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Given an image u_0, the aim of minimising the Mumford-Shah functional is to find a decomposition of the image domain into sub-domains and a piecewise smooth approximation u of u_0 such that u varies smoothly within each sub-domain. Since the Mumford-Shah functional is highly non- smooth, regularizations such as the Ambrosio-Tortorelli approximation can be considered, which is one of the most computationally efficient approximations of the Mumford-Shah functional for image segmentation. While very impressive numerical results have been achieved in a large range of applications when minimising the functional, no analytical results are currently available for minimizers of the functional in the piece- wise smooth setting, and this is the goal of this work. Our main result is the Γ-convergence of the Ambrosio-Tortorelli approximation of the Mumford-Shah functional for piecewise smooth approximations. This requires the introduction of an appropriate function space. As a consequence of our Gamma-convergence result, we can infer the convergence of minimizers of the respective functionals. 

Abstract In this paper, a model for defects in nematic liquid crystals that was introduced in Zhang et al. (Physica D Nonlinear Phenom 417:132828, 2021) is studied. In the literature, the setting of many models for defects is the function space SBV (special functions of bounded variation). However, the model considered herein regularizes the director field to be in a Sobolev space by introducing a second vector field tracking the defect. A relaxation result in the case of fixed parameters is proved along with some partial compactness results as the defect width vanishes. 

This article offers various mathematical contributions to the behavior of thin films. The common thread is to view thin film behavior as the variational limit of a three-dimensional domain with a related behavior when the thickness of that domain vanishes. After a short review in Section 1 of the various regimes that can arise when such an asymptotic process is performed in the classical elastic case, giving rise to various well-known models in plate theory (membrane, bending, Von Karmann, etc…), the other sections address various extensions of those initial results. Section 2 adds brittleness and delamination and investigates the brittle membrane regime. Sections 4 and 5 focus on micromagnetics, rather than elasticity, this once again in the membrane regime and discuss magnetic skyrmions and domain walls, respectively. Finally, Section 3 revisits the classical setting in a non-Euclidean setting induced by the presence of a pre-strain in the model.