Calculus of Variations and Geometric Measure Theory

B. Bulanyi - A. Lemenant

Regularity for the planar optimal $p$-compliance problem

created on 21 Nov 2019
modified by bulanyi on 12 Apr 2021


Accepted Paper

Inserted: 21 nov 2019
Last Updated: 12 apr 2021

Journal: ESAIM Control Optim. Calc. Var.
Year: 2021

ArXiv: 1911.09240 PDF


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

Keywords: shape optimization, compliance, Minimizers, regularity theory