Generalised Young Measures and characterisation of gradient Young Measures

Given a function $f\in C({\mathbb{R}}^d)$ of linear growth, we give a new way of representing accumulation points of \begin{equation} \int_Ωf(v_i(z))dμ(z), \end{equation} where $\mu \in {\mathcal{M}}^{+}(\mathrm{\Omega })$, and $(v_i{)}_{i\in \mathbb{N}}\subset L^1(\mathrm{\Omega },\mu )$ is norm bounded. We call such representations “generalised Young Measures”. With the help of the new representations, we then characterise these limits when they are generated by gradients, i.e. when $v_i=D{u}_i$ for $u_i\in W^{1,1}(\mathrm{\Omega },{\mathbb{R}}^m)$, via a set of integral inequalities.