We consider structures of period 2 spanning a two-dimensional waveguide of width 2N . Scattering problems where Neumann conditions are imposed on the boundary of the structure and either Neumann or Dirichlet conditions are applied on the guide walls are decomposed into N +1 independent problems. The existence of at least N trapped modes is proved for the Neumann guide case and for the Dirichlet...

Lamé's formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle under Dirichlet and Neumann boundary conditions are herein extended to the Robin boundary condition. They are shown to form a complete orthonormal system. Various properties of the spectrum and modal functions are explored. 1. Introduction. The eigenstructure of the Laplacian on an equilateral tri...

We derive the radiation pressure force on a non-relativistic moving plate in 1+1 dimensions. We assume that a massless scalar field satisfies either Dirichlet or Neumann boundary conditions (BC) at the instantaneous position of the plate. We show that when the state of the field is invariant under time translations, the results derived for Dirichlet and Neumann BC are equal. We discuss the forc...

An extension of the lower-bound lemma of Boggio is given for the weak forms of certain elliptic operators, which have partially Dirichlet and partially Neumann boundary conditions, and are in general nonlinear. Its consequences and those of an adapted Hardy inequality for the location of the bottom of the spectrum are explored in corollaries wherein a variety of assumptions are placed on the sh...

We consider the method of particular solutions for numerically computing eigenvalues and eigenfunctions of the Laplacian on a smooth, bounded domain Ω in Rn with either Dirichlet or Neumann boundary conditions. This method constructs approximate eigenvalues E, and approximate eigenfunctions u that satisfy ∆u = Eu in Ω, but not the exact boundary condition. An inclusion bound is then an estimate...

WKB p-forms are constructed as approximate solutions to boundary value problems associated with semi-classical Witten Laplacians. Naturally distorted Neumann or Dirichlet boundary conditions are considered. MSC 2000: 58J37 (58J10 58J32 81Q20)

Questions of flux regulation in biological cells raise a renewed interest in the narrow escape problem. The determination of a higher order asymptotic expansion of the narrow escape time depends on determining the singularity behavior of the Neumann Green's function for the Laplacian in a three-dimensional (3D) domain with a Dirac mass on the boundary. In addition to the usual 3D Coulomb singul...

In this paper, we propose a domain embedding method associated with an optimal boundary control problem with boundary observations to solve elliptic problems. We prove that the optimal boundary control problem has a unique solution if the controls are taken in a finite dimensional subspace of the space of the boundary conditions on the auxiliary domain. Using a controllability theorem due to J....

We construct a series of examples of planar isospectral domains with mixed Dirichlet-Neumann boundary conditions. This is a modification of a classical problem proposed by M. Kac.