ERC StG ANGEVA

                Anisotropic geometric variational problems: existence, regularity and uniqueness

                                                  ERC Starting Grant 2022 - Period: 2023 - 2028

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Project ID

Title: Anisotropic geometric variational problems: existence, regularity and uniqueness

Acronym: ANGEVA

Grant agreement ID: 101076411

DOI: 10.3030/101076411


Start date: 1 September 2023

End date: 31 August 2028

Duration:  60 months


Primary coordinator: Antonio De Rosa  

Host institution: Bocconi University  

Funding Agency

Funded under: European Research Council (ERC)

Funding Scheme:  Starting Grant

Call year:  2022   Panel:  PE1   Project number: 101076411

Total cost: € 1 492 700,00

Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

Here you can find a divulgation article and here you can find a divulgation video about the project

Description

      The focus of this project is to advance the theory of anisotropic geometric variational problems. A vast literature is devoted to the study of critical points of the area functional, referred to as minimal surfaces. However, minimizing the surface area is often an idealization in physics. In order to account for preferred inhomogeneous and directionally dependent configurations and to capture microstructures, more general anisotropic energies are often utilized in several important models. Relevant examples include crystal structures, capillarity problems, gravitational fields and homogenization problems. Motivated by these applications, anisotropic energies have attracted an increasing interest in the geometric analysis community. Moreover in differential geometry they lead to the study of Finsler manifolds. Unlike the rich theory for the area functional, very little is understood in the anisotropic setting, as many of the essential techniques do not remain valid. This project aims to develop the tools to prove existence, regularity and uniqueness properties of the critical points of anisotropic functionals, referred to as anisotropic minimal surfaces. In order to show their existence in general Riemannian manifolds, it will be crucial to generalize the min-max theory. This theory plays a crucial role in proving a number of conjectures in geometry and topology. In order to determine the regularity of anisotropic minimal surfaces, I will study the associated geometric nonlinear elliptic partial differential equations (PDEs). Finally, in addition to the stationary configurations, this research will shed light on geometric flows, through the analysis of the related parabolic PDEs. The new methods developed in this project will provide new insights and results even for the isotropic theory: in the size minimization problem, in the vectorial Allen-Cahn approximation of the general codimension Brakke flow, and in the Almgren-Pitts min-max construction.

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