*Published Paper*

**Inserted:** 9 jan 2017

**Last Updated:** 9 jan 2017

**Journal:** Calculus of Variations and Partial Differential Equations

**Volume:** 55

**Number:** 6

**Pages:** 154-197

**Year:** 2016

**Doi:** 10.1007/s00526-016-1085-5

**Abstract:**

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

\[(u, A) \mapsto \int_\Omega 2 f u \, \text{d}x − \int_{\Omega \cap A} \sigma_1\mathcal A u · \mathcal A u \, \text{d}x − \int_{\Omega \setminus A} \sigma_2 \mathcal A u · \mathcal A u \, \text{d}x + \text{Per}(A;\overline\Omega), \] where $\Omega$ is a bounded Lipschitz domain, $A \subset \mathbb R^N$ is a Borel set, $u : \Omega \subset \mathbb R^N \to \mathbb R^d$, $\mathcal A$ 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 $C^1$-hypersurface up to a closed set of zero $\mathcal H^{N−1}$-measure.

**Keywords:**
partial regularity, almost perimeter minimizer, optimal design problem

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