Microstructure Patterns in Continuum Mechanics
Many problems in material sciences are characterized by the spontaneous formation of fine-scale structures, whose topology changes when the control parameters are varied. The different patterns can be understood using nonconvex functionals with a singular perturbation, which physically represents surface-energy terms.
Traditionally, physical understanding has been based on the construction of good test functions, leading to upper bounds on the ground-state energy. I shall discuss recent mathematical progress in the field, mainly focusing on the derivation of matching lower bounds. I shall show how this permitted, in specific examples, to prove optimality of the constructions, leading to predictions for the macroscopic phase diagram of the material.
Sergio Conti, Universität Duisburg-Essen, Germany