Related papers: Two disks maximize the third Robin eigenvalue: pos…

We introduce an analogue of Payne's nodal line conjecture, which asserts that the nodal (zero) set of any eigenfunction associated with the second eigenvalue of the Dirichlet Laplacian on a bounded planar domain should reach the boundary of…

Let $\Omega=\Omega_0\setminus \overline{\Theta}\subset \mathbb{R}^n$, $n\geq 2$, where $\Omega_0$ and $\Theta$ are two open, bounded and convex sets such that $\overline{\Theta}\subset \Omega_0$ and let $\beta<0$ be a given parameter. We…

We consider a Riemannian cylinder endowed with a closed potential 1-form A and study the magnetic Laplacian with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and…

On a bounded Lipschitz domain we consider two selfadjoint operator realizations of the same second order elliptic differential expression subject to Robin boundary conditions, where the coefficients in the boundary conditions are functions.…

We discuss isoperimetric inequalities for the magnetic Laplacian on bounded domains of $\mathbb R^2$ endowed with an Aharonov-Bohm potential. When the flux of the potential around the pole is not an integer, the lowest eigenvalue for the…

We generalize a classical inequality between the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on bounded, planar domains: in 1955, Payne proved that below the $k$-th eigenvalue of the Dirichlet Laplacian…

We prove an improved Pleijel nodal domain theorem for the Robin eigenvalue problem. In particular we remove the restriction, imposed in previous work, that the Robin parameter be non-negative. We also improve the upper bound in the…

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a…

Characterizations of eigenvalues and eigenfunctions of the Laplacian on a product domain are obtained. When zero Dirichlet, Robin or Neumann boundary conditions are specified on each factor, then the eigenfunctions on the product domain are…

The Faber-Krahn inequality in $\mathbb{R}^2$ states that among all open bounded sets of given area the disk minimizes the first Dirichlet Laplacian eigenvalue. There are numerical evidences that for all $N\ge 3$ the first Dirichlet…

This article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter $h$). For the square, our first…

Let $M$ be a compact connected smooth manifold with smooth boundary, and let $\rho$ be a positive continuous function on the boundary which is served as the Robin parameter. In this paper, we study three problems concerning the prescription…

We consider the Laplace operator on a triangle, subject to attractive Robin boundary conditions. We prove that the equilateral triangle is a local maximiser of the lowest eigenvalue among all triangles of a given area provided that the…

We consider the Robin Laplacian in the domains $\Omega$ and $\Omega^\varepsilon$, $\varepsilon >0$, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient $a$ is large enough, the spectrum of the problem in $\Omega$…

In this paper, we prove two results for the Robin eigenvalue problem. One is an upper bound for the ratio of the first two eigenvalues which can be used to recover the PPW conjecture proved by M.S.Ashbaugh and R.D.Benguria, the other is a…

This paper deals with eigenelements of the Laplacian in bounded domains, under Robin boundary conditions, without any assumption on the sign of the Robin parameter. We quantify the asymptotics of the variation of simple eigenvalues under…

We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet-Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary…

We consider the Laplacian on a class of smooth domains $\Omega\subset \mathbb{R}^{\nu}$, $\nu\ge 2$, with attractive Robin boundary conditions: \[ Q^\Omega_\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text{ on }…

In this work, optimal rigidity results for eigenvalues on K\"ahler manifolds with positive Ricci lower bound are established. More precisely, for those K\"ahler manifolds whose first eigenvalue agrees with the Ricci lower bound, we show…

We consider the Courant-sharp eigenvalues of the Robin Laplacian for bounded, connected, open sets in $\mathbb{R}^n$, $n \geq 2$, with Lipschitz boundary. We prove Pleijel's theorem which implies that there are only finitely many…