Related papers: The Payne conjecture for Dirichlet and Buckling ei…

We prove a lower bound on the eigenvalues $\lambda_k$, $k\in\mathbb{N}$, of the Dirichlet Laplacian of a bounded domain $\Omega\subset\mathbb{R}^n$ of volume $V$: $$ \lambda_k \geq C_n\bigg( \delta\frac{k}{V}\bigg)^{2/n} $$ where $\delta$…

A plate is rigid if its admissible displacement fields inducing vanishing two-dimensional strain tensors must vanish. We prove that the nonlinear model of Kirchhoff-Love for such a plate has a solution for any applied forces and boundary…

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. For any $\epsilon\in (0,1)$ we show that for any Dirichlet eigenvalue $\lambda_k(\Omega)>\Lambda(\epsilon,\Omega)$, it holds \begin{align*} k&\le…

Let $(M^{n}, g)$ be a closed connected Einstein space, $n=dim M ,$ and $\kappa_{0} $ be the lower bound of the sectional curvature. In this paper, we prove Udo Simon's conjecture: on closed Einstein spaces, $n\geq 3,$ there is no eigenvalue…

Among all $C^{\infty}$ bounded domains with equal volume, we show that the second eigenvalue of the Robin plate is uniquely maximized by an open ball, so long as the Robin parameter lies within a particular range of negative values. Our…

The Dirichlet eigenvalues of the Laplacian on a triangle that collapses into a line segment diverge to infinity. In this paper, to track the behavior of the eigenvalues during the collapsing process of a triangle, we establish a…

In this paper, we use a weighted isoperimetric inequality to give a lower bound on the first Dirichlet eigenvalue of the Laplacian on a bounded domain inside a Euclidean cone. Our bound is sharp, in that only sectors realize it. This result…

Let $M$ and $N$ be two closed hyperbolic Riemann surfaces. The Ehrenpreis Conjecture (proved by Kahn-Markovic) asserts that for any $\epsilon>0$ there are finite covers $M_\epsilon \to M$, and $N_\epsilon \to N$, such that the Teichmuller…

It has been known since Erdos that the sum of $1/(n\log n)$ over numbers $n$ with exactly $k$ prime factors (with repetition) is bounded as $k$ varies. We prove that as $k$ tends to infinity, this sum tends to 1. Banks and Martin have…

We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold $P$ of $CP^n$ which depends only on the radius of the tube, the degrees of the polynomials defining $P$ and the first eigenvalue of some model…

A plasmon of a bounded domain $\Omega\subset\mathbb{R}^n$ is a non-trivial bounded harmonic function on $\mathbb{R}^n\setminus\partial\Omega$ which is continuous at $\partial\Omega$ and whose exterior and interior normal derivatives at…

The probability that a random permutation in $S_n$ is a derangement is well known to be $\displaystyle\sum\limits_{j=0}^n (-1)^j \frac{1}{j!}$. In this paper, we consider the conditional probability that the $(k+1)^{st}$ point is fixed,…

We provide a direct proof of Weyl's law for the buckling eigenvalues of the biharmonic operator on a wide class of domains of $\mathbb R^d$ including bounded Lipschitz domains. The proof relies on asymptotically sharp lower and upper bounds…

It is pointed out that the entropy of a membrane which is quantized perturbatively around a background position of fixed radius in a black hole spacetime is equal to the Bekenstein-Hawking entropy, if 1) the membrane surface is the horizon…

We discuss connections between certain well-known open problems related to the uniform measure on a high-dimensional convex body. In particular, we show that the "thin shell conjecture" implies the "hyperplane conjecture". This extends a…

We study the Dirichlet problem for first order hyperbolic quasi-linear functional PDEs on a simply connected bounded domain of $\R^2$, where the domain has an interior outflow set and a mere inflow boundary. While the question of existence…

We consider the problem of minimising the $k$th eigenvalue, $k \geq 2$, of the ($p$-)Laplacian with Robin boundary conditions with respect to all domains in $\mathbb{R}^N$ of given volume $M$. When $k=2$, we prove that the second eigenvalue…

We prove that the isoperimetric inequality due to Hersch-Payne-Schiffer for the n-th nonzero Steklov eigenvalue of a bounded simply-connected planar domain is sharp for all n=1,2,... The equality is attained in the limit by a sequence of…

It is natural to consider continuous dependence of the $n$-th eigenvalue on $d$-dimensional ($d\geq2$) Sturm-Liouville problems after the results on $1$-dimensional case by Kong, Wu and Zettl [14]. In this paper, we find all the boundary…

We analyse the buckling stability of a thin, viscous sheet when subject to simple shear, providing conditions for the onset of the dominant out-of-plane modes using two models: (i) an asymptotic theory for the dynamics of a viscous plate…