Related papers: Estimates on the first two buckling eigenvalues on…

This paper is concerned with an optimisation problem of Robin Laplacian eigenvalue with respect to an indefinite weight, which is formulated as a shape optimisation problem thanks to the known bang-bang distribution of the optimal weight…

Sharp upper bounds for the first eigenvalue of the Laplacian on a surface of a fixed area are known only in genera zero and one. We investigate the genus two case and conjecture that the first eigenvalue is maximized on a singular surface…

This paper presents some new results on the eigenvalues of the spheroidal wave equation. We study the angular and Coulomb spheroidal wave equation as a special case of a more general linear Hamiltonian system depending on three parameters.…

We obtain sharp upper bounds for the first two nonzero Steklov eigenvalues among bounded domains in Euclidean spaces of dimension $d \geq 7$ under a natural normalization involving volume and boundary measure. These bounds are derived from…

We give a new lower bound for the first gap $\lambda_2 - \lambda_1$ of the Dirichlet eigenvalues of the Schr{\"o}dinger operator on a bounded convex domain $\Omega$ in R$^n$ or S$^n$ and greatly sharpens the previous estimates. The new…

We compute estimates for eigenvalues of a class of linear second-order elliptic differential operators in divergence form (with Dirichlet boundary condition) on a bounded domain in a complete Riemannian manifold. Our estimates are based…

We prove the long-standing Payne conjecture that the $k^{\text{th}}$ eigenvalue in the buckling problem for a clamped plate is not less than the ${k+1}^{\text{st}}$ eigenvalue for the membrane of the same shape which is fixed on the…

In this paper we consider in a bounded domain $\Omega \subset \mathbb{R}^N$ with smooth boundary an eigenvalue problem for the negative $(p,q)$-Laplacian with a Steklov type boundary condition, where $p\in (1,\infty)$, $q\in (2,\infty)$ and…

We give simple new proofs of two well-known results for the Schr\"odinger operator: first, the Brunn--Minkowski inequality for Dirichlet eigenvalues and, second, the log-concavity of the first Dirichlet eigenfunction. Our proof of the first…

We theoretically explain the complete sequence of shapes of deflated spherical shells. Decreasing the volume, the shell remains spherical initially, then undergoes the classical buckling instability, where an axisymmetric dimple appears,…

We prove P\'olya's conjecture for the eigenvalues of the Dirichlet Laplacian on annular domains. Our approach builds upon and extends the methods we previously developed for disks and balls. It combines variational bounds, estimates of…

We consider Steklov eigenvalues of nearly hyperspherical domains in $\mathbb{R}^{d + 1}$ with $d\ge 3$. In previous work, treating such domains as perturbations of the ball, we proved that the Steklov eigenvalues are analytic functions of…

Expository article on the problem of determining the maximum number of equiangular lines with a fixed angle, and the associated problem of second eigenvalue multiplicity in graphs.

This is a continuation of our previous work arXiv:1601.05617 on trace and inverse trace of Steklov eigenvalues. More new inequalities for the trace and inverse trace of Steklov eigenvalues are obtained.

The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension d greater or equal to 3. Apart…

We obtain the simplicity of the first Neumann eigenvalue of convex thin domain with boundary in $R^n$ and compact thin manifolds with non-negative Ricci curvature. For convex thin domain in $R^2$, we get the simplicity of the first k…

We consider Steklov eigenvalues of three-dimensional, nearly-spherical domains. In previous work, we have shown that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term…

We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity three. We also analyze…

We discuss stability of the first eigenvalue of the 1-Laplacian under perturbations of the domain.

The author extends previous results to general classes of equations under weaker assumptions obtained in 2016 by Bao, Dong and Jiao concerning the study of the regularity of solutions for the first initial-boundary value problem for…