Abstract
Relaxation problems for a functional of the type $G(u)=\int_\Omega g(x,\nabla u)\,{\rm d}x$ are analyzed, where $\Omega$ is a bounded smooth subset of $\R^N$ and $g$ is a Carath\’eodory function, when the admissible $u$ are forced to satisfy a pointwise gradient constraint of the type $\nabla u(x)\in C(x)$ for a.e. $x\in\Omega$, $C(x)$ being, for every $x\in\Omega$, a bounded convex subset of $\R^N$. The relaxed functionals $\overline{G_{PC^1(\Omega)}}$, and $\overline{G_{W^{1,\infty}(\Omega)}}$ of $G$ obtained letting $u$ vary in $PC^1(\Omega)$, the set of the piecewise $C^1$-functions in $\Omega$, and in $W^{1,\infty}(\Omega)$ respectively in the definition of $G$ are considered. Identity and integral representation results are proved under continuity-type assumptions on $C$, together with the description of the common density by means of convexification arguments. Classical relaxation results are extended to the case of the continuous variable dependence of $C$, and the non-identity features described in the measurable dependence case by De Arcangelis, Monsurr\‘o and Zappale (2004) are shown to be non-occurring. Proofs are based on the properties of certain limits of multifunctions, and on an approximation result for functions $u$ in $W^{1,\infty}(\Omega)$, with $\nabla u(x)\in C(x)$ for a.e. $x\in\Omega$, by $PC^1(\Omega)$ ones satisfying the same condition. Results in more general settings are also obtained.
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De Arcangelis, R., Zappale, E. The Relaxation of Some Classes of Variational Integrals with Pointwise Continuous-Type Gradient Constraints. Appl Math Optim 51, 251–277 (2005). https://doi.org/10.1007/s00245-004-0811-0
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DOI: https://doi.org/10.1007/s00245-004-0811-0