Numerical methods for scattering problems by curved gratings at high frequencies

A new numerical method is derived for the calculation of high-frequency asymptotic expansions of the scalar wave scattered by curved periodic structures. Optimal error estimates for this method are established. © 1997 B. G. Teubner Stuttgart-John Wiley & Sons Ltd.     

Dissipative Maxwell's equations at low frequencies

A study of the low-frequency behaviour of solutions to dissipative Maxwell's equations with discontinuous coefficients is given.     

Quantum scattering at low energies

For a class of negative slowly decaying potentials, including V(x):=−γ|x|^−μ with 0<μ<2, we study the quantum mechanical scattering theory in the low-energy regime. Using appropriate modifiers of the Isozaki-Kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the Hamiltonian, including the energy 0. We show that the modified scattering matrices S(λ) are well-defined and strongly continuous down to the zero energy threshold. Similarly, we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces. These results are used to derive (oscillatory) asymptotics of the standard short-range and Dollard type S-matrices for the subclasses of potentials where both kinds of S-matrices are defined. For potentials whose leading part is −γ|x|^−μ we show that the location of singularities of the kernel of S(λ) experiences an abrupt change from passing from positive energies λ to the limiting energy λ=0. This change corresponds to the behaviour of the classical orbits. Under stronger conditions one can extract the leading term of the asymptotics of the kernel of S(λ) at its singularities.     

Landscape of wave focusing and localization at low frequencies

High-contrast scattering problems are special among classical wave systems as they allow for strong wave focusing and localization at low frequencies. We use an asymptotic framework to develop a landscape theory for high-contrast systems that resonate in a subwavelength regime. Our from-first-principles asymptotic analysis yields a characterization in terms of the generalized capacitance matrix, giving a discrete approximation of the three-dimensional scattering problem. We develop landscape theory for the generalized capacitance matrix and use it to predict the positions of three-dimensional wave focusing and localization in random and non-periodic systems of subwavelength resonators.     

Nonlinear scattering theory at low energy: Sequel

We extend the nonlinear scattering theory and apply it to Schrödinger and Klein-Gordon equations with various kinds of nonlinear terms, including some nonlocal ones.     

On the scattering frequencies of time-dependent potentials

This paper discusses the scattering frequencies associated with the scalar wave equation and a time-periodic, real, potential. It is shown that the scattering frequencies form a discrete set in the complex plane and depend continuously on the potential. Existence of the scattering frequencies is proved for periodic potentials which are perturbations of a time independent, nonnegative, potential.     

Splitting of resonant and scattering frequencies under shape deformation

It is well known that the main difficulty in solving eigenvalue problems under shape deformation relates to the continuation of multiple eigenvalues of the unperturbed configuration. These eigenvalues may evolve, under shape deformation, as separated, distinct eigenvalues, and the splitting may only become apparent at high orders in their Taylor expansion. In this paper, we address the splitting problem in the evaluation of resonant and scattering frequencies of the two-dimensional Laplacian operator under boundary variations of the domain. By using surface potentials we show that the eigenvalues are the characteristic values of meromorphic operator-valued functions that are of Fredholm type with index 0. We then proceed from the generalized Rouché's theorem to investigate the splitting problem.     

Direct and inverse multichannel scattering problems

We consider small oscillations of a system of pairwise interacting particles in an external field near a stable equilibrium. The system is assumed to consist of finitely many channels, i.e., semi-infinite linear chains of particles, attached to a scatterer, which is a finite system of interacting particles. Direct and inverse scattering problems are considered. In particular, an algorithm finding characteristics of the channels on the basis of scattering data is given.     

On the scattering frequencies of the laplace operator for exterior domains

In this paper we show that the so-called scattering frequencies of the Laplace operator over an exterior domain, subject to Robin or Dirichlet boundary condition, cannot lie in certain portions of the upper half-plane. The excluded sets depend only on the type of boundary condition and the radius of the smallest sphere containing the scattering obstacle.     

Decay rates at low and high frequencies for a plate equation with feedback concentrated in interior curves

In this paper we study the stabilization of plate vibrations by means of piezoelectric actuators. In this situation the geometric control condition of Bardos, Lebeau and Rauch [6] is not satisfied. We prove that we have exponential stability for the low frequencies but not for the high frequencies. We give an explicit decay rate for regular initial data at high frequencies while clarifying the behavior of the constant which intervenes in this estimation there function of the frequency of cut n. The method used is based on some trace regularity which reduces stability to some observability inequalities for the corresponding undamped problem. Moreover, we show numerically at low frequencies, that the optimal location of the actuator is the center of the domain Ω. Research supported by the RIP program of Oberwolfach Institut and by the Tunisian Ministry for Scientific Research and Technology (MRST) under Grant 02/UR/15-01. Research supported by the RIP program of Oberwolfach Institut. (Received: September 17, 2003; revised: February 26, 2004)     

Decay rates at low and high frequencies for a plate equation with feedback concentrated in interior curves

In this paper we study the stabilization of plate vibrations by means of piezoelectric actuators. In this situation the geometric control condition of Bardos, Lebeau and Rauch [6] is not satisfied. We prove that we have exponential stability for the low frequencies but not for the high frequencies. We give an explicit decay rate for regular initial data at high frequencies while clarifying the behavior of the constant which intervenes in this estimation there function of the frequency of cut n. The method used is based on some trace regularity which reduces stability to some observability inequalities for the corresponding undamped problem. Moreover, we show numerically at low frequencies, that the optimal location of the actuator is the center of the domain Ω.     

Constrained optimization problems in Banach space

This article is devoted to constrained minimization problems in Banach space. A necessary condition is derived and is compared with other similar results. In the process, correlation of some of the existing results is established.     

Variational problems in space flight mechanics

The effect of weight limitations on the parameters of the optimum motion of a variable-mass body in a gravitational field is considered. An analysis is made of the optimum motion for three possible propulsion systems: limited exit velocity, limited power, and limited thrust. Solutions are presented for some typical cases. The problems of selecting the optimum weight parameters, finding the optimum propulsion system controls, and determining the optimum trajectory are solved simultaneously.