Variational Methods for Discontinuous Structures

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This volume presents the Proceedings of the International Workshop on Variational Methods for Discontinuous Structures, organized by the Dipartimento di Matematica Francesco Brioschi at Milano Politecnico and the International School for Advanced Studies (SISSA) in Trieste. Held at Villa Erba Antica (Cernobbio) from July 4-6, 2001, the conference addressed the evolution of the calculus of variations, which traditionally focused on continuous structures and smooth solutions. Recent advancements have introduced sophisticated tools to explore discontinuities, acknowledging that many variational problems exhibit singularities, often revealing the most intriguing aspects of solutions. The conference highlighted developments in differential and variational modeling related to image segmentation, occlusion, texture synthesis in image analysis, micro-magnetic materials, dimension reduction in elasticity and plasticity, phase transitions, and crystalline shape evolution, with both theoretical and numerical analyses presented. Additional discussions centered on specific calculus of variations challenges, including the variational theory of weak or lower-dimensional structures, optimal transport problems with free Dirichlet regions, higher-order variational problems, and symmetrization within the BV framework.