A Riemannian Extension of a Quadrature Surface Free Boundary Problem: Stability and Optimality

@misc{djite2026riemannian,
  title = {{A Riemannian Extension of a Quadrature Surface Free Boundary Problem: Stability and Optimality}},
  author = {Ababacar Sadikhe Djite and Diaraf Seck},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29462},
  eprint = {2608.29462},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29462},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - A Riemannian Extension of a Quadrature Surface Free Boundary Problem: Stability and Optimality
AU  - Ababacar Sadikhe Djite
AU  - Diaraf Seck
PY  - 2026
DA  - 2026/08/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.29462
UR  - https://arxiv.org/abs/2608.29462
AB  - We study a quadrature surface free boundary problem on a smooth compact finite-dimensional Riemannian manifold (M,g). The problem is formulated as a shape optimization problem involving a Dirichlet problem for the Laplace--Beltrami operator, with a geometric condition on the free boundary. Under suitable uniform geometric assumptions on the admissible class, we establish its compactness and prove the stability of the corresponding Dirichlet problems, including strong convergence of the associated states in H_0^1(M). We derive the first-order optimality condition for the associated shape functional. Finally, we establish a Riemannian comparison principle for the second fundamental forms and the mean curvatures of tangent boundaries at a contact point. These results provide a rigorous extension of the quadrature surface free boundary framework from the Euclidean setting to compact Riemannian manifolds.
ER  -