We obtain a measure representation for a functional arising in the context of optimal design problems under linear growth conditions. The functional in question corresponds to the relaxation with respect to a pair $(\chi,u)$, where $\chi$ is the characteristic function of a set of finite perimeter and $u$ is a function of bounded deformation, of an energy with a bulk term depending on the symmetrized gradient as well as a perimeter term.

We consider the problem of minimizing the energy $$ E(u):= \int_{\Omega}|\nabla u(x)|^2 \, {\rm d}x + \int_{S_u \cap \Omega}\left (1 + |[u](x)|\right) \, {\rm d}H^{N - 1}(x)$$ among all functions u ∈ SBV²(Ω) for which two level sets $\{u = l_i\}$ have prescribed Lebesgue measure $\alpha_i$. Subject to this volume constraint the existence of minimizers for E(.) is proved and the asymptotic behaviour of the solutions is investigated.

We obtain the Γ(L1(Ώ))-limit of the sequence

where Eε is the family of anisotropic perturbations

of the nonconvex functional of vector-valued functions

The proof relies on the blow-up argument introduced by Fonseca and Müller.