The presentation will focus on some new results concerning the regularity for the optimal p-compliance problem with length penalization. In collaboration with Antoine Lemenant, we prove (preprint 2020) that in dimension 2 every solution to the optimal p-compliance problem with length penalization has no loops, is Ahlfors regular and $C^{1,\alpha}$ at $\mathcal{H}^1$-a.e. point for every $p\in (1,+\infty)$, extending some of the results obtained by Antonin Chambolle, Jimmy Lamboley, Antoine Lemenant and Eugene Stepanov (2017). The p-compliance problem can be defined in dimension $N>2$, provided that $p>N-1$, still with a penalization with the one dimensional Hausdorff measure. In this talk I will try to give an overview about the regularity and qualitative properties of minimizers in the general spatial dimension $N \geq 2$.
Maxime Laborde