Calculus of Variations and Geometric Measure Theory
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G. Alberti - S. Müller

A New Approach to Variational Problems with Multiple Scales

created on 19 Sep 1999
modified by paolini on 07 Jun 2013


Published Paper

Inserted: 19 sep 1999
Last Updated: 7 jun 2013

Journal: Comm. Pure Appl. Math.
Volume: 54 (2001)
Number: 7
Pages: 761-825.
Year: 1999

Revised version, March 2001.


We introduce a new concept, the Young measure on micro-patterns, to study singularly perturbed variational problems which lead to multiple small scales. This allows one to extract the relevant information at the macroscopic scale as well as the coarsest microscopic scale, and to eliminate all finer scales. To achieve this we consider for every u the map that associates to each value of the macroscopic variable s a suitable blow-up around s; the limiting problem can then be formulated as a variational problem on the Young measures generated by these blow-up maps. 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.

Keywords: Young Measures, microstructures, invariant measures, singular perturbation problems


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