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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. 

We study the asymptotic Plateau problem in H2 × R for area-minimizing surfaces, and give a fairly complete solution for finite curves. 

ABSTRACT. Let X be a smooth simply connected closed 4- manifold with definite intersection form. We show that any automorphism of the intersection form of X is realized by a dif- feomorphism of X#(S2×S2). This extends and completes Wall’s foundational result from 1964.