Abstract

Given a continuous function f:Rd→Rf:\R^d\to \R with pp-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 ∫Ωf(vi(z)) dμ(z),\begin{aligned}\int_\Omega f(v_i(z))\,d\mu(z),\end{aligned} where μ\mu is a finite positive Borel measure on an open bounded set Ω⊂Rn\Omega\subset \R^n, and (vi)i∈N⊂Lp(Ω,μ)(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 vi=Duiv_i = Du_i for ui∈W1,1(Ω,Rm)u_i\in W^{1,1}(\Omega,\R^m), via a set of integral inequalities.

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Keywords

  • Young measures
  • separable compactifications
  • functions of bounded variation
  • quasi-convexity

Mathematics Subject Classification

49J45, 28B05, 35B05, 35B33, 46G10, 54D35

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Suggested citation

T. Seneci. “Generalised Young Measures and Characterisation of Gradient Young Measures.” Journal of Convex Analysis 31 (2024), No. 3, 889–946. https://doi.org/10.68381/jca31044

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