)$ as $k \to \infty$.

Remark. The following inequality is satisfied:

\begin{equation}\label{eq:1} \limsup_{k \to \infty} \frac{m(\lambda_k)}{k} < \frac{8}{j_0^2} - 6 \cdot 10^{-9} = 1.383320546…, \end{equation}

where $j_0$ is the first zero of the Bessel function $J_0$.

Moreover, $m(\lambda_k) \leq 2k-5$ provided

\begin{equation}\label{2} k > \frac{32 \pi j_0^4 |\Omega|}{(j_0^2-4)^2} \left(\inf \left\{\epsilon:~ |{x \in \Omega: d(x,\partial \Omega) < \epsilon}| \geq \frac{(j_0^2-4)|\Omega|}{2j_0^2}\right\}\right)^{-2}. \end{equation}

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The inequality \eqref{eq:1} follows from Theorem B of ¹ in combination with the result of Pleijel ² and Bourgain ³.

The estimate \eqref{2} follows from Theorem B of ¹ in combination with Theorem 1 (iii) of van den Berg & Gittins ⁴.

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Bibliography

A week ago I became really surprised to realize that there are planar rings (annuli) for which there is an eigenvalue of the Dirichlet Laplacian with the multiplicity at least 3. I was quite sure that the situation for rings is the same as for the disk, where the multiplicity of any eigenvalue is either 1 or 2. The latter result is a consequence of Bourget’s hypothesis which states that no Bessel functions $J_\nu$ and $J_{\nu+m}$ with natural $\nu, m$ have common positive zeros. Moreover, in the well-known survey article of Grebenkov and Nguen ¹, Section 3.2 (in between (3.7) and (3.8)), it is written that eigenvalues (in rings) are either simple or twice degenerate. This assertion appears to be not true.

So, let us consider the ring $A_r := \{x \in \mathbb{R}^2: r<|x|<1\}$ and the eigenvalue problem

\[\left\{ \begin{aligned} -\Delta u &= \lambda u &&{\rm in}\ A_r, \\ u&=0 &&{\rm on }\ \partial A_r, \end{aligned} \right.\] Read more...

Consider the eigenvalue problem

\[\left\{ \begin{aligned} -\Delta u &= \lambda u &&{\rm in}\ \Omega, \\ u&=0 &&{\rm on }\ \partial \Omega, \end{aligned} \right.\]

where $\Omega \subset \mathbb{R}^2$ is a bounded domain. Denote by $\{\lambda_n\}$ the sequence of the corresponding eigenvalues,

\[0 < \lambda_1 < \lambda_2 \leq \dots \leq \lambda_n \to \infty \quad \text{as } n \to \infty,\]

and let $\varphi_n$ be an eigenfunction associated with $\lambda_n$. Let $\mu(\varphi_n)$ be a number of nodal domains of $\varphi_n$. Courant’s theorem asserts that $\mu(\varphi_n) \leq n$ for any $n$. This result was refined by Å. Pleijel as follows.

Theorem (Section 5 in [_¹]). Let $j_{0,1}$ be the first zero of the Bessel function $J_0$. Then

\[Pl(\Omega) := \limsup_{n \to \infty} \frac{\mu(\varphi_n)}{n} \leq \frac{4}{j_{0,1}^2} = 0.69166\ldots\]

The upper bound $\frac{4}{j_{0,1}^2}$ is not sharp, as it was proved, e.g., by Bougain [_²]. Moreover, it was conjectured by Polterovich [_³] that

\[Pl(\Omega) \leq \frac{2}{\pi} = 0.63661\ldots\]

In fact, this conjectured upper bound is achieved for rectangles $\Omega = (0,a) \times (0,b)$ such that $\frac{a^2}{b^2}$ is irrational; see, e.g., [_⁴]. However, it seems that, apart such rectangles, the Pleijel constant $Pl(\Omega)$ have not been found explicitly for any other domain $\Omega$. At least, the question of finding such domains was explicitly posed by Bonnaillie-Noël et al in Section 6.1 of [_⁵].

The aim of the present post is to obtain the explicit expression for $Pl(B)$, where $B$ is a unit disk (ball) in $\mathbb{R}^2$. Disk is the second most natural candidate for such tryings (after irrational rectangles), since we explicitly know all of its eigenvalues and eigenfunctions, and we know that its eigenfunctions have some good multiplicity properties. Our main result is the following.

Theorem.

\[Pl(B) = 8 \, \sup_{x>0} \left\{ x \left(\cos \theta(x)\right)^2 \right\} = 0.4613019\ldots,\]

where $\theta=\theta(x)$ is the solution of the transcendental equation

\[\tan \theta - \theta = \pi x, \quad \theta \in \left(0, \frac{\pi}{2}\right).\]

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In the previous post we discussed that the classical Faber-Krahn inequality for the first eigenvalue of the $p$-Laplacian

\[\lambda_1(\Omega) \geq \lambda_1(B)\]

can be refined in several ways, for instance, in the form

\[\lambda_1(\Omega) \geq \lambda_1(B)(1 + C \alpha),\]

where $C>0$ is some coefficient and $\alpha$ is some deficiency factor.

However, in almost all improvements known for me, the refinement is given in a non-constructive way. That is, at least the coefficient $C$ it not quantified. This causes some troubles if one wants to use such refinements for some particular domains $\Omega$ in order to get better lower estimates for $\lambda_1(\Omega)$. Nevertheless, in the linear case $p=2$ such quantification was done in [_¹] (see also [_²] for an improvement).

In this post, I would like to transpose the arguments from [_¹] to the nonlinear case $p \geq 2$. We will see how $C$ and $\alpha$ look like. Withal, the post could be served as a material to study some new techniques (at least for me :-).

So, our main result is the following.

Theorem. Let $p \geq 2$. Let $\Omega \subset \mathbb{R}^2$ be a bounded simply-connected domain. Let $B$ be a ball such that $\vert\Omega\vert = \vert B\vert$ and $r_0(\Omega)$ be its radius. Let $r_i(\Omega)$ be the inradius of $\Omega$. Define the interior deficiency of $\Omega$ as

\[d_i(\Omega) := 1 - \frac{r_i(\Omega)}{r_0(\Omega)}.\]

Then the following inequality is satisfied:

\begin{equation}\label{eq:FKimp} \lambda_1(\Omega) \geq \lambda_1(B) \left(1 + \frac{\left(\sqrt{p^2+1}-p\right)^p}{(p+1)(p+2)\pi^{p/2}} \, d_i(\Omega)^{p+1}\right). \end{equation}

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We discuss a simple and effective technique for obtaining nonexistence results for nonnegative solutions of the following problem:

\begin{equation}\label{eq:D} -\Delta_p u = g_\lambda(x, u, \nabla u), \quad x \in \Omega, \end{equation} where $p>1$, $\lambda \in \mathbb{R}$ is a parameter, and $\Omega \subset \mathbb{R}^N$ ($N \geq 1$) is a bounded domain whose boundary $\partial \Omega$ is of class $C^{1}$.

Our main assumption on the nonlinearity $g_\lambda$ is the following:

$(\ast)$ There exists $M \subset \mathbb{R}$ such that $g_\lambda(x, t, \eta) > \lambda_1 t^{p-1}$ for all $\lambda \in M$, $x \in \Omega$, $t > 0$, and $z \in \mathbb{R}^N$.

Here $\lambda_1$ is the first eigenvalue of the $p$-Laplacian with the corresponding eigenfunction $\varphi_1$, i.e.,

\[\left\{ \begin{aligned} -\Delta_p \varphi_1 &= \lambda_1 |\varphi_1|^{p-2}\varphi_1, &&x \in \Omega, \\\ \varphi_1 &= 0 &&x \in \partial \Omega \end{aligned} \right.\]

It is known [¹] that $\lambda_1$ is positive, simple and isolated, and $\varphi_1 > 0$ in $\Omega$. Moreover by the regularity result of [²] we have $\varphi_1 \in C^{1,\beta}(\overline{\Omega})$ for some $\beta \in (0,1)$.

Under a weak solution $u$ of \eqref{eq:D}, we mean a function $u \in W^{1,p}(\Omega)$ where $W^{1,p}(\Omega)$ is the standart Sobolev space, which satisfies \begin{equation} \label{eq:weak} \int_{\Omega} |\nabla u|^{p-2} \left( \nabla u, \nabla \phi \right) dx = \int_{\Omega} g_\lambda(x, u, \nabla u) \phi \, dx, \quad \forall \phi \in W_0^{1,p}(\Omega) \backslash { 0}. \end{equation}

Our main result is the following.

Theorem. Assume $(\ast)$ is satisfied. Then for every $\lambda \in M$ problem \eqref{eq:D} has no weak nonnegative solution $u \in C^{1}(\overline{\Omega})$.

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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

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About a year ago I found a pathetic prefase to Notices of the AMS, 57(1), 2010 entitled “The Art of Mathematics” and written by Michael Atiyah. That time, by some reason, I decided to translate it to Russian, and finally I did it. Now, looking back, I’m thinking it was a bit stupid idea. However, in any case the result is available, so, why not publish it here? Maybe one in ten thousand will find it interesting.

Here are .pdf of the original text and the corresponding Russian translation. The latter is also below.

Искусство математики

Майкл Атья

В традиционной дихотомии между Искусством и Наукой, математика осторожно занимает промежуточную позицию. Герман Вейль говорил, что в своих математических работах он всегда стремился к красоте и истине¹, и мы можем рассматривать их как контрастирующие характеристики Искусства и Науки. Математики пытаются понять физический мир, раскопать секреты природы, отыскать истину. Они делают это создавая мысленные конструкции огромного изящества и красоты, руководствуясь своими эстетическими взглядами. С этой точки зрения, математика связывает Искусство и Науку в одно большое предприятие – попытку человека наделить вселенную смыслом.

Мы, математики, можем оценить такое крупное философское единение, однако неспециалисту, непосвященному в наши секреты, Наука и Искусство кажутся чем-то диаметрально противоположным. Наука имеет дело с конкретными фактами бытия, в то время как Искусство существует лишь в человеческом сознании; «красота – в глазах смотрящего». Наука - объективна, Искусство - субъективно. Они обитают в параллельных плоскостях, и никогда не пересекаются.

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So, let’s try to start, and we will see how deep the rabbit hole goes.