Geometric rigidity estimates in variable domains and applications in dimension reduction

Quantitative rigidity results, besides their inherent geometric interest, have played a prominent role in the mathematical study of variational models related to elasticity\plasticity. For instance, the celebrated rigidity estimate of Friesecke, James, and Müller has been widely used in problems related to linearization, discrete-to-continuum or dimension-reduction issues for energies within the framework of nonlinear elasticity. After a quick review of the aforementioned results, we will present appropriate generalizations to the setting of variable domains, where the geometry of the domain comes into play in terms of a suitable surface energy of its boundary. If time permits, we will also discuss applications of this new rigidity estimates in questions related to dimension reduction for elastic materials with free surfaces. This is joint work with Manuel Friedrich and Leonard Kreutz.