Regularity for the planar optimalp-compliance problem

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 $p \neq 2$ 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 $p \neq 2$, 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 $C^{1, \alpha}$ at $\mathcal{H}^{1}$-a.e. point for every $p\in (1, +\infty)$.