Global and Local Behavior of Deformations

Quasiconformal elasticity is a current research trend at the triple point of quasiconformal analysis, calculus of variations, and modeling in solid mechanics. It contributes a unique perspective on analytical and geometrical challenges arising from variational models in continuum mechanics, fostering advancements in both theory and applications. The project Global and Local Behavior of Deformations of Sobolev and Fractional Sobolev Classes (GLoBe) explores two crucial aspects of quasiconformal elasticity: the examination of local properties of nonlocally defined deformations and the behavior of functions with specific local characteristics on a global scale. The first part of the project focuses on fractional Sobolev spaces and includes the investigation of local properties of deformations and the modeling of thin structures in peridynamics. The second part deals with solutions to variational problems when deformations are almost everywhere invertible. In particular, it concerns Lavrentievs phenomenon in classes of limits of Sobolev homeomorphisms and mappings of finite distortion. Using these classes of mappings as admissible deformations, the project proposes a new approach to the mathematical understanding of noninterpenetration of matter and self-contact in elasticity. Specifically, the project includes the study of magnetoelasticity and self-contact for thin elastic films. This project will lay the theoretical and methodological foundations for quasiconformal analysis in the fractional Sobolev setting and for the theory of weak limits of Sobolev homeomorphisms. The analysis will be based on combining of tools from quasiconformal analysis and variational methods for nonlinear elasticity. The key instruments include distributional Jacobian, topological degree, fractional gradient, and Gamma-convergence.