Ambrosio-Tortorelli approach to topological singularities

The exploration of topological singularities is a fascinating task of absolute relevance both from the theoretical and applied point of view.

For example, in physics and materials science they arise from the study of mathematical models for vortices in superconductors, grain boundaries in polycrystals, fractures in solids, and defects in crystals such as disclinations or dislocations.

Furthermore, topological singularities play an important role in the study of more geometric problems such as the Plateau problem and the theory of minimal surfaces.

The goal of TopSing is to study some physical/mechanical problems where singularities appear, through a theoretical approach that opens promising directions also for other, apparently unrelated, situations like the non-parametric Plateau problem in higher codimension.

More specifically, we draw our attention to codimension-two singularities and consider two-dimensional models for fields having point singularities which are relevant in the study of two main problems: 1) Screw Dislocations in crystals and their relation with vortices in superconductors; 2) The non-parametric Plateau problem in codimension-two.The main novelty consists in developing a unified approach, inspired by the classical model by Ambrosio and Tortorelli (AT), that allows to study topological singularities in both contexts listed above.

Furthermore this will provide a model which is easier to handle numerically and thus interesting from the point of view of applications.The project is organised into four main objectives whose common thread is the asymptotic analysis of elliptic functionals á la AT for maps taking values on the unit circle.

To our best knowledge there are no similar results in the literature.

This is due to the non trivial task of constructing a recovery sequence that takes values on the circle, which we aim at achieving by relying on degree theory and by using techniques developed to study the relaxed area.