*Accepted Paper*

**Inserted:** 12 jan 2021

**Last Updated:** 25 aug 2021

**Journal:** Calc. Var. Partial Differential Equations

**Pages:** 51

**Year:** 2021

**Abstract:**

We establish a partial $C^{1,\alpha}$ regularity result for minimizers of the optimal $p$-compliance problem with length penalization in any spatial dimension $N\geq 2$, extending some of the results obtained in $[$Chambolle-Lamboley-Lemenant-Stepanov 17$]$, $[$Bulanyi-Lemenant 21$]$. The key feature is that the $C^{1,\alpha}$ regularity of minimizers for some free boundary type problem is investigated with a free boundary set of codimension $N-1$. We prove that every optimal set cannot contain closed loops, cannot contain quadruple points, and it is $C^{1,\alpha}$ regular at $\mathcal{H}^{1}$-a.e. point for every $p\in (N-1,+\infty)$.

**Keywords:**
shape optimization, $p$-compliance, regularity theory

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