On the Dimension of the Singular Set in Optimization Problems with Measure Constraint

We prove estimates on the dimension of the singular part of the free boundary for solutions to shape optimization problems with measure constraint. 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 recently published by G. Buttazzo, F. P. Maiale, D. Mazzoleni, G. Tortone and B. Velichkov [Regularity of the optimal sets for a class of integral shape functionals, arxiv 2212.09118 (2022)] to this class of domain variation.