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Zbl 1112.49014
De Arcangelis, Riccardo
On the relaxation of some classes of pointwise gradient constrained energies. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 24, No. 1, 113-137 (2007). ISSN 0294-1449

Summary: The integral representation problem on $BV(\Omega)$ for the $L^1 (\Omega)$-lower semicontinuous envelope $\overline F$ of the functional $F:u\in W^{1,\infty} (\Omega)\mapsto\int_\Omega f(\nabla u)dx$ is approached when $f$ is a Borel function, not necessarily convex, with values in $[0,+\infty]$. The presence of the value $+\infty$ in the image of $f$ involves a pointwise gradient constraint on the admissible configurations, since those generating the relaxation process must satisfy the condition $\nabla u(x)\in\text{dom}\,f$ for a.e. $x\in\Omega$. The main novelty relies in the absence of any convexity assumption on the domain of $f$. For every convex bounded open set $\Omega$, $\overline F$ is represented on the whole $BV(\Omega)$ as an integral of the calculus of variations by means of the convex lower semicontinuous envelope of $f$. Due to the lack of the convexity properties of dom\,$f$, the classical integral representation techniques, based on measure theoretic arguments, seem not to work properly, thus an alternative approach is proposed. Applications are given to the relaxation of Dirichlet variational problems and to first order differential inclusions.

MSC 2000:: *49J45 Optimal control problems inv. semicontinuity and convergence
49J24 Optimal control problems with differential inclusions (existence)
49J40 Variational methods including variational inequalities

Keywords: relaxation; pointwise gradient constraints; nonconvex variational problems; $BV$ spaces; first order differential inclusions

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