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Comparing Neumann and Dirichlet eigenvalues of the Laplacian on a bounded domain $\Omega\subseteq\Rbb^n$ is a topic that goes back at least to the work of P\'olya \cite{polya}. We study the effect of the isoperimetric ratio of $\Omega$ on…

We give a new proof of an isoperimetric inequality for a family of closed surfaces, which have Gaussian curvature identically equal to one wherever the surface is smooth. These surfaces are formed from a convex, spherical polygon, with each…

We study the log-concavity of the first Dirichlet eigenfunction of the Laplacian for convex domains. For positively curved surfaces satisfying a condition involving the curvature and its second derivatives, we show that the first…

We prove a theorem that can be thought of as a common generalization of the Discrete Nodal Theorem and (one direction of) Cheeger's Inequality for graphs. special case of this result will assert that if the second and third eigenvalues of…

In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison…

Let $H(\Om_0)=-\Delta+V$ be a Schr\"odinger operator on a bounded domain $\Om_0\subset \mathbb R^d$ with Dirichlet boundary conditions. Suppose that the $\Om_\ell$ ($\ell \in \{1,...,k\}$) are some pairwise disjoint subsets of $\Om_0$ and…

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^d$. Under certain conditions on the coefficients of $L$, we first establish the existence of a unique Green's…

In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general…

This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet…

We consider the Laplace operator with Dirichlet boundary conditions on a domain in R^d and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling…

We provide quantitative improvements to the Berezin-Li-Yau inequality and the Kr\"oger inequality, in $\mathbb{R}^n$, $n\ge 2$. The improvement on Kr\"oger's inequality resolves an open question raised by Weidl from 2006. The improvements…

We prove the existence of classical solutions to the Dirichlet problem for a class of fully nonlinear elliptic equations of curvature type on Riemannian manifolds. We also derive new second derivative boundary estimates which allows us to…

This paper is concerned with the location of nodal sets of eigenfunctions of the Dirichlet Laplacian in thin tubular neighbourhoods of hypersurfaces of the Euclidean space of arbitrary dimension. In the limit when the radius of the…

In this paper, we prove that the Extended Courant Property fails to be true for certain smooth, strictly convex domains with Neumann boundary condition: there exists a linear combination of a second and a first Neumann eigenfunctions, with…

We consider the problem of minimising the $k$-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are…

This paper is concerned with the Dirichlet eigenvalue problem for Laplace operator in a bounded domain with periodic perforation in the case of small volume. We obtain the optimal quantitative error estimates independent of the spectral…

In this paper, we continue the study of eigenfunctions on triangles initiated by the first author in \cite{Chr-tri} and \cite{Chr-simp}. The Neumann data of Dirichlet eigenfunctions on triangles enjoys an equidistribution law, being…

In this paper we prove the propagation of singularities for the wave equation on differential forms with natural (i.e. relative or absolute) boundary conditions on Lorentzian manifolds with corners, which in particular includes a…

We establish integral formulas and sharp two-sided bounds for the Ricci curvature, mean curvature and second fundamental form on a Riemannian manifold with boundary. As applications, sharp gradient and Hessian estimates are derived for the…

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…