FRG: Collaborative Research: Vectorial and geometric problems in the calculus of variations

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This project focuses on mathematical methods for problems that appear naturally in physics and engineering, for example, non-linear elasticity, phase transitions and interaction energies. Apart from being fascinating mathematical problems that require new ideas and methods, their study is also of great importance for a deeper understanding of the physical phenomena themselves. For instance, understanding the best way to crumple a paper is a natural problem in elasticity which has several applications in material sciences, while nonlocal interaction energies appear in physics and chemistry as a way to describe the energy of nuclei. Our mathematical research will shed new light on the range of energies under which nuclei are stable. The project will also have important impact on human resource development. We propose a system of personnel exchanges between our home institutions, designed to provide a rich training experience to students and postdocs. We propose to coordinate our activities and to have an emphasis month every year in one of the home universities of the PIs. We also propose summer schools designed to bring members of the group together (including graduate students and postdocs) to stimulate scientific progress, while exposing the fruits of our research to a broader audience and helping to educate and attract a new generation of researchers to this exciting area of emerging mathematical challenges and ideas.The goals of this project include developing new general methods to study variational problems both in a vectorial setting and in the case of non-local interaction energies. The problems we plan to study include: variational problems in nonlinear elasticity related to crumpling; existence of gradient flows for quasiconvex energy functionals and variational problems involving surface tension and nonlocal energies. We focus on central issues in the calculus of variations, such as proving existence of minimizers of energy functionals and understanding their (possible) uniqueness and regularity. While much is known in the scalar case, very few results are available in the vectorial setting (both in the static and evolutionary cases) and for geometric problems involving non-local energies. Through a common effort, bringing together the investigators and collaborators for this Focused Research Group Grant, we expect some general methods to emerge, allowing us to obtain new substantial results. In addition to its purely mathematical interest, this project will improve the understanding of the phenomena that these models attempt to reflect.