V5B6 - Selected Topics in Analysis and Calculus of Variations

An Introduction to $\Gamma$-convergence

Many problems in applied mathematics involve minimizing an energy that depends on a small parameter. This parameter may arise from a discretization or approximation procedure, or describe the microscopic structure of a material. Rather than studying each problem separately, it is often more useful to understand the limiting behavior of the corresponding energies as the parameter vanishes. $\Gamma$-convergence, introduced by De Giorgi, is the natural notion of convergence for variational problems. It provides a powerful framework for analyzing the asymptotic behavior of minimization problems and ensures, under suitable assumptions, the convergence of minimizers and minimum values. This lecture offers an introduction to $\Gamma$-convergence and its applications. 

We begin with the abstract theory and its fundamental properties before turning to integral functionals on Sobolev spaces. We then explore how $\Gamma$-convergence can be used to study a variety of important problems, including homogenization, discrete-to-continuum limits, phase-transition models, and dimension reduction problems. 

A preliminary list of contents: 

  • The Calculus of Variations and $\Gamma$-convergence 
  • Homogenization
  • Phase-transition models
  • Dimension reduction and Linearization

The lecture takes place on Monday and our first session will be held on October 12 at 16:15 in Seminar Room N0.008, Endenicher Allee 60 / Neubau.
 

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