Differential Inclusions in SBV ₀ (Ω) and Applications to the Calculus of Variations

We study necessary and sufficient conditions for the existence of solutions in

of a variational problem involving only bulk energy. Related to that we study the problem of finding

u \in SBV_0(\Omega) u ∈ S B V 0 ( Ω )

such that

\nabla u (x)\in E,\text{ a.e. in } \Omega, ∇ u ( x ) ∈ E ,  a.e. in  Ω ,

subject to the condition

\int \nabla u = \zeta_0 |\Omega|, ∫ ∇ u = ζ 0 ∣ Ω ∣ ,

where

E\subseteq \mathbb{R}^N E ⊆ R N

is a given set and

\zeta_0 \in ζ 0 ∈

int co

E E

is prescribed