Related papers: Maximization of the second Laplacian Eigenvalue on…
In this paper, we establish two $p$-eigenvalue pinching sphere theorems, for the \( p \)-Laplacian, $p>1$. The first result states that if the first non-zero $p$-eigenvalue of a closed Riemannian $n$-manifold with sectional curvature…
An integral inequality is derived for compact submanifolds (with or without boundary) in the unit sphere. This result leads to a characterization of spheres.
In this paper, under suitable geometric constraints, we have successfully obtained characterizations for the extremum values of the functional of mixed eigenvalues of the Laplacian on triangles (or trapezoids) in the Euclidean plane…
We study a $(k+1)$-dimensional hyperbolic space of a negative constant sectional curvature $\kappa=-1/\rho^2$. Let $\lambda$ be a real eigenvalue and $f_{\lambda} (x)$ be an eigenfunction of the hyperbolic Laplacian assuming a non-zero…
Given a Riemmanian manifold, we provide a new method to compute a sharp upper bound for the first eigenvalue of the Laplacian for the Dirichlet problem on a geodesic ball of radius less than the injectivity radius of the manifold. This…
We prove that, for a Finsler space, if the weighted Ricci curvature is bounded below by a positive number and the diam attains its maximal value, then it is isometric to a standard Finsler sphere. As an application, we show that the first…
For a domain $\Omega$ contained in a hemisphere of the $n$-dimensional sphere $\SS^n$ we prove the optimal result $\lambda_2/\lambda_1(\Omega) \le \lambda_2/\lambda_1(\Omega^{\star})$ for the ratio of its first two Dirichlet eigenvalues…
In this paper, we study the optimization of the first Laplacian eigenvalue on axisymmetric doubly connected domains under positive Robin boundary conditions. Under additional geometric constraints, we prove that spherical shells maximize…
We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We…
Based on a recent work of Mancini-Thizy [28], we obtain the nonexistence of extremals for an inequality of Adimurthi-Druet [1] on a closed Riemann surface $(\Sigma,g)$. Precisely, if $\lambda_1(\Sigma)$ is the first eigenvalue of the…
We prove sharp upper bounds for the first and second non-trivial eigenvalues of the Neumann Laplacian in two classes of domains: parallelograms and domains of constant width. This gives in particular a new proof of an isoperimetric…
This note discusses a relation between the multiplicity m of the second eigenvalue {\lambda}2 of a Laplacian on a graph G, tight mappings of G and a discrete analogue of Courant's nodal line theorem. For a certain class of graphs, we show…
In this paper, we deal with the Steklov-Dirichlet eigenvalue problem for the Laplacian in annular domains. More precisely, we consider $\Omega_r = \Omega_0 \setminus \overline{B}_r$, where $\Omega_0 \subset \mathbb{R}^n$, $n \geq 2$, is an…
The celebrated conjecture by Payne, P\'{o}lya and Weinberger (1956) states that for the fixed membrane problem, the ratio of the first two eigenvalues, $\lambda_2/\lambda_1$, is maximized by a disk. A more general dimensional version of…
In this paper we study the first Steklov-Laplacian eigenvalue with an internal fixed spherichal obstacle. We prove that the spherical shell locally maximizes the first eigenvalue among nearly spherical sets when both the internal ball and…
In this article, we prove that for an embedded minimal hypersurface $\Sigma^{m}$ in $S^{m+1}$, the first eigenvalue $\lambda_1$ of the Laplacian operator on $\Sigma$ satisfies: $$\lambda_1> \frac{m}{2}+G(m, |A|_{\max}, |A|_{\min} ) ,$$…
One isoperimetric inequality for the fundamental sloshing eigenvalue is derived under the assumption that containers have vertical side walls and either finite or infinite depth. It asserts that among all such containers, whose free…
We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering…
We investigate the question of sharp upper bounds for the Steklov eigenvalues of a hypersurface of revolution of the Euclidean space with two boundary components isometric to two copies of $\mathbb{S}^{n-1}$. For the case of the first non…
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 $\Omega=\Omega_0 \setminus \bar{B}_{R_1}$,…