  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 $p \in (1 ,+\infty)$.

Optimization and Control;Analysis of PDEs