Periodic Manifolds with Spectral Gaps

We investigate spectral properties of the Laplace operator on a class of non-compact Riemannian manifolds. For a given number $N$ we construct periodic (i.e. covering) manifolds such that the essential spectrum of the corresponding Laplacian has at least $N$ open gaps. We use two different methods. First, we construct a periodic manifold starting from an infinite number of copies of a compact manifold, connected by small cylinders. In the second construction we begin with a periodic manifold which will be conformally deformed. In both constructions, a decoupling of the different period cells is responsible for the gaps.Comment: 21 pages, 3 eps-figures, LaTe

Minimization variational principles for acoustics, elastodynamics, and electromagnetism in lossy inhomogeneous bodies at fixed frequency

The classical energy minimization principles of Dirichlet and Thompson are extended as minimization principles to acoustics, elastodynamics and electromagnetism in lossy inhomogeneous bodies at fixed frequency. This is done by building upon ideas of Cherkaev and Gibiansky, who derived minimization variational principles for quasistatics. In the absence of free current the primary electromagnetic minimization variational principles have a minimum which is the time-averaged electrical power dissipated in the body. The variational principles provide constraints on the boundary values of the fields when the moduli are known. Conversely, when the boundary values of the fields have been measured, then they provide information about the values of the moduli within the body. This should have application to electromagnetic tomography. We also derive saddle point variational principles which correspond to variational principles of Gurtin, Willis, and Borcea.Comment: 32 pages 0 figures (Previous version omitted references

Layered fractal fibers and potentials

We study spectral asymptotic properties of conductive layered-thin-fibers of invasive fractal nature. The problem is formulated as a boundary value problem for singular elliptic operators with potentials in a quasi-filling geometry for the fibers. The methods are those of variational singular homogenization and M-convergence. We prove that the spectral measures of the differential problems converge to the spectral measure of a non-trivial self-adjoint operator with fractal terms

Mini-Workshop: Nonlinear Least Squares in Shape Identification Problems

This mini-workshop brought together mathematicians engaged in shape optimization and in inverse problems in order to address a specific class of problems on the reconstruction of geometries. Such a problem can be formulated as an inverse problem forgetting the shape point of view or as minimization with respect to the shape

Extracting discontinuity using the probe and enclosure methods

This is a review article on the development of the probe and enclosure methods from past to present, focused on their central ideas together with various applications.Comment: 121 pages, minor modificatio

Computational Multiscale Methods

Almost all processes in engineering and the sciences are characterised by the complicated relation of features on a large range of nonseparable spatial and time scales. The workshop concerned the computer-aided simulation of such processes, the underlying numerical algorithms and the mathematics behind them to foresee their performance in practical applications

Computational Electromagnetism and Acoustics

The challenge inherent in the accurate and efficient numerical modeling of wave propagation phenomena is the common grand theme in both computational electromagnetics and acoustics. Many excellent contributions at this Oberwolfach workshop were devoted to this theme and a wide range of numerical techniques and algorithms were mustered to tackle this challenge