SubsidieMeestersAnisotropic geometric variational problems: existence, regularity and uniqueness
This project aims to develop tools for analyzing anisotropic geometric variational problems, focusing on existence, regularity, and uniqueness of anisotropic minimal surfaces in Riemannian manifolds.
Projectdetails
Introduction
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.
Motivation
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
- Homogenization problems
Motivated by these applications, anisotropic energies have attracted increasing interest in the geometric analysis community. Moreover, in differential geometry, they lead to the study of Finsler manifolds.
Challenges
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.
Methodology
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.
To determine the regularity of anisotropic minimal surfaces, I will study the associated geometric nonlinear elliptic partial differential equations (PDEs).
Additional Insights
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, including:
- Solving the size minimization problem
- The vectorial Allen-Cahn approximation of the general codimension Brakke flow
- The Almgren-Pitts min-max construction
Financiële details & Tijdlijn
Financiële details
| Subsidiebedrag | € 1.492.700 |
| Totale projectbegroting | € 1.492.700 |
Tijdlijn
| Startdatum | 1-9-2023 |
| Einddatum | 31-8-2028 |
| Subsidiejaar | 2023 |
Partners & Locaties
Projectpartners
- UNIVERSITA COMMERCIALE LUIGI BOCCONIpenvoerder
Land(en)
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