Geodesic
Differential Inclusions in SBV0(Ω) and Applications to the Calculus of Variations
Journal of convex analysis, Tome 14 (2007) no. 3, pp. 465-477.

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We study necessary and sufficient conditions for the existence of solutions in $SBV_0(\Omega)$ of a variational problem involving only bulk energy. Related to that we study the problem of finding $u \in SBV_0(\Omega)$ such that $$\nabla u (x)\in E,\text{ a.e. in } \Omega,$$ subject to the condition $$\int \nabla u = \zeta_0 |\Omega|,$$ where $E\subseteq \mathbb{R}^N$ is a given set and $\zeta_0 \in $ int co $E$ is prescribed. 
@article{JCA_2007_14_3_JCA_2007_14_3_a1,
     author = {J. Matias},
     title = {Differential {Inclusions} in {SBV\protect\textsubscript{0}(\ensuremath{\Omega})} and {Applications} to the {Calculus} of {Variations}},
     journal = {Journal of convex analysis},
     pages = {465--477},
     publisher = {mathdoc},
     volume = {14},
     number = {3},
     year = {2007},
     url = {http://geodesic.mathdoc.fr/item/JCA_2007_14_3_JCA_2007_14_3_a1/}
}
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J. Matias. Differential Inclusions in SBV0(Ω) and Applications to the Calculus of Variations. Journal of convex analysis, Tome 14 (2007) no. 3, pp. 465-477. http://geodesic.mathdoc.fr/item/JCA_2007_14_3_JCA_2007_14_3_a1/