Project description
Variational analysis of anisotropic minimal surfaces
The study of geometric variational problems is one of the most fascinating topics in the calculus of variations. Solutions of geometric variational problems describe equilibrium configurations of physical systems. Minimal surfaces, namely surfaces with zero mean curvature, are critical points of the area functional and constitute a large class of solutions of geometric variational problems. A broader class is given by anisotropic minimal surfaces, which are critical points of anisotropic functionals. Anisotropic functionals are integrals of directionally dependent density functions over a surface. Motivated by several applications such as crystal structures, capillarity problems, gravitational fields and homogenisation problems, the EU-funded ANGEVA project will study anisotropic functionals, seeking to develop tools to prove existence, regularity and uniqueness properties of anisotropic minimal surfaces.
Objective
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.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
- natural sciences mathematics pure mathematics topology
- natural sciences mathematics pure mathematics geometry
- natural sciences mathematics pure mathematics mathematical analysis differential equations partial differential equations
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Keywords
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Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
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Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
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Call for proposal
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(opens in new window) ERC-2022-STG
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20136 Milano
Italy
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