On the dimension of the singular set in optimization problems with measure constraint
Abstract
In this paper, we prove estimates on the dimension of the singular part of the free boundary for solutions to shape optimization problems with measure constraints. The focus is on the heat conduction problem studied by Aguilera, Caffarelli, and Spruck and the one-phase Bernoulli problem with measure constraint introduced by Aguilera, Alt and Caffarelli. To estimate the Hausdorff dimension of the singular set, we introduce a new formulation of the notion of stability for the one-phase problem along volume-preserving variations, which is preserved under blow-up limits. Finally, the result follows by applying the program developed in [Buttazzo et al. 2022] to this class of domain variation.
Cite
@article{arxiv.2310.06591,
title = {On the dimension of the singular set in optimization problems with measure constraint},
author = {Dario Mazzoleni and Giorgio Tortone and Bozhidar Velichkov},
journal= {arXiv preprint arXiv:2310.06591},
year = {2023}
}
Comments
17 pages. arXiv admin note: text overlap with arXiv:2212.09118