Existence of free discontinuity problems in SBD(Ω)SBD(Ω)

In this paper, we consider the variational problems for the following two functionals in SBD(Ω)SBD(Ω): F1(u)=∫Ωf1(x,…u)dx+αHN−1(Ju)+β∫Jug(x,[u]⊙νu)dHN−1−∫Ωh⋅udx.F2(u)=∫Ωf2(x,u,…u)dx+αHN−1(Ju)+β∫Jug(x,[u]⊙νu)dHN−1−∫Ωh⋅udx where f1:Ω×MsymN×N→[0,∞) be a Carathéodory function satisfying a weak convexity property and growth assumptions of order s>1s>1. f2f2 and gg satisfy convexity property and lower semicontinuity. h∈Lp(Ω,RN),p≥Nh∈Lp(Ω,RN),p≥N satisfies the compatibility condition. α,β>0α,β>0. Here we do not assume that u(x)∈H(x)u(x)∈H(x) a.e. in ΩΩ, {H(x)}x∈Ω{H(x)}x∈Ω be a uniformly bounded family of closed subsets of RNRN.