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MiS Preprint
60/1999
A new approach to variational problems with multiple scales
Giovanni Alberti and Stefan Müller
Abstract
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 $\varepsilon$. This allows one to extract, in the limit $\varepsilon \to 0$, the relevant information at the macroscopic scale as well as the coarsest microscopic scale (say $\varepsilon^\alpha$), and to eliminate all finer scales. To achieve this we consider rescaled functions $R^\varepsilon_A x(t) :=x(\hbar + \varepsilon^\alpha t $ viewed as maps of the macroscopic variable $\hbar \in \oslash $ with values in a suitable function space. The limiting problem can then be formulates as a variational problem on the Young measures generated by $R^\varepsilon x$. As an illustration we study a one-dimensional model that describe 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.
