Calculus of Variations and Geometric Measure Theory

A. Braides - L. Truskinovsky

Asymptotic expansions by $\Gamma$-convergence

created by braidesa on 10 Jun 2007
modified on 22 Jun 2023


Published Paper

Inserted: 10 jun 2007
Last Updated: 22 jun 2023

Journal: Cont. Mech. Therm.
Volume: 20
Pages: 21-62
Year: 2008


We propose a new framework aimed at constructing approximations of different order to static mechanical models with variational structure. Our starting point is a parameterized family of functionals (a theory) and we are interested in approximating the global minima of the energy when a secondary small parameter goes to zero. The goal is to develop a set of increasingly accurate asymptotic variational models allowing one to deal with the cases when this secondary parameter is small but finite. At the basis of our approach is the idea of $\Gamma$-equivalence, allowing one to divide the given set of theories into classes of asymptotic equivalence with respect to the small parameter. Since $\Gamma$-convergence may be nonuniform within a theory we pose a problem of finding a uniform approximation. To achieve this goal we propose a method based on rectifying the singular points in the parameter space by using the blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas with physically meaningful examples covering broad set of subjects from homogenization and dimension reduction to fracture and phase transitions. The analysis of many of the examples is new and presents an independent interest. In particular, we give considerable attention to the problem of transition from discrete to continuum when the internal and external scales are not well separated and one has to deal with the so-called size or scale effects.

Keywords: Gamma-convergence, Asymptotic expansions, uniform approximation