Overview of Gamma-Convergence

Motivation and introduction

The calculus of variations is a field of mathematics that concerns studying integral functionals, where functions belonging to a certain function class are plugged into an integral formula. A real number will be spit out. A minimizer is a function in the function class for which the integral takes the smallest possible value. Some of the central questions in the calculus of variations include:

1. When does a minimizer exist?
2. When is a minimizer unique?
3. If a sequence of functions approaches a minimum, when and how does the sequence actually converge to a minimizer?
4. How do sequences of functionals behave in a limit?

The goal of this article will be to unpack the last of these four questions, where we consider families of functionals and how these functionals may converge.

The cover of one of my favorite texts on the calculus of variations

Before diving into specifics, let me make something clear: gamma-convergence is a convergence of functions, not a convergence of real numbers. This is despite the fact that the statements within the definition of gamma-convergence are inequalities related to real numbers.

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