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 26 Apr 2008


Published Paper

Inserted: 10 jun 2007
Last Updated: 26 apr 2008

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 minimuma 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