Consider the Fucik eigenvalue problem

$$\left\{ \begin{aligned} -\Delta u &= \alpha (u^+)^{p-1} - \beta (u^-)^{p-1}, &&x \in \Omega,\\ u&=0, &&x \in \partial\Omega, \end{aligned} \right.$$

where $u = u^+ - u^-$, and $u^\pm := \max{\pm u, 0}$.

In [¹] it is proved that the first nontrivial curve of the Fucik spectrum can be described as a set of points $(s + c(s), c(s))$, where $s \in \mathbb{R}$ and $c(s)$ defined by

$$c(s) = \inf_{\gamma \in \Gamma} \max_{u \in \gamma[-1,1]} \left(\int_{\Omega}|\nabla u|^p \, dx - s \int_{\Omega}|u^+|^p \, dx \right).$$

Here

$$\Gamma := \{\gamma \in C([-1,1], S):~ \gamma(-1) = -\varphi_1,~ \gamma(1) = \varphi_1 \},$$

where $S :=\{w \in W_0^{1,p}:~ \|w\|_{L^p}=1\}$ and $\varphi_1$ is the first eigenfunction.

There is another characterization of the first nontrivial curve of the Fucik spectrum. Namely, consider

$$\alpha^*(\beta) := \inf\left\{ \frac{\int_{\Omega}|\nabla u^-|^p \, dx}{\int_{\Omega}|u^-|^p \, dx}:~ u \in W_0^{1,p},~ u^\pm \not\equiv 0,~ \frac{\int_{\Omega}|\nabla u^+|^p \, dx}{\int_{\Omega}|u^+|^p \, dx} = \beta \right\}.$$

Note that the admissible set for this minimization problem is nonempty for all $\beta > \lambda_1(p)$. This definition is, in essence, the same as of Theorem 1.2 in [²] for the linear case $p=2$ (see also [³]), and it was pointed out in that works that for $p>1$ this definition is also ok. Let us prove this fact explicitly.

Proposition. The set of points $(\alpha^*(\beta), \beta)$ is the first nontrivial curve of the Fucik spectrum.

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In this post we would like to show the impossibility of having one certain lower bound for the first eigenvalue $\nu_1(\Omega)$ of the following problem:

$$\left\{ \begin{aligned} -\Delta u &= \nu u, &&x \in \Omega,\\ u&=0, &&x \in \Gamma_0,\\ \frac{\partial u}{\partial n}&=0, &&x \in \partial \Omega \setminus \Gamma_0. \end{aligned} \right.$$

Here $\Omega \subset \mathbb{R}^N$ is a bounded domain whose boundary $\partial \Omega$ consists of two connected components: the outer one - $\Gamma_0$, and the inner one - $\partial \Omega \setminus \Gamma_0$.

Let $\Omega^\#$ be the spherical shell (i.e., difference of two balls centred at the same point) of the same measure as $\Omega$ and such that the outer boundary of $\Omega^\#$ has the same $(N-1)$-measure as $\Gamma_0$.

It was proved by Payne & Weinberger [¹] that in the planar case $N=2$, $\nu_1(\Omega)$ satisfies the following reversed Faber-Krahn inequality:

$$\nu_1(\Omega) \leq \nu_1(\Omega^\#).$$

This result was generalized to the $N$-dimensional case, as well as to the $p$-Laplacian version of the problem, by Anoop & Kumar [²], under the assumption that $\Gamma_0$ is a sphere.

In Section 4 of the same paper [²], it is asked whether there exists a universal constant $C>0$ such that the following lower bound for $\nu_1(\Omega)$ takes place:

$$C \nu_1(\Omega^\#) \leq \nu_1(\Omega).$$

Let us show that such $C$ does not exists, in general. To this end, we will construct a sequence of equimeasurable domains $\Omega_n$ with equimeasurable outer boundary, such that $\nu_1(\Omega_n) \to 0$.

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In a recently published (but submitted a long time ago) article [¹] by S. Kolonitskii and myself, we studied the dependence of the least critical levels of the energy functional

$$E[u] = \frac{1}{p} \int_\Omega |\nabla u|^p \, dx - \int_\Omega F(u) \, dx$$

upon domain perturbations driven by a family of diffeomorphisms

$$\Phi_t(x) = x + t R(x), \quad R \in C^1(\mathbb{R}^N, \mathbb{R}^N), \quad |t|<\delta.$$

Here $F$ satisfies certain rather classical conditions.

Let us take an arbitrary minimizer $v_0$ of $E$ over the Nehari manifold $\mathcal{N}(\Omega)$ and consider a function $v_t(y) := v_0(\Phi_t^{-1}(y))$, $y \in \Omega_t$. One of the main results of our paper is the following Hadamard-type formula:

$$\left. \frac{\partial E[\alpha(v_t) v_t]}{\partial t} \right|_{t=0} = - \frac {p-1} p \int_{\partial \Omega} \left| \frac{\partial v_0}{\partial n} \right|^p \left<R, n\right> \, d\sigma,$$

where $\alpha(v_t) \in \mathbb{R}$ is a normalization coefficient such that $\alpha(v_t) v_t \in \mathcal{N}(\Phi_t(\Omega))$, and $n$ is the outward unit normal vector to $\partial \Omega$.

In the particular case $F(u) = |u|^q$, $q \in [1, p^*)$, it can easily be checked that the finding of the least critical level of $E$ can be restated as the finding of minimum for the problem

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The Picone inequality (identity) is a well-known tool with wide applications in PDEs, see, e.g., [¹]. In my paper with M. Tanaka [²], Proposition 8, we found one form of the Picone inequality which appears to be quite useful for studying problmes with the $(p,q)$-Laplacian. In this post, I would like to provide a slight generalization of Proposition 8 from [²].

Theorem. Let $1 < q < p < \infty$ and $\alpha, \beta > 0$. Assume that $u>0$ and $\varphi \geq 0$ are some differentiable functions in a domain $\Omega$. Then

$$|\nabla u|^{p-2} \nabla u \nabla \left(\frac{\varphi^p}{\alpha u^{p-1} + \beta u^{q-1}}\right) \leq \frac{1}{\alpha C} |\nabla \varphi|^p$$

and

$$|\nabla u|^{q-2} \nabla u \nabla \left(\frac{\varphi^p}{\alpha u^{p-1} + \beta u^{q-1}}\right) \leq \frac{1}{\beta}|\nabla (\varphi^{p/q})|^q,$$

where $C = 1$ if $p \leq q+1$, and $C= \frac{(q-1)^{p-2} (p-q)}{(p-2)^{p-2}}$ if $p \geq q+1$.

In particular, if $\mu>0$, then

$$(|\nabla u|^{p-2} + \mu |\nabla u|^{q-2}) \nabla u \nabla \left(\frac{\varphi^p}{\alpha u^{p-1} + \beta u^{q-1}}\right) \leq \frac{1}{\alpha C} |\nabla \varphi|^p + \frac{\mu}{\beta}|\nabla (\varphi^{p/q})|^q.$$

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Several times I found myself looking for an explicit expression of the $p$-Laplacian in polar coordinates. Usual Laplace operator considered in polar coordinates can be very useful if one works with radial domains. So, in some problems it can be quite natural to be interested in the corresponding expression for the $p$-Laplacian. However, such an expression appears to be quite bulky, which makes it complicated to apply. Nevertheless, to find easily this expression in future, I decided to post it here.

Let $u = u(x,y) = u(r,\theta)$, where $r>0$ and $\theta \in (-\pi, \pi)$. Then we have

\begin{align} \notag \Delta_p u(x,y) &= \left(u_r^2 + \frac{u_\theta^2}{r^2} \right)^\frac{p-4}{2} \newline \notag &\times \left( (p-1) u_r^2 u_{rr} + \frac{u_r^3}{r} + \frac{2(p-2) u_r u_\theta u_{r\theta}}{r^2} + \frac{u_r^2 u_{\theta \theta}}{r^2} + \frac{u_{rr} u_\theta^2}{r^2} - \frac{(p-3) u_r u_\theta^2}{r^3} + \frac{(p-1) u_\theta^2 u_{\theta \theta}}{r^4} \right). \end{align}

In particular, if $u = u(x,y) = u(r)$, then $u_\theta = u_{\theta \theta}=0$, and we get the usual

$$\Delta_p u(x,y) = (p-1) |u_r|^{p-2} u_{rr} + \frac{|u_r|^{p-2} u_r}{r}.$$

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Let us take any odd natural $n$ and denote by $H(n)$ the number of ways how $n$ can be represented as the product of factors larger or equal than $3$, where the order of factors matters. For instance, if $n=27$, then $H(n)=4$, since

$$n = 3 \cdot 3 \cdot 3 = 9 \cdot 3 = 3 \cdot 9 = 27.$$

It is known that $H(n) < n^{\eta}$, where $\eta = 1.37779\dots$ is the unique positive real zero of $\left(1-\frac{1}{2^s}\right) \zeta(s)=2$, see Theorem 5 in [¹]. Our aim is to show that this upper bound is optimal.

Lemma. For any $\varepsilon>0$ there exist infinitely many odd $n$ such that $H(n) > n^{\eta-\varepsilon}$.

Proof. We will argue along the same lines as in Section 3 of [²], where the similar optimality was obtained for all (even) numbers. First, we notice that

$$\left(1-\frac{1}{2^s}\right) \zeta(s) = \sum_{m=1,~ m ~\text{odd}}^{\infty} \frac{1}{m^s}.$$

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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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