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ANGEVA

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Anisotropic geometric variational problems: existence, regularity and uniqueness
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, minim... 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 solving the size minimization problem, in the vectorial Allen-Cahn approximation of the general codimension Brakke flow, and in the Almgren-Pitts min-max construction. ver más
31/08/2028
UB
1M€
Perfil tecnológico estimado
Duración del proyecto: 69 meses Fecha Inicio: 2022-11-22
Fecha Fin: 2028-08-31

Línea de financiación: concedida

El organismo HORIZON EUROPE notifico la concesión del proyecto el día 2022-11-22
Línea de financiación objetivo El proyecto se financió a través de la siguiente ayuda:
ERC-2022-STG: ERC STARTING GRANTS
Cerrada hace 2 años
Presupuesto El presupuesto total del proyecto asciende a 1M€
Líder del proyecto
UNIVERSITA COMMERCIALE LUIGI BOCCONI No se ha especificado una descripción o un objeto social para esta compañía.
Perfil tecnológico TRL 4-5