  We present a way to study a wide class of optimal design problems with a
perimeter penalization. More precisely, we address existence and regularity
properties of saddle points of energies of the form $$ (u,A) \quad \mapsto
\quad \int_\Omega 2fu \; \text{d}x \; - \int_{\Omega \cap A} \sigma_1 \mathscr
A u \cdot \mathscr A u \; \text{d}x \; - \int_{\Omega \setminus A}
\sigma_2\mathscr A u\cdot \mathscr A u \; \text{d}x \; + \;
\text{Per}(A;\overline \Omega),$$ where $\Omega$ is a bounded Lipschitz domain,
$A\subset \mathbb R^N$ is a Borel set, $u:\Omega \subset \mathbb R^N \to
\mathbb R^d$, $\mathscr A$ is an operator of gradient form, and $\sigma_1,
\sigma_2$ are two not necessarily well-ordered symmetric tensors. The class of
operators of gradient form includes scalar- and vector-valued gradients,
symmetrized gradients, and higher order gradients. Therefore, our results may
be applied to a wide range of problems in elasticity, conductivity or
plasticity models.
  In this context and under mild assumptions on $f$, we show for a solution
$(w,A)$, that the topological boundary of $A \cap \Omega$ is locally a
$\rm{C}^1$-hypersurface up to a closed set of zero $\mathscr H^{N-1}$-measure.
