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A boundary integral equation for Calderón's inverse conductivity problem.

Kari Astala, Lassi Päivärinta (2006)

Collectanea Mathematica

Towards a constructive method to determine an L∞-conductivity from the corresponding Dirichlet to Neumann operator, we establish a Fredholm integral equation of the second kind at the boundary of a two dimensional body. We show that this equation depends directly on the measured data and has always a unique solution. This way the geometric optics solutions for the L∞-conductivity problem can be determined in a stable manner at the boundary and outside of the body.

Anisotropic inverse problems and Carleman estimates

David Dos Santos Ferreira (2007/2008)

Séminaire Équations aux dérivées partielles

This note reports on recent results on the anisotropic Calderón problem obtained in a joint work with Carlos E. Kenig, Mikko Salo and Gunther Uhlmann [8]. The approach is based on the construction of complex geometrical optics solutions to the Schrödinger equation involving phases introduced in the work [12] of Kenig, Sjöstrand and Uhlmann in the isotropic setting. We characterize those manifolds where the construction is possible, and give applications to uniqueness for the corresponding anisotropic...

Degenerate Eikonal equations with discontinuous refraction index

Pierpaolo Soravia (2006)

ESAIM: Control, Optimisation and Calculus of Variations

We study the Dirichlet boundary value problem for eikonal type equations of ray light propagation in an inhomogeneous medium with discontinuous refraction index. We prove a comparison principle that allows us to obtain existence and uniqueness of a continuous viscosity solution when the Lie algebra generated by the coefficients satisfies a Hörmander type condition. We require the refraction index to be piecewise continuous across Lipschitz hypersurfaces. The results characterize the value...

Exemples d’instabilités pour des équations d’ondes non linéaires

Guy Métivier (2002/2003)

Séminaire Bourbaki

Le but de l’exposé est de donner un guide de lecture pour un article de Gilles Lebeau où il est montré que le problème de Cauchy pour l’équation d’onde surcritique ( ∂ t 2 - Δ x ) u + u p = 0 est mal posé au sens de Hadamard dans l’espace d’énergie, pour p ≥ 7 en dimension 3. La preuve repose sur des constructions d’optique géométrique et des analyses d’instabilité dans des régimes fortement non linéaires. On donnera les étapes de l’analyse en essayant de les situer dans leur contexte plus général : construction de solutions...

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