Variational methods for discontinuous structures

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This volume presents the Proceedings of the International Workshop "Variational Methods For Discontinuous Structures," held at Villa Erba Antica (Cernobbio) from July 4-6, 2001. Organized by the Dipartimento di Matematica Francesco Brioschi of Milano Politecnico and the International School for Advanced Studies (SISSA) of Trieste, the workshop addressed the evolution of the calculus of variations, which traditionally focused on continuous structures and smooth solutions. Recently, the introduction of sophisticated tools has enabled the study of discontinuities, where singularities often represent the most intriguing aspects of solutions. The conference highlighted recent advancements in this area, featuring talks on topics such as differential and variational modeling in image analysis, variational descriptions of micro-magnetic materials, dimension reduction in elasticity and plasticity, phase transitions, and the evolution of crystalline shapes. Theoretical and numerical analyses were provided for many models. Additional discussions covered specific challenges in the calculus of variations, including weak or lower-dimensional structures, optimal transport problems, higher-order variational issues, and symmetrization in the BV framework. This volume includes contributions from 12 of the 16 invited speakers, showcasing original results in rapidly evolving fields.