Related papers: A note on the first eigenvalue of spherically symm…
We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell-Laplace operator of a domain $\Omega$, involving only geometrical informations. We provide such an upper bound, by generalizing Brock's…
We study compactly free boundary minimal submanifolds in spherical caps $\Br$ and their geometric spectral properties. Following the foundational work of Fraser-Schoen \cite{FS2012}, Lima-Menezes \cite{LM23} established the connection…
In constant curvatures spaces, there are a lot of characterizations of geodesic balls as optimal domain for shape optimization problems. Although it is natural to expect similar characterizations in rank one symmetric spaces, very few is…
On a family of arithmetic hyperbolic 3-manifolds of squarefree level, we prove an upper bound for the sup-norm of Hecke-Maass cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the…
We obtain upper bounds for the Steklov eigenvalues $\sigma_k(M)$ of a smooth, compact, connected, $n$-dimensional submanifold $M$ of Euclidean space with boundary $\Sigma$ that involve the intersection indices of $M$ and of $\Sigma$. One of…
In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or…
We discuss a sharp lower bound for the first positive eigenvalue of the sublaplacian on a closed, strictly pseudoconvex pseudo-hermitian manifold of dimension $2m+1\geq 5$. We prove that the equality holds iff the manifold is equivalent to…
We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter…
A spherical conical metric $g$ on a surface $\Sigma$ is a metric of constant curvature $1$ with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone…
In this paper, we study the Gauss map of a free boundary minimal surface. The main theorem asserts that if components of the Gauss map are eigenfunctions of the Jacobi-Steklov operator, then the surface must be rotationally symmetric.
This work is an extension of a result given by Kuttler and Sigillito (SIAM Rev $10$:$368-370$, $1968$) on a star-shaped bounded domain in $\mathbb{R}^2$. Let $\Omega$ be a star-shaped bounded domain in a hypersurface of revolution, having…
We prove explicit upper and lower bounds for the Poisson hierarchy, the averaged $L^1$-moment spectra $\{\dfrac{\mathcal{A}_k\left(B_R^M\right)}{\text{vol}\left(S_R^M\right)}\}_{k=1}^\infty$, and the torsional rigidity…
In this paper we exhibit deformations of the hemisphere $S^{n+1}_+$, $n\geq 2$, for which the ambient Ricci curvature lower bound $\text{Ric}\geq n $ and the minimality of the boundary are preserved, but the first Laplace eigenvalue of the…
For the biharmonic Steklov eigenvalue problem considered in this paper, we show that among all bounded Euclidean domains of class $C^{1}$ with fixed measure, the ball maximizes the first positive eigenvalue.
The paper is devoted to the study of fine properties of the first eigenvalue on negatively curved spaces. First, depending on the parity of the space dimension, we provide asymptotically sharp harmonic-type expansions of the first…
We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries perturb in some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner…
In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few years due to the seminal work of Fraser and Schoen on the…
In this paper we express the eigenvalues of a sort of real heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From these prescribed eigenvalues we compute also…
Let $\om $ be a bounded domain in an $n$-dimensional Euclidean space $\Bbb R^n$. We study eigenvalues of an eigenvalue problem of a system of elliptic equations: $$ \{\aligned &\Delta {\mathbf u}+ \alpha{\rm grad}(\text{div}{\mathbf…
We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero…