Existence of regular minimizers for the Riemannian quadrature surface problem

Establish whether minimizers of the Riemannian quadrature surface shape functional are regular, specifically of class C^2 with associated Dirichlet solutions in C^2(C), under the assumptions of the quadrature surface free boundary problem.

Background

The paper derives the quadrature surface solution through a shape optimization problem whose minimizers are assumed in the proofs to have C2 boundaries and whose associated Dirichlet solutions are assumed to possess the required C2 regularity. The authors explicitly note that regularity is not established in general.

They indicate that free-boundary regularity theory, such as the Alt–Caffarelli theory adapted to Riemannian manifolds, may apply under additional assumptions on the source function, for example when the source is bounded away from zero. Until such regularity is proved or suitable hypotheses are identified, the result may need to be interpreted in a weak sense.

References

The proofs above assume that the minimizers $\Omega$ and $\Omega*$ are of class $C2$ and that $u_\Omega \in C2(C)$. In general, the existence of regular minimizers is not guaranteed; however, under standard assumptions on $f$ (e.g., $f$ bounded away from zero), one can invoke the free boundary regularity theory (Alt--Caffarelli) adapted to Riemannian manifolds.

— Necessary and Sufficient Condition of Existence for the Quadrature Surfaces Free Boundary Problem in Riemannian Geometry  (2609.08245 - Barkatou, 8 Sep 2026) in Section 7.1, “Regularity Issues”