A new approach to variational problems with multiple scales
Giovanni Alberti, Stefan G. Müller · Communications on Pure and Applied Mathematics · 2001
Abstract We introduce a new concept, the Young measure on micropatterns, to study singularly perturbed variational problems that lead to multiple small scales depending on a small parameter ε. This allows one to extract, in the limit ε → 0, the relevant information at the macroscopic scale as well as the coarsest microscopic scale (say εα) and to eliminate all finer scales. To achieve this we consider rescaled functions R x (t) := x (s + εαt) viewed as maps of the macroscopic variable s ∈ Ω with values in a suitable function space. The limiting problem can then be formulated as a variational problem on the Young measures generated by Rεx. As an illustration, we study a one‐dimensional model that describes the competition between formation of microstructure and highest gradient regularization. We show that the unique minimizer of the limit problem is a Young measure supported on sawtooth functions with a given period. © 2001 John Wiley & Sons, Inc.