Regularity for the planar optimal

In this paper we prove a partial C1,α regularity result in dimension N = 2 for the optimal p-compliance problem, extending for p≠2 some of the results obtained by Chambolle et al. (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 is C1,α at ℌ1-a.e. point for every p ∈ (1, +∞)