A new approach to variational problems with multiple scales

Giovanni Alberti and Stefan Müller

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Submission date: 04. Oct. 1999
Pages: 52
published in: Communications on pure and applied mathematics, 54 (2001) 7, p. 761-825 
DOI number (of the published article): 10.1002/cpa.1013
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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 . This allows one to extract, in the limit , the relevant information at the macroscopic scale as well as the coarsest microscopic scale (say ), and to eliminate all finer scales. To achieve this we consider rescaled functions viewed as maps of the macroscopic variable with values in a suitable function space. The limiting problem can then be formulates as a variational problem on the Young measures generated by . 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.