A new approach to variational problems with multiple scales

We introduce a new concept, the Young measure on micro-patterns, to study singularly perturbed variational problems which lead to multiple small scales depending on a small parameter $\epsilon$. This allows one to extract, in the limit $\epsilon\to 0$, the relevant information at the macroscopic scale as well as the coarsest microscopic scale (say $\epsilon^\alpha$), and to eliminate all finer scales. To achieve this we consider rescaled functions $v_s(t) := x (s + \epsilon^\alpha t)$ viewed as maps of the macroscopic variable $s\in\Omega$ with values in a suitable function space. The limiting problem can then be formulated as a variational problem on the Young measures generated by $v_s$. 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.