Traditionally, many phenomena in nature, science and engineering are modelled with differential equations and local variational principles. Locality in this context means that the behaviour of an object depends only on its immediate neighbourhood. However, there are situations where the usual local modelling falls short because longer-range interactions and global effects have to be taken into account. This gives rise to nonlocal models, whose benefits are very prominent in many application areas, partly because they provide effective ways to bridge between different length scales and have shown to lead to refined predictions. Some of those areas are materials science, which is the focus of this proposal, as well as imaging and machine learning.

This project will contribute to a better understanding of nonlocality by addressing a new relevant class of variational problems, where the functionals are integrals depending on nonlocal gradients. These problems differ from standard hyperelasticity, which involves usual gradients, and from peridynamics, where the nonlocality is not expressed with a gradient. Problems with nonlocal gradients are not yet well explored, despite their advantageous properties. In starting from functionals that involve a specific nonlocal gradient, the overall goal of our research programme is twofold:

•     We aim to provide the theoretical foundations to establish these functionals as suitable energies in a refined nonlocal model for hyper elasticity. Potential benefits refer to accuracy and the possibility of going beyond purely elastic behaviour by accounting for singular effects. Our justification of the new model comprises several facets, including relations with local and linear models, a discussion of suitable local boundary values, and non-interpenetration of matter.
•    We propose to develop a variational theory in a broad framework of nonlocal gradients, considering also different types of boundary conditions. This involves the introduction of new function spaces and requires a careful investigation of fundamental structural properties of nonlocal gradients that differ from those of their local counterparts.

Despite impressive developments in the last years, many conceptual questions still remain open and new techniques need to be established to address relevant types of nonlocal structures. Apart from the foundations already laid by the applicants and others in recent works, we will borrow and adapt some tools from other classical areas of mathematical analysis.

15.04.2025: We are exited to anounce that our PhD position has been filled. Felix Seifert is joining the project team in Eichstätt, having recently finished his Masters degree in Mathematics at Friedrich-Schiller-University Jena. We are looking forward to working together. Welcome to the team, Felix!

12.04.2025: CRM News has published a nice article about the School on New Frontiers in Homogenisation and Fractional Calculus, where Carolin Kreisbeck gave a talk. You can find the article here.

19.09.2024: We have currently two job openings within this project, a 3-year PhD and a 2.5-year postdoc position. The PhD position will be at KU Eichstätt-Ingolstadt, the postdoc position at Universidad Autónoma de Madrid. The intended starting date is March 2025 or as soon as possible thereafter. If you are interested in becoming part of our international team, we would be excited to receive your application!