PDE and applications seminar | Nancy

Anisotropic Calderon’s inverse problem asks if the data given by voltage-to-current measurements on the boundary of a conducting domain can be used to uniquely determine the anisotropic conductivity in the interior of the domain. Geometric reformulated, this problem becomes: given a compact Riemannian manifold (M, g) with boundary, does the full knowledge of the Dirichlet-to-Neumann map (corresponding to the metric Laplacian -\Delta_g) determine the Riemannian metric g up to isometries fixing the boundary? In this talk, I will explain a positive answer at high frequencies, that is we will show that the D-t-N map of -\Delta_g – \lambda^2 for \lambda large enough determines the lens data, i.e. the exit points and directions of incoming geodesics (scattering data), together with travel times; under favourable geometric assumptions, this is known to determine g up to isometries. Joint work with S. Sahoo and G. Uhlmann.

Plateau’s problem is a notorious problem in Calculus of Variations and Geometric Measure Theory. In this presentation, I will introduce a phase-field approximation of Plateau’s problem, based on the coupling of the Ambrosio–Tortorelli energy with a geodesic distance penalization, which encodes the topological constraints. I will then justify this approach through a Γ-convergence result towards a formulation of Plateau’s problem in codimension one, and analyze the functional by establishing existence and regularity results for minimizers. From an analytical perspective, I will also present an analysis of the limit problem and provide a characterization of quasi-minimizers in terms of John domains. Finally, this approach is implemented in a numerical framework to approximate solutions of Plateau’s problem in various configurations, illustrating the efficiency and flexibility of the proposed model.

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Nous montrons avec I.Shafrir qu’une solution localement minimisante non constante de $R^3$ à  valeurs dans $R^2$ de l’équation de Ginzburg-Landau a une énergie qui croît au moins comme celle du filament de vorticité. Nous conjecturons d’ailleurs que le filament de vorticité est l’unique solution localement minimisante.

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