Preprint 60/1999

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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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 tex2html_wrap_inline12. This allows one to extract, in the limit tex2html_wrap_inline14, the relevant information at the macroscopic scale as well as the coarsest microscopic scale (say tex2html_wrap_inline16), and to eliminate all finer scales. To achieve this we consider rescaled functions tex2html_wrap_inline18 viewed as maps of the macroscopic variable tex2html_wrap_inline20 with values in a suitable function space. The limiting problem can then be formulates as a variational problem on the Young measures generated by tex2html_wrap_inline22. 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.

03.07.2017, 01:40