Related papers: A Payne-Weinberger eigenvalue estimate for wedge d…

We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a…

We exploit the connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians. First, we find a graphical relation between their smallest positive eigenvalues, which allows us to deduce a recipe for translating…

While the classical Faber-Krahn inequality shows that the ball uniquely minimizes the first Dirichlet eigenvalue of the Laplacian in the continuum, this rigidity may fail in the discrete setting. We establish quantitative fluctuation…

We establish inequalities for the eigenvalues of Schr\"{o}dinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related…

In this paper we derive Lieb-Thirring estimates for eigenvalues of Dirichlet Laplacians below the threshold of the essential spectrum on asymptotically Archimedean spiral-shaped regions.

We generalize a classical inequality between the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on bounded, planar domains: in 1955, Payne proved that below the $k$-th eigenvalue of the Dirichlet Laplacian…

We prove sharp Dirichlet eigenvalue inequalities for planar triangles. We settle a conjecture of Laugesen and Siudeja by showing that the equilateral triangle uniquely minimizes a scale-invariant functional of the first Dirichlet…

A sharp lower bound for the first Dirichlet eigenvalue of the $p$-laplacian in Gaussian space is derived for sets with prescribed generalized torsional rigidity. The result provides an extension of the classical spectral inequality due to…

In this paper, we study the first two eigenvalues of the buckling problem on spherical domains. We obtain an estimate on the second eigenvalue in terms of the first eigenvalue, which improves one recent result obtained by Wang-Xia in [7].

We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the…

We study estimates involving the principal Dirichlet eigenvalue associated to a smoothly bounded domain in a complete Riemannian manifold and L1-norms of exit time moments of Brownian motion. Our results generalize a classical inequality of…

We obtain local estimates, also called propagation of smallness or Remez-type inequalities, for analytic functions in several variables. Using Carleman estimates, we obtain a three sphere-type inequality, where the outer two spheres can be…

In [SWW], S. Seto, L. Wang and G. Wei proved that the gap between the first two Dirichlet eigenvalues of a convex domain in the unit sphere is at least as large as that for an associated operator on an interval with the same diameter,…

We prove Berezin--Li--Yau inequalities for the Dirichlet and Neumann eigenvalues on domains on the sphere $\mathbb{S}^{d-1}$. The case of $\mathbb{S}^{2}$ is treated in greater detail, including the vector Dirichlet Laplacian and the Stokes…

We obtain sharp lower bounds for the first eigenvalue of four types of eigenvalue problem defined by the bi-Laplace operator on compact manifolds with boundary and determine all the eigenvalues and the corresponding eigenfunctions of a…

We study the efficiency of the first Dirichlet eigenfunction $u$ on bounded convex domains $\Omega \subset \mathbb{R}^N$, defined as the ratio between the mean value of $u$ on $\Omega$ and its maximum value. By exploiting improved…

The equivalence between the inequalities of Babu\v{s}ka-Aziz and Friedrichs for sufficiently smooth bounded domains in the plane has been shown by Horgan and Payne 30 years ago. We prove that this equivalence, and the equality between the…

In this note we prove a version of the classical Schwarz lemma for the first eigenvalues of the Laplacian with Dirichlet boundary data. A key ingredient in our proof is an isoperimetric inequality for the first eigenfunction, due to Payne…

In this paper we obtain a Hadamard type formula for simple eigenvalues and an analog to the Rayleigh-Faber-Krahn inequality for a class of nonlocal eigenvalue problems. Such class of equations include among others, the classical nonlocal…

We prove sharp bilinear estimates for Dirichlet or Neumann eigenfunctions in domains in the plane. These are the natural analog of earlier estimates for the boundaryless case of Burq, G\'erard, and Tzvetkov.