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Journal of Convex Analysis 31 (2024), No. 2, 689--708
Copyright Heldermann Verlag 2024

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

Dario Mazzoleni
Dipartimento di Matematica, University of Pavia, Pavia, Italy

Giorgio Tortone
Dipartimento di Matematica, UniversitÓ di Pisa, Pisa, Italy

Bozhidar Velichkov
Dipartimento di Matematica, UniversitÓ di Pisa, Pisa, Italy

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.

Keywords: Free boundary regularity, optimal shapes, one-phase Bernoulli problem, global stable solutions, dimension reduction, critical dimension.

MSC: 35R35, 49Q10, 35B65, 49N60, 35N25.

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