Related papers: A sharp estimate for the first Robin-Laplacian eig…

For a geodesic ball with non-negative Ricci curvature and mean convex boundary, it is known that the first Dirichlet eigenvalue of this geodesic ball has a sharp lower bound in term of its radius. We show a quantitative explicit inequality,…

Let $m$ be a bounded function and $\alpha$ a nonnegative parameter. This article is concerned with the first eigenvalue $\lambda\_\alpha(m)$ of the drifted Laplacian type operator $\mathcal L\_m$ given by $\mathcal L\_m(u)=…

We consider the first eigenvalue of the magnetic Laplacian with zero magnetic field on simply connected compact surfaces and we establish isoperimetric inequalities and upper bounds in terms of a bound on the gaussian curvature. As a…

Let $B_1$ be a ball in $\mathbb{R}^N$ centred at the origin and $B_0$ be a smaller ball compactly contained in $B_1$. For $p\in(1, \infty)$, using the shape derivative method, we show that the first eigenvalue of the $p$-Laplacian in…

In this paper we prove the existence of a maximum for the first Steklov-Dirichlet eigenvalue in the class of convex sets with a fixed spherical hole under volume constraint. More precisely, if $\Omega=\Omega_0 \setminus \bar{B}_{R_1}$,…

This paper is motivated by the maximization of the $k$-th eigenvalue of the Laplace operator with Neumann boundary conditions among domains of ${\mathbb R}^N$ with prescribed measure. We relax the problem to the class of (possibly…

In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by some issues arising in population dynamics. The question is to determine optimal spatial arrangements of favorable and…

In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary…

This article deals with the inverse problem of determining the unbounded real-valued electric potential of the Robin Laplacian on a bounded domain of dimension 3 or greater, by incomplete knowledge of its boundary spectral data. Namely, 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…

We complete the picture of sharp eigenvalue estimates for the p-Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator $\Delta_p$ when the Ricci curvature is bounded from below…

Let $\lambda_i(\Omega,V)$ be the $i$th eigenvalue of the Schr\"odinger operator with Dirichlet boundary conditions on a bounded domain $\Omega \subset \R^n$ and with the positive potential $V$. Following the spirit of the…

We establish rigorous quantitative inequalities for the first eigenvalue of the generalized $p$-Robin problem, for both the classical diffusion absorption case, where the Robin boundary parameter $\alpha$ is positive, and the…

In this paper we study the main properties of the first eigenvalue and its eigenfunctions of a class of highly nonlinear elliptic operators in a bounded Lipschitz domain, assuming a Robin boundary condition. Moreover, we prove a Faber-Krahn…

We study the asymptotic behavior of individual eigenvalues of the Laplacian in domains with outward peaks for large negative Robin parameters. A large class of cross-sections is allowed, and the resulting asymptotic expansions reflect both…

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…

We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the…

Robin problem for the Laplacian in a bounded planar domain with a smooth boundary and a large parameter in the boundary condition is considered. We prove a two-sided three-term asymptotic estimate for the negative eigenvalues. Furthermore,…

Previous works provided several counterexamples to monotonicity of the lowest eigenvalue for the magnetic Laplacian in the two-dimensional case. However, the three-dimensional case is less studied. We use the results obtained by Helffer,…

Motivated by relativistic materials, we develop a numerical scheme to support existing or state new conjectures in the spectral optimisation of eigenvalues of the Dirac operator, subject to infinite-mass boundary conditions. We study the…