Calculus of Variations and Geometric Measure Theory

D. Mazzoleni - G. Tortone - B. Velichkov

On the dimension of the singular set in optimization problems with measure constraint

created by mazzoleni on 11 Oct 2023



Inserted: 11 oct 2023
Last Updated: 11 oct 2023

Year: 2023

ArXiv: 2310.06591 PDF


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.