The winter school started at VIASM on 31/10/2016. The two main speakers are Prof. Chiara Zanini, Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino and Dr. Marita Thomas, Weierstrass Institute, Berlin (Germany). The school is organized until 04/11/2016.
The lectures provide an introduction to variational convergence methods. 1) Focuses on the method of Gamma-convergence in static and evolution problems with applications in continuum mechanics, addressing both rate-dependent and rate-independent systems as well as coupled processes. 2) Is devoted to the topic of homogenization of periodic structures, focusing on second order linear elliptic operators, monotone operators, methods of two-scale convergence, G-convergence and H-convergence. The lectures are suited for master- and PhD-students in applied mathematics, mechanics and master- and PhD-students in engineering with an interest in analytical tools for applications.
The lectures of Prof. Chiara Zanini are about the Introduction to Homogenization and G-convergence which are applied in Periodic structures. These structures are employed to describe composite materials, which appear in many fields like Mechanics, Physics, Chemistry and Engineering. The typical situation is that the physical parameters (conductivity, elasticity coefficients…) are discontinuous and oscillating between the different values characterizing each of the components. When the components are intimately mixed, these parameters oscillate very rapidly and the microscopic structure becomes complicated. Homogenization is a mathematical tool which, roughly speaking connects the length scales associated with the microscopic and the macroscopic phenomena. Indeed, the idea is to get a good approximation of the macroscopic behavior of such a heterogeneous material by means of a homogeneous material, whose overall response is close to that of the composite (periodic) material. In other words, starting from Partial Differential Equations of Physics describing a heterogeneous material with a fine periodic structure, Homogenization deals with the asymptotic analysis when the parameter describing the fineness of the microscopic structure tends to zero.
The lectures of Dr. Marita Thomas focus on Gamma-convergence methods proposed by De Giorgi in the 1970s. The framework of Gamma-convergence provides a general and flexible tool to describe the asymptotic behavior of minimum problems for families of functionals in the calculus of variations. In particular, it provides exactly the criteria needed to ensure that cluster points of minimizers of the approximating functional are minimizers of the limit functional.
In the first day of the school, there are more than 20 participants.