Related papers: On the number of Courant-sharp Dirichlet eigenvalu…

We obtain spectral inequalities and asymptotic formulae for the discrete spectrum of the operator $\frac12\, \log(-\Delta)$ in an open set $\Omega\in\Bbb R^d$, $d\ge2$, of finite measure with Dirichlet boundary conditions. We also derive…

Beurling density plays a key role in the study of frame-spectrality of normalized Lebesgue measure restricted to a set. Accordingly, in this paper, the authors study the $s$-Beurling densities of regular maximal orthogonal sets of a class…

We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part,…

For every fixed dimension $d$ and sufficiently large $n$, we determine the maximum possible diameter of a strongly connected $d$-dimensional simplicial complex on $n$ vertices. This improves on a sequence of previous results and settles a…

This paper is devoted to the study of directed graphs with extremal properties relative to certain metric functionals. We characterize up to isomorphism critical digraphs with infinite values of diameter, quasi-diameter, radius and…

Let $\Omega$ be an open set in Euclidean space $\R^m,\, m=2,3,...$, and let $v_{\Omega}$ denote the torsion function for $\Omega$. It is known that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian…

We show upper bounds on the maximal dimension $d$ of Hilbert cubes $H=a_0+\{0,a_1\}+\cdots + \{0, a_d\}\subset S \cap [1, N]$ in several sets $S$ of arithmetic interest such as the squares, powerful numbers and pure powers.

The purpose of this article is to establish new lower bounds for the sums of powers of eigenvalues of the Dirichlet fractional Laplacian operator $(-\Delta)^{\alpha/2}|_{\Omega}$ restricted to a bounded domain $\Omega\subset{\mathbb R}^d$…

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in…

We prove that the first eigenvalue of the Dirichlet Laplacian for a triangle in the plane is bounded above by $\pi^2 L^2\over 9A^2$, where $L$ is the perimeter and $A$ is the area of this triangle. We show that the \mbox{constant 9} is…

For an algebraic number $\alpha$ we denote by $M(\alpha)$ the Mahler measure of $\alpha$. As $M(\alpha)$ is again an algebraic number (indeed, an algebraic integer), $M(\cdot)$ is a self-map on $\overline{\mathbb{Q}}$, and therefore defines…

In this paper we develop a detailed study on maximum and comparison principles for a degenerate elliptic system. Explicit lower bounds for principal eigenvalues of this system in terms of the measure of $\Omega$ are also proved.

We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and {to} the chordal one as well, analogous to classical…

In this note we study boundedness of a large class of maximal operators in Sobolev spaces that includes the spherical maximal operator. We also study the size of the set of Lebesgue points with respect to convergence associated with such…

This paper is about a shape optimization problem related to the Dirichlet Laplacian eingevalues in the Euclidean plane. More precisely we study the shape of the minimizer in the class of open sets of constant width. We prove that the disk…

We prove the existence of an open set $\Omega\subset\mathbb{S}^2$ for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large…

On a compact metric graph, we consider the spectrum of the Laplacian defined with a mix of standard and Dirichlet vertex conditions. A Cheeger-type lower bound on the gap $\lambda_2 - \lambda_1$ is established, with a constant that depends…

This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space with its first variation given by either a Radon measure or a function in some Lebesgue space. Pointwise decay results for the quadratic…

We provide an upper estimate for the eigenvalues of the curl curl operator on a bounded, three-dimensional Euclidean domain in terms of eigenvalues of the Dirichlet Laplacian. The result complements recent inequalities between curl curl and…

We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain $\Omega\subset\RR^n$ with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E > 0$ and the spectrum $\{E_j \}$ in terms of…