Related papers: On the first Dirichlet Laplacian eigenvalue of reg…

We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator $$ L(u)=\Delta u-\langle\nabla u,x\rangle\,, $$ as a function of the domain, is convex with respect to the Minkowski addition, and we characterize…

The main result of this paper is a sharp upper bound on the first positive eigenvalue of Dirac operators in two dimensional simply connected $C^3$-domains with infinite mass boundary conditions. This bound is given in terms of a conformal…

In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the…

This paper studies the size of the minimal gap between any two consecutive eigenvalues in the Dirichlet and in the Neumann spectrum of the standard Laplace operator on the Sierpinski gasket. The main result shows the remarkable fact that…

A celebrated inequality by Payne relates the first eigenvalue of the Dirichlet Laplacian to the first eigenvalue of the buckling problem. Motivated by the goal of establishing a quantitative version of this inequality, we show that Payne's…

We show that the ball does not maximize the first nonzero Steklov eigenvalue among all contractible domains of fixed boundary volume in $\mathbb{R}^n$ when $n \geq 3$. This is in contrast to the situation when $n=2$, where a result of…

We prove a lower bound on the first eigenvalue of the fractional Dirichlet-Laplacian of order $s$ on planar open sets, in terms of their inradius and topology. The result is optimal, in many respects. In particular, we recover a classical…

We study a system of quasilinear eigenvalue problems with Dirichlet boundary conditions on complete compact Riemannian manifolds. In particular, Cheng comparison estimates and inequality of Faber-Krahn for the first eigenvalue of a…

Inequalities between Dirichlet and Neumann eigenvalues of the Laplacian and of other differential operators have been intensively studied in the past decades. The aim of this paper is to introduce differential forms and the de Rham complex…

The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $\alpha/L(\Omega)$, and $\alpha$ lies between…

We establish two-sided estimates for the fundamental frequency (the lowest eigenvalue) of the Laplacian in an open subset G of R^n with the Dirichlet boundary condition. This is done in terms of the interior capacitary radius of G which is…

Sharp upper bounds for the first eigenvalue of the Laplacian on a surface of a fixed area are known only in genera zero and one. We investigate the genus two case and conjecture that the first eigenvalue is maximized on a singular surface…

This paper is devoted to a comparison between the normalized first (non-trivial) Neumann eigenvalue $|\Omega| \mu_1(\Omega)$ for a Lipschitz open set $\Omega$ in the plane, and the normalized first (non-trivial) Steklov eigenvalue…

We study the ground state energy of the Neumann magnetic Laplacian on planar domains. For a constant magnetic field we consider the question whether, under an assumption of fixed area, the disc maximizes this eigenvalue. More generally, we…

In this paper, we prove an upper bound for the first Robin eigenvalue of the $p$-Laplacian with a positive boundary parameter and a quantitative version of the reverse Faber-Krahn type inequality for the first Robin eigenvalue of the…

Let $M$ be an $n$-dimensional closed orientable submanifold in an $N$-dimensional space form. When $1<p \le \frac n2 + 1$, we obtain an upper bound for the first nonzero eigenvalue of the $p$-Laplacian in terms of the mean curvature of $M$…

We investigate how the lowest eigenvalue of a magnetic Laplacian depends on the geometry of a planar domain with a disk shaped hole, where the magnetic field is generated by a singular flux. Under Dirichlet boundary conditions on the inner…

The fundamental gap conjecture proved by Andrews and Clutterbuck in 2011 provides the sharp lower bound for the difference between the first two Dirichlet Laplacian eigenvalues in terms of the diameter of a convex set in $\mathbb{R}^N$. The…

A computer-assisted proof is proposed for the Laplacian eigenvalue minimization problems over triangular domains under diameter constraints. The proof utilizes recently developed guaranteed computation methods for both eigenvalues and…

For a bounded N-dimensional domain with Lipschitz boundary we extend Korn's first inequality to incompatible tensor fields. For compatible tensor fields our estimate reduces to a non-standard variant of the well known Korn's first…