Variational models for phase transitions An approach via Γ-convergence

This paper is an extended version of the lecture delivered at the Summer School on Differential Equations and Calculus of Variations (Pisa, September 16-28, 1996). That lecture was conceived as an introduction to the theory of Γ-convergence and in particular to the Modica-Mortola theorem; I have tried to reply the style and the structure of the lecture also in the written version. Thus first come few words on the definition and the meaning of Γ-convergence, and then we pass to the theorem of Modica and Mortola. The original idea was to describe both the mechanical motivations which underlay this result and the main ideas of its proof. In particular I have tried to describe a guideline for the proof which would adapt also to other theorems on the same line. I hope that this attempt has been successful. Notice that I never intended to give a detailed and exhaustive description of the many results proved in this field through the recent years, not even of the main ones. In particular the list of references is not meant to be complete, neither one should assume that the contributions listed here are the most relevant or significant. This paper is organized as follows: