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In this paper, continuing our earlier article [CGIKO], we study qualitative properties of solutions of a certain eigenvalue optimization problem. Especially we focus on the study of the free boundary of our optimal solutions on general domains.
This paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on Q-rank 1 locally symmetric spaces and Riemannian manifolds generalizing these. This generalization is the interior of a manifold with boundary, where the boundary has the structure of a tower of fibre bundles. The class of operators… (More)
We study the surface plasmon modes of an arbitrarily shaped nanoparticle in the electrostatic limit. We first deduce an eigenvalue equation for these modes, expressed in terms of the Dirichlet-Neumann operators. We then use the properties of these pseudo-differential operators for deriving the limit of the high-order modes.
Let (Gε) ε>0 be a family of 'ε-thin' Riemannian manifolds modeled on a finite metric graph G, for example, the ε-neighborhood of an embedding of G in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the Laplace-Beltrami operator on Gε as ε → 0, for various boundary conditions. We obtain complete asymptotic… (More)
Access to security spaces and the verification of credit cards require, ideally, a simple and inexpensive system that combines accuracy with a high resistance to compromise. We have been investigating such a system. This system incorporates a novel fingerprint input arrangement that permits the use of optical pattern recognition for fingerprint verification… (More)
We show how 'test' vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for networks implies that Cheeger's constant… (More)
A thin tube is an n-dimensional space which is very thin in n−1 directions , compared to the remaining direction, for example the ε-neighborhood of a curve or an embedded graph in R n for small ε. The Laplacian on thin tubes and related operators have been studied in various contexts, with different goals but overlapping techniques. In this survey we… (More)
We discuss aspects of the L 2 –Stokes theorem on certain manifolds with singularities. We show that the L 2 –Stokes theorem does not hold on real projective varietes, even for isolated singularities. For a complex projec-tive variety of complex dimension n, with isolated singularities, we show that the Laplacians of the de Rham and Dolbeault complexes are… (More)
Let V ⊂ R N be a compact real analytic surface with isolated singularities, and assume its smooth part V0 is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on R N. We prove: 1. Each point of V has a neighborhood which is quasi-isometric (naturally and 'almost isometrically') to a union of metric cones and horns, glued… (More)
R. B. Melrose's b-calculus provides a framework for dealing with problems of partial differential equations that arise in singular or degenerate geometric situations. This article is a somewhat informal short course introducing many of the basic ideas of this world, assuming little more than a basic analysis and manifold background. As examples, classical… (More)