ao -d yn /9 40 50 01 2 M ay 1 99 4 Spectral Duality for Planar Billiards J.-P. Eckmann1;2 and C.-A. Pillet1 Dépt. de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland Section de Mathématiques, Université de Genève, CH-1211 Genève 4, Switzerland Abstract. For a bounded open domain with connected complement in R2 and piecewise smooth boundary, we consider the Dirichlet Lapla...

We review some recent results on eigenvalues of fractional Laplacians and fractional Schrödinger operators. We discuss, in particular, Lieb–Thirring inequalities and their generalizations, as well as semi-classical asymptotics.

Let G be a graph on n vertices with Laplacian matrix L and let b be a binary vector of length n. The pair (L,b) is controllable if the smallest L-invariant subspace containing b is of dimension n. The graph G is called essentially controllable if (L,b) is controllable for every b / ∈ ker(L), completely uncontrollable if (L,b) is uncontrollable for every b, and conditionally controllable if it i...

In this paper we find out the evolution formula for first nonzero eigenvalue of weighted p-Laplacian operator acting on space functions under Cotton flow a closed Riemannian 3-manifold M3.

This is a continuation of the paper [3]. We consider the Dirichlet Laplacian in a family of unbounded domains {x ∈ R, 0 < y < h(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We show that the number of eigenvalues lying below the essential spectrum indefinitely grows as → 0, and find the two-term asymptotics in → 0 of each e...

Let Λ ⊂ R be a domain consisting of several cylinders attached to a bounded center. One says that Λ admits a threshold resonance if there exists a nontrivial bounded function u solving −∆u = νu in Λ and vanishing at the boundary, where ν is the bottom of the essential spectrum of the Dirichlet Laplacian in Λ. We derive a sufficient condition for the absence of threshold resonances in terms of t...

This is a continuation of the paper [3]. We consider the Dirichlet Laplacian in a family of unbounded domains {x ∈ R, 0 < y < h(x)}. The main assumption is that x = 0 is the only point of global maximum of the positive, continuous function h(x). We show that the number of eigenvalues lying below the essential spectrum indefinitely grows as → 0, and find the twoterm asymptotics in → 0 of each ei...

where ⊂Rn (n≥ 2) is a bounded smooth domain, s ∈ (0, 1), (– )s denotes the fractional Laplacian, λ is a real parameter, the nonlinear term f satisfies superlinear and subcritical growth conditions at zero and at infinity. When λ≤ 0, we prove the existence of a positive solution, a negative solution and a sign-changing solution by combing minimax method with invariant sets of descending flow. Wh...

We are interested in finite cones of fixed height 1 parametrized by their opening angle. We study the eigenpairs of the Dirichlet Laplacian in such domains when their apertures tend to 0. We provide multi-scale asymptotics for eigenpairs associated with the lowest eigenvalues of each fiber of the Dirichlet Laplacian. In order to do this, we investigate the family of their one-dimensional Born-O...

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 Euclidean space. This inequality controls the kth eigenvalue by the lower eigenvalues, independently of the particular geometry of the domain. Our inequality is shar...