Regularity for free interface variational problems in a general class of gradients

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 (Formula presented.) ,where Ω is a bounded Lipschitz domain, A⊂ RN is a Borel set, u: Ω ⊂ RN→ Rd, A is an operator of gradient form, and σ1, σ2 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∩ Ω is locally a C 1-hypersurface up to a closed set of zero HN-1-measure.