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We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance…

In this paper, we considered the spectrum of the Dirichlet Laplacian $\Delta_\epsilon$ on $\Omega_\epsilon=\{(x,y): -l_1<x<l_2, 0<y<\epsilon h(x)]\}$ where $ l_1,l_2>0$ and $h(x)$ is a positive analytic function having $0$ the only point…

This paper is concerned with eigenvalue problems for non-symmetric elliptic operators with large drifts in bounded domains under Dirichlet boundary conditions. We consider the minimal principal eigenvalue and the related principal…

We consider the Dirichlet Laplacian in a domain two three-dimensional parallel layers having common boundary and coupled by a window. The window produces the bound states below the essential spectrum; we obtain two-sided estimates for them.…

On the unit tangent bundle of a compact Riemannian surface, we consider a natural sub-Riemannian Laplacian associated with the canonical contact structure. In the large eigenvalue limit, we study the escape of mass at infinity in the…

Many of the interesting patterns seen in recent multi-frequency Faraday experiments can be understood on the basis of three-wave interactions (resonant triads). In this paper we consider two-frequency forcing and focus on a resonant triad…

We present an analysis of wave propagation and reflection in an acoustic waveguide with random sound soft boundary and a turning point. The waveguide has slowly bending axis and variable cross section. The variation consists of a slow and…

We consider small perturbations of the Laplace operator in a multi-dimensional cylindrical domain by second order differential operators with periodic coefficients. We show that under certain non-degeneracy conditions such perturbations can…

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 analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in…

We investigate the influence of an electric field on trapped modes arising in a two-dimensional curved quantum waveguide ${\bf \Omega}$ i.e. bound states of the corresponding Laplace operator $-\Delta\_{{\bf \Omega}}$. Here the curvature of…

The considered Robin problem can formally be seen as a small perturbation of a Dirichlet problem. However, due to the sign of the impedance value, its associated eigenvalues converge point-wise to $-\infty$ as the perturbation goes to zero.…

Given an Euclidean domain with very mild regularity properties, we prove that there exist perturbations of the Dirichlet Laplacian of the form $-(I+S_\epsilon)\Delta$ with $\|S_\epsilon\|_{L^2\to L^2}\leq \epsilon$ whose high energy…

We consider the eigenvalues of an elliptic operator for systems with bounded, measurable, and symmetric coefficients. We assume we have two non-empty, open, disjoint, and bounded sets and add a set of small measure to form the perturbed…

We study the spectrum of the Robin Laplacian with a complex Robin parameter $\alpha$ on a bounded Lipschitz domain $\Omega$. We start by establishing a number of properties of the corresponding operator, such as generation properties, local…

We establish accurate eigenvalue asymptotics and, as a by-product, sharp estimates of the splitting between two consecutive eigenvalues, for the Dirichlet magnetic Laplacian with a non-uniform magnetic field having a jump discontinuity…

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double…

The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is…

We consider the Robin Laplacian in the domains $\Omega$ and $\Omega^\varepsilon$, $\varepsilon >0$, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient $a$ is large enough, the spectrum of the problem in $\Omega$…

The aim of this paper is investigating the existence of solutions of some semilinear elliptic problems on open bounded domains when the nonlinearity is subcritical and asymptotically linear at infinity and there is a perturbation term which…