Related papers: A sharp estimate for the first Robin-Laplacian eig…
We prove two upper bounds for the Steklov eigenvalues of a compact Riemannian manifold with boundary. The first involves the volume of the manifold and of its boundary, as well as packing and volume growth constants of the boundary and its…
We show that as the ratio between the first Dirichlet eigenvalues of a convex domain and of the ball with the same volume becomes large, the same must happen to the corresponding ratio of isoperimetric constants. The proof is based on the…
In this paper, we study the first eigenvalue of a nonlinear elliptic system involving $p$-Laplacian as the differential operator. The principal eigenvalue of the system and the corresponding eigenfunction are investigated both analytically…
We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two…
A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\geq2$, there exists a constant $c>0$, depending only on $n$,…
We prove a quantitative Faber-Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions. The asymmetry term involves the square power of the Fraenkel asymmetry, multiplied by a constant depending on…
We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic…
We prove asymptotically optimal upper bounds for the eigenvalues of the Wentzel-Laplace operator on Riemannian manifolds with Ricci curvature bounded below. These bounds depend highly on the geometry of the boundary in addition to the…
In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a bounded domain (with smooth boundary) in a given complete (not compact a priori) Riemannian manifold with Ricci bounded below . For this, we…
This paper deals with the numerical optimization of the first three eigenvalues of the Laplace-Beltrami operator of domain in the Euclidean sphere in $\mathbb{R}^3$ with Neumann boundary conditions. We address two approaches : the first one…
The reverse isoperimetric problem asks for existence and properties of bounded convex sets in a Riemannian manifold which maximise the perimeter under all those sets of fixed volume which roll freely in a ball of some given radius. If the…
In this paper, we obtain optimal upper bounds for all the Neumann eigenvalues in two situations (that are closely related). First we consider a one-dimensional Sturm-Liouville eigenvalue problem where the density is a function $h(x)$ whose…
Upper bounds of the first non-trivial eigenvalue $\lambda_1$ of the Laplace operator of a compact submanifold $M^n$ of Euclidean space $\R^{m+1}$, by means of a new technique, are obtained. Each of the upper bounds of $\lambda_1$ depends on…
By means of a family of counter-examples, it is shown that the Reilly upper bound for the first eigenvalue of the Laplace operator for a compact submanifold in Euclidean space does not work for $n$-dimensional compact spacelike submanifolds…
An eigenvalue problem arising in optimal insulation related to the minimization of the heat decay rate of an insulated body is adapted to enforce a positive lower bound imposed on the distribution of insulating material. We prove the…
For a bounded open set $\Omega \subset \mathbb{R}^n$ with the same volume as the unit ball, the classical Faber-Krahn inequality says that the first Dirichlet eigenvalue $\lambda_1(\Omega)$ of the Laplacian is at least that of the unit ball…
We prove lower bound for the first closed or Neumann nonzero eigenvalue of the Laplacian on a compact quaternion-K\"ahler manifold in terms of dimension, diameter, and scalar curvature lower bound. It is derived as large time implication of…
We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary…
We discuss several properties of eigenvalues and eigenfunctions of the $p$-Laplacian on a ball subject to zero Dirichlet boundary conditions. Among main results, in two dimensions, we show the existence of nonradial eigenfunctions which…
We prove that every nodal domain of an eigenfunction of the Laplacian of eigenvalue $\lambda$ on a $d$-dimensional closed Riemannian manifold contains a ball of radius $c\lambda^{-1/2}(\log\lambda)^{-(d-2)/2}$. This ball is centered at a…