Materials Instabilities: Regularity Theory and the Calculus of Variations

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The general objective of this project is the development of techniques in the calculus of variations and in the regularity theory for systems of partial differential equations. The applications that guide this work come mainly fromthe areas of continuum mechanics of liquids and solids. Tools from geometric measure theory, a theory for the study of generalizations of surfaces, will play animportant role. The analysis will be undertaken for time-dependent problems, including in some cases certain idealizations where inertial effects are neglected.The main focus will be on the search for the effective description of materials with complicated composition, on the study of regularity properties of solutions in the bulk and at interfaces, and on the prediction of their oscillatory or hysteretic behavior as time evolves. The research will be on the mathematical description and prediction of a range ofphenomena that occur in material science. This includes phase transformations (between liquid and solid phases), the structure, nucleation and growth of such phases, and fracture and defects in solids. An important part of this work willbe the study of mathematical models for novel man-made materials such as shape memory alloys, ferroelectric, magnetic and magnetostrictive materials, composites, liquid crystals, and thin films. Although unexpected, the mathematicaltheory also provides new methods for image segmentation in computer vision. The mathematical challenges lie in the description of the dynamics and evolution of microscopic structures and of phenomena that occur at vastly different temporal or spatialscales. They require recently developed mathematical tools and the introduction of new mathematical techniques.