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Journal of Convex Analysis 14 (2007), No. 3, 465--477
Copyright Heldermann Verlag 2007

Differential Inclusions in SBV0(Ω) and Applications to the Calculus of Variations

José Matias
Dep. de Matemática, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal


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.

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