Projects

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. 

Details: 

• Grant agreement ID: 101076411 

• DOI: 10.3030/101076411 

• Start date: 1 September 2023

• End date: 31 August 2028

• Duration:  60 months

• Funding Scheme:  Starting Grant

• Call year:  2022   Panel:  PE1   Project number: 101076411

• Total cost: € 1 492 700,00

Description: This project stems out from the strong collaboration of several researchers that in recent years developed new ideas, techniques and methods in the study of gradient flows and evolution problems, optimal transport, metric analysis and geometric optimization problems. All these topics are indeed strictly intertwined. Many evolution problems of interest in pure and applied Mathematics are energy-driven and the possibility to identify a gradient flow structure is of fundamental importance in the understanding of energy dissipation rates and of trends to equilibrium. In this respect, particularly in the last two decades, optimal transport has been a striking source of tools for studying evolution, due on one hand to the identification of new families of natural metrics, on the other hand to the possibility to lift the space dynamics to probability measures in the space of curves. Since lack of smoothness is an intrinsic feature of many of these models, their analysis leads in a natural way to the investigation of metric measure structures, with the development of new calculus tools and synthetic theories for geometries that can also be very far from being Riemannian.

Details: 

• Grant 2017TEXA3H_002  (2019-2023) PRIN