38 research outputs found
The wave front set of oscillatory integrals with inhomogeneous phase function
A generalized notion of oscillatory integrals that allows for inhomogeneous
phase functions of arbitrary positive order is introduced. The wave front set
of the resulting distributions is characterized in a way that generalizes the
well-known result for phase functions that are homogeneous of order one.Comment: 12 pages, published versio
Trace ideals for Fourier integral operators with non-smooth symbols II
We consider Fourier integral operators with symbols in modulation spaces and
non-smooth phase functions whose second orders of derivatives belong to certain
types of modulation space. We establish continuity and Schatten-von Neumann
properties of such operators when acting on modulation spaces.Comment: 25 page
Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian
We consider the Landau Hamiltonian (i.e. the 2D Schroedinger operator with
constant magnetic field) perturbed by an electric potential V which decays
sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian
consists of clusters of eigenvalues which accumulate to the Landau levels.
Applying a suitable version of the anti-Wick quantization, we investigate the
asymptotic distribution of the eigenvalues within a given cluster as the number
of the cluster tends to infinity. We obtain an explicit description of the
asymptotic density of the eigenvalues in terms of the Radon transform of the
perturbation potential V.Comment: 30 pages. The explicit dependence on B and V in Theorem 1.6 (i) -
(ii) indicated. Typos corrected. To appear in Communications in Mathematical
Physic
Approximation of Fourier Integral Operators by Gabor multipliers
A general principle says that the matrix of a Fourier integral operator with
respect to wave packets is concentrated near the curve of propagation. We prove
a precise version of this principle for Fourier integral operators with a
smooth phase and a symbol in the Sjoestrand class and use Gabor frames as wave
packets. The almost diagonalization of such Fourier integral operators suggests
a specific approximation by (a sum of) elementary operators, namely modified
Gabor multipliers. We derive error estimates for such approximations. The
methods are taken from time-frequency analysis.Comment: 22. page
Abstract composition laws and their modulation spaces
On classes of functions defined on R^2n we introduce abstract composition
laws modelled after the pseudodifferential product of symbols. We attach to
these composition laws modulation mappings and spaces with useful algebraic and
topological properties.Comment: 19 page
Operateurs differentiels et conjugaison par des operateurs integraux de Fourier
SIGLET 55598 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc
Sur une formule de dérivée de forme dans la théorie de Brunn-Minkowski
We extend a formula for the computation of the shape derivative of an integral cost functional with respect to a class of convex domains, using the so called support functions and gauge functions to express it. This is a priori a formula in shape optimization theory. However, the result also happens to be an extension of a well known formula from the Brunn-Minkowski theory of convex bodies.Nous généralisons une formule donnant la dérivée de forme d'une fonctionnelle coût par rapport à une classe de domaines convexes, en utilisant ce que l'on appelle fonctions support et fonctions jauge pour l'exprimer. C'est a priori une formule intervenant en optimisation de formes. Cependant, il se trouve qu'elle généralise aussi une formule bien connue dans la théorie de Brunn-Minkowski des corps convexes
On a shape derivative formula for a family of star-shaped domains
In this work, we consider again the shape derivative formula for a volume cost functional which we studied in preceding papers where we used the Minkowski deformation and the support functions in the convex setting. Here, we extend it to some non convex domains, namely the star-shaped ones. The formula also happens to be an extension of a well known formula in the Brunn-Minkowski theory. Finally, we illustrate the formula by applying it to the computation of the shape derivative for a shape optimization problem and by giving an algorithm based on the gradient method
On a numerical shape optimization approach for a class of free boundary problems
This paper is devoted to a numerical method for the approximation of a class of free boundary problems of Bernoulli's type, reformulated as optimal shape design problems with appropriate shape functionals. We show the existence of the shape derivative of the cost functional on a class of admissible domains and compute its shape derivative by using the formula proposed in [5, 6], that is, by means of support functions. On the numerical level, this allows us to avoid the tedious computations of the method based on vector fields. A gradient method combined with boundary element method are performed for the approximation of this problem, in order to overcome the re-meshing task required by the finite element method. Finally, we present some numerical results and simulations concerning practical applications, showing the effectiveness of the proposed approach