Static and Dynamic Behavior of Complex Systems via Variational Methods
Speaker
Nadia Ansini
Sapienza University of Rome
Time
2025-12-23 Tuesday 09:30 ~ 10:30
Abstract
In this seminar, I will present several results concerning the static and dynamic behavior of different complex systems within the framework of the calculus of variations. Complex systems are characterized by the presence of one or more significant scales arising from the microscopic description of the underlying phenomena.
The mathematical modeling of complex systems has attracted considerable interest in the international scientific community, as it naturally involves a wide range of disciplines including physics, biology, materials science, and even the social sciences. Despite their diverse origins, these systems often share an intrinsically multiscale nature.
Variational methods provide an effective tool for describing systems at equilibrium and for deriving macroscopic models through multiscale analysis. At the same time, they can be applied to the study of various dynamics, historically not considered variational in nature, which can nevertheless be reformulated as gradient flows with respect to suitable metrics and analyzed through the minimizing movement scheme.
In the static setting, I will briefly review several results on perforated domains, thin films, and phase transitions. I will then present recent results on a class of functionals modeled on convolution energies with kernels having finite p-moments. This result can be applied to obtain periodic homogenization results, to study applications to functionals defined on point-clouds, and to the study of limits of the related gradient flows.
Next, I will discuss recent advances on evolutionary problems, such as gradient flows with oscillatory potentials (modeling microstructure evolution) and nonlinear diffusion models with degenerate mobility. The last problem can be studied as gradient flows with respect to the Wasserstein metric and find interesting applications.
In both the static and dynamic contexts, I will highlight and compare the different variational techniques employed.
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