Regularity for the planar optimal p-compliance problem

TL;DR

This paper establishes partial regularity results for the optimal p-compliance problem in two dimensions, showing that optimal sets are smooth at most points and have no loops, extending previous work for p ≠ 2.

Contribution

It extends regularity results for the optimal p-compliance problem to all p in (1, ∞) in 2D, using a novel compactness approach due to the lack of monotonicity estimates.

Findings

Optimal sets are $C^{1,eta}$ at almost every point.

Optimal sets have no loops.

Optimal sets are Ahlfors regular.

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