# Bienvenue - Laboratoire Jacques-Louis Lions

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Chiffres janvier 2022

### Bogdan Bulanyi

Lundi 23 mars 2020

Bogdan Bulanyi (Université Paris Diderot)
Regularity for the optimal p-compliance problem

Résumé :
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 2019) 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 $\mathcalH^1$-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 of minimizers in dimension 2 and in higher dimensions in which the Ahlfors regularity, absence of loops and a partial $C^1,\alpha$ regularity for solutions are also valid.