On the Quadrature Surface Free Boundary Problem on Riemannian Manifolds: Sufficient Condition of Existence
Abstract
We study the quadrature surface free boundary problem on a compact Riemannian manifold (M,g) of dimension N ≥ 2. Given a positive source f ∈ L2g(M) with compact support K, and k>0, one seeks a domain Ω⊃ K on which the overdetermined problem −Δg u = f in Ω, u=0 and |∇g u|g = k on ∂Ω admits a solution. Following the geometric and variational framework of Djité--Seck, we prove that the integral condition (NS)g ∫C f dv(g) > k |∂ C|g, where C is the totally convex hull of K, is \emph{sufficient} for the existence of a solution strictly containing C. We then show, by an explicit counterexample on the round sphere SnR, that (NS)g is \emph{not necessary} in general under nonnegative curvature: the phenomenon arises because the perimeter is not monotone with respect to inclusion in positive curvature. Finally, we identify an additional geometric hypothesis---perimeter monotonicity within a normal convex neighborhood of C---under which necessity is restored, and we discuss the critical equality case ∫C f = k|∂ C|g.
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BibTeX
@misc{barkatou2026quadrature,
title = {{On the Quadrature Surface Free Boundary Problem on Riemannian Manifolds: Sufficient Condition of Existence}},
author = {Mohammed Barkatou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08245},
eprint = {2609.08245},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08245},
note = {CC0},
}
RIS
TY - UNPB
TI - On the Quadrature Surface Free Boundary Problem on Riemannian Manifolds: Sufficient Condition of Existence
AU - Mohammed Barkatou
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.08245
UR - https://arxiv.org/abs/2609.08245
AB - We study the quadrature surface free boundary problem on a compact Riemannian manifold (M,g) of dimension N ≥ 2. Given a positive source f ∈ L^2_g(M) with compact support K, and k>0, one seeks a domain Ω⊃ K on which the overdetermined problem −Δ_g u = f in Ω, u=0 and |∇_g u|_g = k on ∂Ω admits a solution. Following the geometric and variational framework of Djité--Seck, we prove that the integral condition (NS)_g ∫_C f dv(g) > k |∂ C|_g, where C is the totally convex hull of K, is \emph{sufficient} for the existence of a solution strictly containing C. We then show, by an explicit counterexample on the round sphere S^n_R, that (NS)_g is \emph{not necessary} in general under nonnegative curvature: the phenomenon arises because the perimeter is not monotone with respect to inclusion in positive curvature. Finally, we identify an additional geometric hypothesis---perimeter monotonicity within a normal convex neighborhood of C---under which necessity is restored, and we discuss the critical equality case ∫_C f = k|∂ C|_g.
ER -