Partial regularity for the optimal $p$-compliance problem with length penalization

Abstract

We establish a partial $C^{1,\alpha }$ regularity result for minimizers of the optimal $p$-compliance problem with length penalization in any spatial dimension $N\ge 2$, extending some of the results obtained in [Chambolle-Lamboley-Lemenant-Stepanov 17], [Bulanyi-Lemenant 20]. The key feature is that the $C^{1,\alpha }$ regularity of minimizers for some free boundary type problem is investigated with a free boundary set of codimension $N-1$. We prove that every optimal set cannot contain closed loops, and it is $C^{1,\alpha }$ regular at ${\mathcal{H}}^1$-a.e. point for every $p\in (N-1,+\infty )$.

42 pages, 2 figures. arXiv admin note: text overlap with arXiv:1911.09240

References in corpus (2)

• Approximation of length minimization problems among compact connected sets
• On the importance of the connectedness assumption in the statement of the optimal $p$-compliance problem