Regularity for the planar optimal p-compliance problem

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 $p≠2$, 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 $p\in (1,+\mathrm{\infty })$.