Elastic solids with geometric boundary

Elastic bodies are characterized by their ability to resist external forces and return to their original shape once the force is removed. Examples include steel springs and rubber balls. Mathematically, the deformation of these materials can be described by their property to minimize the stored energy. For a deformation to be physically admissible, it must satisfy certain constraints, such as the non- interpenetration of matter. In certain complex or composite materials, the driving physical effects may depend both on the behavior of the interior of the elastic body and on properties of its surface, such as geometry and curvature. An important example is given by red blood cells, whose shape can be described by a model that favors negatively curved regions. The goal of this project is to develop a rigorous mathematical theory for physically admissible deformations of such materials that seamlessly fits into the established concept of minimal stored energy. Special attention is given to the description of so-called singular objects. On the geometric side, this includes non-smooth surfaces that may have sharp corners. On the other hand, the deformation itself can exhibit abrupt behavior, for example, when cavities form in a rubber-like material under tension. The systematic examination of these processes within the project combines classical principles of the calculus of variations with methods from geometry, in particular geometric measure theory. Key aspects of the study include the bulk-surface interaction, the properties of minimizers, and the application of the theory to dynamic processes.