Related papers: Domain deformations and eigenvalues of the Dirichl…

For a bounded domain $\Omega$ with a piecewise smooth boundary in a complete Riemannian manifold $M$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal…

In this paper, we investigate critical points of the Laplacian's eigenvalues considered as functionals on the space of Riemmannian metrics or a conformal class of metrics on a compact manifold. We obtain necessary and sufficient conditions…

The present paper is devoted to geometric optimization problems related to the Neumann eigenvalue problem for the Laplace-Beltrami operator on bounded subdomains $\Omega$ of a Riemannian manifold $(\mathcal{M},g)$. More precisely, we…

For a bounded domain $\Omega$ with a piecewise smooth boundary in an $n$-dimensional Euclidean space $\mathbf{R}^{n}$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for…

We consider the first eigenvalue $\lambda_1(\Omega,\sigma)$ of the Laplacian with Robin boundary conditions on a compact Riemannian manifold $\Omega$ with smooth boundary, $\sigma\in\bf R$ being the Robin boundary parameter. When $\sigma>0$…

In this paper, we compute the second variation of the first Dirichlet eigenvalue on extremal domains in general Riemannian manifolds and establish a criterion for stability. We classify the stable extremal domains in the 2-sphere and…

We study an eigenvalue problem for the Laplacian on a compact K\"{a}hler manifold. Considering the $k$-th eigenvalue $\lambda_{k}$ as a functional on the space of K\"{a}hler metrics with fixed volume on a compact complex manifold, we…

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 revisit the eigenvalue problem of the Dirichlet Laplacian on bounded domains in complete Riemannian manifolds. By building on classical results like Li-Yau's and Yang's inequalities, we derive upper and lower bounds for eigenvalues. For…

We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where…

In this paper we study some relationships between the first Dirichlet eigenvalue $\Lambda(\Omega)$ and the torsional rigidity $T(\Omega)$ of a domain $\Omega$. We consider the problem of optimizing the product $\Lambda(\Omega)T(\Omega)$…

It is widely known that the spectrum of the Dirichlet Laplacian is stable under small perturbations of a domain, while in the case of the Neumann or mixed boundary conditions the spectrum may abruptly change. In this work we discuss an…

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 the existence of nontrivial unbounded domains $\O$ in the Euclidean space $\R^d$ for which the Dirichlet eigenvalue problem for the Laplacian on $\Omega$ admits sign-changing eigenfunctions with constant Neumann values on $\partial…

We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains $\Omega\subset\mathbb{C}$ using conformal transformations of the original problem to the weighted eigenvalue problem for the Dirichlet…

We consider Laplacian eigenfunctions on a domain $\Omega \subset \mathbb{R}^d$. Under Neumann boundary conditions, the first eigenfunction is constant and the others have mean value 0. The situation is different for Dirichlet boundary…

Let $M$ be a compact Riemannian manifold with or without boundary, and let $-\Delta $ be its Laplace-Beltrami operator. For any bounded scalar potential $q$, we denote by $\lambda\_i(q)$ the $i$-th eigenvalue of the Schr\"{o}dinger type…

A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains (a.k.a. a Morse--Smale complex). This partition is generated by gradient flow lines of the eigenfunction, which bound the…

We start by revisiting the derivation of the variational formulae for the functional assigning to a bounded regular domain in a Riemannian manifold its first Dirichlet eigenvalue and extend it to (not necessarily bounded) domains in certain…

We explain how the spectrum of a closed embedded surface $\Sigma \subset \mathbb{R}^3$ relates to the Dirichlet spectrum of the bounded domain $\Omega \subset \mathbb{R}^3$ with $\partial \Omega = \Sigma$. We prove that there exists a…