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
$$\begin{aligned} (u,A) \quad \mapsto \quad \int _\Omega 2fu \,\mathrm {d}x - \int _{\Omega \cap A} \sigma _1\mathscr {A}u\cdot \mathscr {A}u \, \,\mathrm {d}x - \int _{\Omega {\setminus } A} \sigma _2\mathscr {A}u\cdot \mathscr {A}u \, \,\mathrm {d}x + \text {Per }(A;\overline{\Omega }), \end{aligned}$$
where
\(\Omega \) is a bounded Lipschitz domain,
\(A\subset \mathbb {R}^N\) is a Borel set,
\(u:\Omega \subset \mathbb {R}^N \rightarrow \mathbb {R}^d\),
\(\mathscr {A}\) 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
\(\mathrm {C}^1\)-hypersurface up to a closed set of zero
\(\mathscr {H}^{N-1}\)-measure.