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The Helmholtz equation $ - \nabla \cdot (a \nabla u) - \omega^2 u = f$ is considered in an unbounded wave-guide $\Omega := \mathbb{R} \times S \subset \mathbb{R}^d$, where $S \subset \mathbb{R}^{d-1}$ is a bounded domain. The coefficient $a$ is strictly elliptic and (locally) periodic in the unbounded direction $x_1\in \mathbb{R}$. For non-singular frequencies $\omega$, we show the existence of a solution $u$. While previous proofs of such results were based on operator theory, our proof uses...
We study a scalar elliptic problem in the data driven context. Our interest is to study the relaxation of a data set that consists of the union of a linear relation and single outlier. The data driven relaxation is given by the union of the linear relation and a truncated cone that connects the outlier with the linear subspace.
In periodic homogenization problems, one considers a sequence \((u^\eta)_\eta \) of solutions to periodic problems and derives a homogenized equation for an effective quantity $\hat u$. In many applications, $\hat u$ is the weak limit of $(u^\eta)_\eta$, but in some applications $\hat u$ must be defined differently. In the homogenization of Maxwell's equations in periodic media, the effective magnetic field is given by the geometric average of the two-scale limit. The notion of a geometric av...
We prove a limiting absorption principle for sesquilinear forms on Hilbert spaces and apply the abstract result to a Helmholtz equation with radiation condition. The limiting absorption principle is based on a Fredholm alternative. It is applied to Helmholtz-type equations in a truncated waveguide geometry. We analyse a problem with radiation conditions on truncated domains, recently introduced in [4]. We improve the previous results by treating the limit δ→0 .
We study time-harmonic Maxwell's equations in meta-materials that use either perfect conductors or high-contrast materials. Based on known effective equations for perfectly conducting inclusions, we calculate the transmission and reflection coefficients for four different geometries. For high-contrast materials and essentially two-dimensional geometries, we analyze parallel electric and parallel magnetic fields and discuss their potential to exhibit transmission through a sample of meta-mate...
We investigate the one-dimensional non-equilibrium Richards equation with play-type hysteresis. It is known that regularized versions of this equation permit traveling wave solutions that show oscillations and, in particular, the physically relevant effect of a saturation overshoot. We investigate here the non-regularized hysteresis operator and combine it with a positive τ-term. Our result is that the model has monotone traveling wave solutions. These traveling waves describe the behavior o...
A first order model for the transmission of waves through a sound-hard perforation along an interface is derived. Mathematically, we study the Neumann problem for the Helmholtz equation in a complex geometry, the domain contains a periodic array of inclusions of size ε > 0 along a co-dimension 1 manifold. We derive effective equations that describe the limit as ε → 0. At leading order, the Neumann sieve perforation has no effect; the corrector is given by a Helmholtz equation on the unpertur...
We investigate the one-dimensional non-equilibrium Richards equation with play-type hysteresis. It is known that regularized versions of this equation permit traveling wave solutions that show oscillations and, in particular, the physically relevant effect of a saturation overshoot. We investigate here the non-regularized hysteresis operator and combine it with a positive τ-term. Our result is that the model has monotone traveling wave solutions. These traveling waves describe the behavior o...
We study time-harmonic Maxwell's equations in meta-materials that use either perfect conductors or high-contrast materials. Based on known effective equations for perfectly conducting inclusions, we calculate the transmission and reflection coefficients for four different geometries. For high-contrast materials and essentially two-dimensional geometries, we analyze parallel electric and parallel magnetic fields and discuss their potential to exhibit transmission through a sample of meta-mate...
We investigate the long time behavior of waves in crystals. Starting from a linear wave equation on a discrete lattice with periodicity ε > 0, we derive the continuum limit equation for time scales of order ε^(-2). The effective equation is a weakly dispersive wave equation of fourth order. Initial values with bounded support result in ring-like solutions and we characterize the dispersive long-time behavior of the radial profiles with a linearized KdV equation of third order.
We present a new numerical scheme to solve the Helmholtz equation in a wave-guide. We consider a medium that is bounded in the x2-direction, unbounded in the x1-direction and ε-periodic for large |x1|, allowing different media on the left and on the right. We suggest a new numerical method that is based on a truncation of the domain and the use of Bloch wave ansatz functions in radiation boxes. We prove the existence and a stability estimate for the infinite dimensional version of the ...
We consider the energetic description of a visco-plastic evolution and derive an existence result. The energies are convex, but not necessarily quadratic. Our model is a strain gradient model in which the curl of the plastic strain contributes to the energy. Our existence results are based on a time-discretization, the limit procedure relies on Helmholtz decompositions and compensated compactness.
We study the time harmonic Maxwell equations in a meta-material consisting of perfect conductors and void space. The meta-material is assumed to be periodic with period η > 0; we study the behaviour of solutions ( E^η ,H^η ) in the limit η → 0 and derive an effective system. In geometries with a non-trivial topology, the limit system implies that certain components of the effective fields vanish. We identify the corresponding effective system and can predict, from topological propertie...
We analyze the time harmonic Maxwell's equations in a geometry containing perfectly conducting split rings. We derive the homogenization limit in which the typical size of the rings tends to zero. The split rings act as resonators and the assembly can act, effectively, as a magnetically active material. The frequency dependent effective permeability of the medium can be large and/or negative.
Meta-materials are assemblies of small components. Even though the single component consists of ordinary materials, the meta-material may behave effectively in a way that is not known from ordinary materials. In this text, we discuss some meta-materials that exhibit unusual properties in the propagation of sound or light. The phenomena are based on resonance effects in the small components. The small (sub-wavelength) components can be resonant to the wave-length of an external field if ...
We study connections between four different types of results that are concerned with vector-valued functions u : Ω→ℝ³ of class L²(Ω) on a domain Ω ⊂ ℝ³: Coercivity results in H^1(Ω) relying on div and curl, the Helmholtz decomposition, the construction of vector potentials, and the global div-curl lemma.
We analyze the propagation of waves in unbounded photonic crystals, the waves are described by a Helmholtz equation with x-dependent coefficients. The scattering problem must be completed with a radiation condition at infinity, which was not available for x-dependent coefficients. We develop an outgoing wave condition with the help of a Bloch wave expansion. Our radiation condition admits a (weak) uniqueness result, formulated in terms of the Bloch measure of solutions. We use the new r...
We derive the homogenization limit for time harmonic Maxwell's equations in a periodic geometry with periodicity length η > 0. The considered meta-material has a singular sub-structure: the permittivity coefficient in the inclusions scales like η⁻² and a part of the substructure (corresponding to wires in the related experiments) occupies only a volume fraction of order η²; the fact that the wires are connected across the periodicity cells leads to contributions in the effective system....