*Submitted Paper*

**Inserted:** 21 nov 2019

**Last Updated:** 3 dec 2019

**Year:** 2019

**Abstract:**

In this paper we prove a partial $C^{1,\alpha}$ regularity result in dimension $N=2$ for the optimal $p$-compliance problem, extending for $p\not = 2$ some of the results obtained by A. Chambolle, J. Lamboley, A. Lemenant, E. Stepanov (2017). Because of the lack of good monotonicity estimates for the $p$-energy when $p\not = 2$, we employ an alternative technique based on a compactness argument leading to a $p$-energy decay at any flat point. We finally obtain that every optimal set has no loop, is Ahlfors regular, and $C^{1,\alpha}$ at $\mathcal{H}^1$-a.e. point for every $p \in (1 ,+\infty)$.

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

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