Generalised Young Measures and Characterisation of Gradient Young Measures

Given a continuous function

with

p p

-growth, we extend the framework developed by J.-J. Alibert and G. Bouchitté [Non-uniform integrability and generalized Young Measure, J. Convex Analysis 4 (1997) 129–148] obtaining a new way of representing accumulation points of

\begin{aligned}\int_\Omega f(v_i(z))\,d\mu(z),\end{aligned} ∫ Ω f ( v i ( z ) )   d μ ( z ) ,

where

\mu μ

is a finite positive Borel measure on an open bounded set

\Omega\subset {\mathbb R}^n Ω ⊂ R n

, and

(v_i)_{i\in \N}\subset L^p(\Omega,\mu) ( v i ) i ∈ N ⊂ L p ( Ω , μ )

is norm bounded. We call such representations generalised Young Measures. With the help of the new representation, we then characterise these limits when they are generated by gradients, that is, when

v_i = Du_i v i = D u i

for

u_i\in W^{1,1}(\Omega,\mathbb R^m) u i ∈ W 1 , 1 ( Ω , R m )

, via a set of integral inequalities.