Elliptic systems with sharp transition on optimal sets – ESSTOS

This project lies at the interface between the calculus of variations and non linear analysis applied to condensed matter physics on the one hand, and geometric measure theory and transport models in urban planning on the other hand. We are particularly interested in models where the energy concentrates on singular sets of positive codimension. Our research program includes two main axes. First, we will study the fine structure of singularities for stable solutions of phase field models. To this aim, we will concentrate on the question of the classification of minimizing entire solutions in some models coming from liquid crystals and micromagnetics. This includes fundamental questions such as the existence, symmetry and uniqueness of solutions. Furthermore, in a second axis of our project, we will explore transport models that encompass two different features of urban planning of transport infrastructures: first, economies of scale that lead to preference of joint transport on a 1D network, and, on the other hand, congestion that we want to avoid due to its multiple disadvantages. We will develop new branched transport models with congestion, and we will study properties of solutions and of the underlying network.