Consider the Neumann eigenvalue problem

$$\left\{ \begin{aligned} -\Delta \, u &= \mu u &&\text{in } \Omega, \\ \frac{\partial u}{\partial \nu} &= 0 &&\text{on } \partial \Omega, \end{aligned} \right.$$

where $\Omega \subset \mathbb{R}^2$ is a bounded convex domain. Recently, Filonov in [¹] obtained the following lower bound on the eigenvalue counting function:

$$N_{\mathcal{N}}(\Omega,\mu) \geq \frac{|\Omega| \lambda}{2 \sqrt{3} j_0^2},$$

where $j_0^2$ is the first positive zero of the Bessel function $J_0$.

Sketchily, the approach of [¹] is the following: first we densely pack equal disks in $\mathbb{R}^2$, and then choose those whose centers lie in $\Omega$. If a disk $B$ is a subset of $\Omega$, then we consider the first Dirichlet eigenfunction in $B$. If $B \setminus \Omega$ is nonempty, then we consider the restriction of the first Dirichlet eigenfunction in $B$ to $\Omega \cap B$. Using these functions, we construct the test subspace and estimate $\mu_k(\Omega)$ from above by a factor coming from the number of disks and the first Dirichlet eigenvalue $\lambda_1(B)$, which leads to a required lower bound for $N_{\mathcal{N}}(\Omega,\mu)$.

The tricky point here is the consideration of the case when $B \setminus \Omega$ is nonempty. In this case, roughly speaking, one needs to justify that

$$\tau_1(\Omega \cap B) \leq \lambda_1(B),$$

where $\tau_1(\Omega \cap B)$ is the first eigenvalue in $\Omega \cap B$ under the zero Dirichlet boundary conditions on $\overline{\Omega} \cap \partial B$ and zero Neumann boundary conditions on the remaining part of $\partial (\Omega \cap B)$. This fact follows from Lemma 2.1 in [¹] which states a certain integral property of Bessel functions. Here the convexity of $\Omega$ is employed. (Note that the fact remains true if $\Omega$ is merely star-shaped with respect to the center of $B$. However, it is hard to weaken the convexity in general, since the position of $B$ with respect to $\Omega$ is not given constructively.)

It is tempting to anticipate that one could substitute disks by hexagons in the approach above, and hence improve the upper bound for $\mu_k(\Omega)$, thereby improving the lower bound for $N_{\mathcal{N}}(\Omega,\mu)$. To do it rigorously, one has to prove the inequality

$$\tau_1(\Omega \cap H) \leq \lambda_1(H),$$

where $H$ is a hexagon such that $H \setminus \Omega \neq \emptyset$.

Unfortunately, it seems that the inequality cannot be true, in general. Let $H$ be a hexagon with the side $1$ centered at $(0,0)$, and let $\Omega$ be a large triangle spanned on the points $(0,0)$, $(-20,-2)$, $(20,-2)$, see figure below.

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Mathematica gives the following values for the corresponding eigenvalues:

$$\tau_1(\Omega \cap H) \approx 7.20569 \quad \text{and} \quad \lambda_1(\Omega) \approx 7.15548.$$

I admit that calculations might be completely wrong for $\tau_1(\Omega \cap H)$, but $\lambda_1(\Omega)$ is calculated more-less ok. Playing with parameters (vertices of the triangle $\Omega$), I also observe continuous dependence of $\tau_1(\Omega \cap H)$ on them. The inequality holds for some parameters, and does not hold for others. This indirectly indicates that the calculation can be reliable.

Conclusion: it is not that easy to enhance the estimate from [¹] using the same strategy.

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Let us see (visually) how the derivative of $\sin_p$ with respect to $p$ looks like. Recall that the generalized $p$-trigonometric function $\sin_p(x)$ for $x \in (0,\pi_p/2)$ can be defined as the inverse of the function

$$\frac{\pi_p}{2} \frac{B(y^p,1/p,1-1/p)}{B(1/p,1-1/p)}, \quad y \in (0,1),$$

where $B(y^p,1/p,1-1/p)$ and $B(1/p,1-1/p)$ stand for the incomplete and complete Beta functions, respectively. See, e.g., Eq. (2.15) in [¹]. Here, $\pi_p = \frac{2\pi}{p \sin(\pi/p)}$.

Consider the function $\sin_p(\pi_p x/2)$ for $x \in (0,1)$ which is then defined as the inverse of

$$\frac{B(y^p,1/p,1-1/p)}{B(1/p,1-1/p)}, \quad y \in (0,1).$$

Using the relation between the derivatives (wrt a parameter) of a function and its inverse, and by launching, say, Mathematica, we can obtain the following figures. (Of course, if I didn’t mess up with the code). The blue graphs are the graphs of $\sin_p(\pi_p x/2)$ for $x \in (0,1)$, and the orange ones are the graphs of the corresponding derivative wrt $p$.

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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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