Regularity for the planar optimal p-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 $p\ne 2$ 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 $p\ne 2$, 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 $p\in (1,+\infty )$.

56 pages

References in corpus (1)

• Approximation of length minimization problems among compact connected sets

Cited by in corpus (2)

• On the importance of the connectedness assumption in the statement of the optimal $p$-compliance problem
• Partial regularity for the optimal $p$-compliance problem with length penalization