Generalised Young Measures and Characterisation of Gradient Young Measures

Given a continuous function $f:\R^d\to \R$ with $p$-growth, we extend the framework developed by J.-J. Alibert and G. Bouchitt{é} [Non-uniform integrability and generalized {Y}oung {M}easure, J. Con\-vex Analysis 4 (1997) 129–148] obtaining a new way of representing accumulation points of $$\begin{aligned}\int_\Omega f(v_i(z))\,d\mu(z),\end{aligned}$$ where $\mu$ is a finite positive Borel measure on an open bounded set $\Omega\subset \R^n$, and $(v_i)_{i\in \N}\subset L^p(\Omega,\mu)$ 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 $v_i = Du_i$ for $u_i\in W^{1,1}(\Omega,\R^m)$, via a set of integral inequalities.