### Oscillation and concentration effects described by Young measures which control discontinuous functions

DOI: http://dx.doi.org/10.12775/TMNA.2008.006

#### Abstract

We study oscillation and concentration effects for

sequences of compositions $\{ f(u^\nu)\}_{\nu\in\mathbb N}$ of

$\mu$-measurable functions $u^\nu\colon \Omega\rightarrow{\mathbb R}^{m}$ where

$\Omega$ is the compact subset of ${\mathbb R}^n$

and $f$ is the (possibly) discontinuous function.

The limits are described in terms of Young measures which can

control discontinuous functions recently introduced in [A. Kałamajska, < i> On Young measures controlling discontinuous functions< /i> , J. Conv.

Anal. < b> 13< /b> (2006), no. 1, 177–192].

sequences of compositions $\{ f(u^\nu)\}_{\nu\in\mathbb N}$ of

$\mu$-measurable functions $u^\nu\colon \Omega\rightarrow{\mathbb R}^{m}$ where

$\Omega$ is the compact subset of ${\mathbb R}^n$

and $f$ is the (possibly) discontinuous function.

The limits are described in terms of Young measures which can

control discontinuous functions recently introduced in [A. Kałamajska, < i> On Young measures controlling discontinuous functions< /i> , J. Conv.

Anal. < b> 13< /b> (2006), no. 1, 177–192].

#### Keywords

Young measures; DiPerna-Majda measures; weak convergence; discontinuous functions

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