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 $μ\in \mathcal{M}^+(Ω)$, and $(v_i)_{i\in \mathbb{N}}\subset L^1(Ω,μ)$ 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 $v_i = Du_i$ for $u_i\in W^{1,1}(Ω,\mathbb{R}^m)$, via a set of integral inequalities.

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