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Elastic wave equation

Yves Colin de Verdière (2006/2007)

Séminaire de théorie spectrale et géométrie

The goal of this talk is to describe the Lamé operator which drives the propagation of linear elastic waves. The main motivation for me is the work I have done in collaboration with Michel Campillo’s group from LGIT (Grenoble) on passive imaging in seismology. From this work, several mathematical problems emerged: equipartition of energy between S - and P - waves, high frequency description of surface waves in a stratified medium and related inverse spectral problems.We discuss the following topics:What...

Enhanced electrical impedance tomography via the Mumford–Shah functional

Luca Rondi, Fadil Santosa (2001)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the problem of electrical impedance tomography where conductivity distribution in a domain is to be reconstructed from boundary measurements of voltage and currents. It is well-known that this problem is highly illposed. In this work, we propose the use of the Mumford–Shah functional, developed for segmentation and denoising of images, as a regularization. After establishing existence properties of the resulting variational problem, we proceed by demonstrating the approach in several...

Enhanced Electrical Impedance Tomography via the Mumford–Shah Functional

Luca Rondi, Fadil Santosa (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the problem of electrical impedance tomography where conductivity distribution in a domain is to be reconstructed from boundary measurements of voltage and currents. It is well-known that this problem is highly illposed. In this work, we propose the use of the Mumford–Shah functional, developed for segmentation and denoising of images, as a regularization. After establishing existence properties of the resulting variational problem, we proceed by demonstrating the approach in several...

Équations d'évolution non linéaires : solutions bornées et périodiques

Alain Haraux (1978)

Annales de l'institut Fourier

Soit ∂ φ un sous-différentiel (non coercif) dans un espace de Hilbert.On étudie l’existence de solutions bornées ou périodiques pour l’équation d u d t + ∂ φ ( u ( t ) ) ∋ f ( t ) , t ≥ 0 . Deux solutions périodiques éventuelles diffèrent d’une constante. Si f est périodique et ( I ˙ + ∂ φ ) - 1 compact, toute trajectoire bornée est asymptote pour t → + ∞ à une trajectoire périodique.

Error estimates for distributed parameter identification in parabolic problems with output least squares and Crank-Nicolson method

Tommi Kärkkäinen (1997)

Applications of Mathematics

The identification problem of a functional coefficient in a parabolic equation is considered. For this purpose an output least squares method is introduced, and estimates of the rate of convergence for the Crank-Nicolson time discretization scheme are proved, the equation being approximated with the finite element Galerkin method with respect to space variables.

Estimates in the Hardy-Sobolev space of the annulus and stability result

Imed Feki (2013)

Czechoslovak Mathematical Journal

The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space H k , ∞ ; k ∈ ℕ * of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L. Baratchart and M. Zerner, On the recovery of functions from pointwise boundary values in a Hardy-Sobolev class of the disk, J. Comput. Appl. Math. 46 (1993), 255–269 and by S. Chaabane and I. Feki, Optimal logarithmic estimates in Hardy-Sobolev spaces...

Exact boundary observability for quasilinear hyperbolic systems

Tatsien Li (2008)

ESAIM: Control, Optimisation and Calculus of Variations

By means of a direct and constructive method based on the theory of semi-global C1 solution, the local exact boundary observability is established for one-dimensional first order quasilinear hyperbolic systems with general nonlinear boundary conditions. An implicit duality between the exact boundary controllability and the exact boundary observability is then shown in the quasilinear case.

Flaw identification in elastic solids: theory and experiments.

A. Gesualdo, F. Guarracino, V. Mallardo, V. Minutolo, L. Nunziante (1997)

Extracta Mathematicae

In this work the problem of identificating flaws or voids in elastic solids is addressed both from a theoretical and an experimental point of view. Following a so called inverse procedure, which is based on appropriately devised experiments and a particular bounding of the strain energy, a gap functional for flaw identification is proposed.

Fourier diffraction theorem for the tensor fields

Alexander Leonidovich Balandin (2023)

Applications of Mathematics

The paper is devoted to the electromagnetic inverse scattering problem for a dielectric anisotropic and magnetically isotropic media. The properties of an anisotropic medium with respect to electromagnetic waves are defined by the tensors, which give the relation between the inductions and the fields. The tensor Fourier diffraction theorem derived in the paper can be considered a useful tool for studying tensor fields in inverse problems of electromagnetic scattering. The method is based on the...

Generalized Backscattering and the Lax-Phillips Transform

Melrose, Richard, Uhlmann, Gunther (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 35P25, 35R30, 58J50.Using the free-space translation representation (modified Radon transform) of Lax and Phillips in odd dimensions, it is shown that the generalized backscattering transform (so outgoing angle w = Sq in terms of the incoming angle with S orthogonal and Id-S invertible) may be further restricted to give an entire, globally Fredholm, operator on appropriate Sobolev spaces of potentials with compact support. As a corollary we show that the...

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