Journal of Convex Analysis 31 (2024), No. 3, 889--946
Copyright Heldermann Verlag 2024
Generalised Young Measures and Characterisation of Gradient Young Measures
Tommaso Seneci
Mathematical Institute, University of Oxford, Oxford, England
seneci@maths.ox.ac.uk
[Abstract-pdf]
\def\R{\mathbb R} Given a continuous function $f:\R^d\to \R$ with $p$-growth, we extend the framework developed by J.-J.\,Alibert and G.\,Bouchitt{\'e} [{\it 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{align*} \int_\Omega f(v_i(z))\,d\mu(z), \end{align*} 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 \emph{generalised Young Measures}.\\[1mm] 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.
Keywords: Young measures, separable compactifications, functions of bounded variation, quasi-convexity.
MSC: 49J45, 28B05, 35B05, 35B33, 46G10, 54D35.
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