Related papers: Fourier transform, null variety, and Laplacian's e…

We present a new formulation of Fourier transform in the picture of the $\kappa$-algebra derived in the framework of the $\kappa$-generalized statistical mechanics. The $\kappa$-Fourier transform is obtained from a $\kappa$-Fourier series…

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two…

We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\Omega \subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues…

Let $\Omega$ be an open, bounded domain in the plane with connected and smooth boundary, and $\omega$ an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue $\mu > 0$. If the boundary value of $\omega$ is a…

For a non-empty compact set $E$ in a proper subdomain $\Omega$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $\Omega$ by $d(E)$ and $d(E,\partial\Omega),$ respectively. The quantity…

Inside a fixed bounded domain $\Omega$ of the plane, we look for the best compact connected set $K$, of given perimeter, in order to maximize the first Dirichlet eigenvalue $\lambda_1(\Omega\setminus K)$. We discuss some of the qualitative…

We prove that, if $\Omega$ is an open bounded domain with smooth and connected boundary, for every $p \in (1, + \infty)$ the first Dirichlet eigenvalue of the normalized $p$-Laplacian is simple in the sense that two positive eigenfunctions…

The Faber-Krahn deficit $\delta\lambda$ of an open bounded set $\Omega$ is the normalized gap between the values that the first Dirichlet Laplacian eigenvalue achieves on $\Omega$ and on the ball having same measure as $\Omega$. For any…

For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $\lambda _1(\Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ (with…

Comparing Neumann and Dirichlet eigenvalues of the Laplacian on a bounded domain $\Omega\subseteq\Rbb^n$ is a topic that goes back at least to the work of P\'olya \cite{polya}. We study the effect of the isoperimetric ratio of $\Omega$ on…

The generalized Cartan-Hadamard conjecture says that if $\Omega$ is a domain with fixed volume in a complete, simply connected Riemannian $n$-manifold $M$ with sectional curvature $K \le \kappa \le 0$, then the boundary of $\Omega$ has the…

We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain $\Omega\subset \mathbb{R}^N$, where the bang-bang weight equals a positive constant $\overline{m}$ on a ball $B\subset\Omega$ and a negative…

Given a bounded Euclidean domain $\Omega$, we consider the sequence of optimisers of the $k^{\rm th}$ Laplacian eigenvalue within the family consisting of all possible disjoint unions of scaled copies of $\Omega$ with fixed total volume. We…

This paper is dedicated to the regularity of the optimal sets for the second eigenvalue of the Dirichlet Laplacian. Precisely, we prove that if the set $\Omega$ minimizes the functional \[ \mathcal…

We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution \[ \mu_\omega = \mathop{\circledast}_{k=1}^{\infty} \left( \frac{\delta_0 + \delta_{\lambda_1 \lambda_2 \ldots \lambda_{k-1}…

Consider an operator that takes the Fourier transform of a discrete measure supported in $\mathcal{X}\subset[-\frac 12,\frac 12)^d$ and restricts it to a compact $\Omega\subset\mathbb{R}^d$. We provide lower bounds for its smallest singular…

Let $\Omega$ be a bounded Lipshcitz domain in $\mathbb{R}^n$ and we study boundary behaviors of solutions to the Laplacian eigenvalue equation with constant Neumann data. \begin{align} \label{cequation0} \begin{cases} -\Delta u=cu\quad…

Inspired by a recent result of Funano's, we provide a sharp quantitative comparison result between the first nontrivial eigenvalues of the Neumann Laplacian on bounded convex domains $\Omega_{1} \subset \Omega_{2}$ in any dimension $d$…

Let $\Omega$ be a bounded, smooth domain. Supposing that $\alpha(p) + \beta(p) = p$, $\forall\, p \in \left(\frac{N}{s},\infty\right)$ and $\displaystyle\lim_{p \to \infty} \alpha(p)/{p} = \theta \in (0,1)$, we consider two systems for the…

A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\geq2$, there exists a constant $c>0$, depending only on $n$,…