An elementary method of calculating an explicit form of Young measures in some special cases

TL;DR

This paper introduces a simple, explicit method for calculating Young measures for specific functions, especially oscillatory ones relevant in optimization and homogenization, avoiding complex functional analysis tools.

Contribution

It provides an elementary, explicit approach to compute Young measures for certain classes of oscillatory functions without advanced functional analysis techniques.

Findings

Applicable to both periodic and nonperiodic oscillatory sequences

Simplifies calculation of Young measures using change of variable theorem

Useful in optimization and homogenization problems involving microstructures

Abstract

We present an elementary method of explicit calculation of Young measures for certain class of functions. This class contains in particular functions of a highly oscillatory nature which appear in optimization problems and homogenization theory. In engineering such situation occurs for instance in nonlinear elasticity (solid-solid phase transition in certain elastic crystals). Young measures associated with oscillating minimizing sequences gather information about their oscillatory nature and therefore about underlying microstructure. The method presented in the paper makes no use of functional analytic tools. There is no need to use generalized version of the Riemann {Lebesgue lemma and to calculate weak* limits of functions. The main tool is the change of variable theorem. The method applies both to sequences of periodic and nonperiodic functions.