On Quadrature Surface Free Boundary Problems for the Riemannian p-Laplacian
Abstract
We study a quadrature-surface free boundary problem driven by the Riemannian p-Laplacian on a smooth compact finite-dimensional Riemannian manifold. We formulate the problem as a shape optimization problem and develop an intrinsic admissible class based on a uniform Riemannian RC-GNP condition~\cite{DS3}. We establish compactness of admissible domains and stability of the associated Dirichlet problems under strong RC-GNP convergence. In particular, the states converge strongly in W1,p and the associated p-torsional energy is continuous. Under additional boundary regularity, we derive the first variation of the shape functional and obtain the free-boundary Euler--Lagrange condition |∇g uΩ|gp=σH∂Ω+kp. We also formulate the contact optimality inequality, prove the Riemannian comparison of second fundamental forms at tangential contact, and obtain a sufficient condition on a reference domain which rules out contact and yields a solution of the free-boundary problem. Explicit radial examples on geodesic balls of the round sphere are included.
Related
- On the p-Laplacian Quadrature Surface Free Boundary Problem on Riemannian Manifolds: Sufficient condition of existence
- A Riemannian Extension of a Quadrature Surface Free Boundary Problem: Stability and Optimality
- Shape Optimization and Bernoulli Free Boundary Problems for the Riemannian p-Laplacian
- On the Quadrature Surface Free Boundary Problem on Riemannian Manifolds: Sufficient Condition of Existence
- Foliated Plateau problems, surface Radon transforms, and boundary area rigidity in dimension three
BibTeX
@misc{djite2026quadrature,
title = {{On Quadrature Surface Free Boundary Problems for the Riemannian $p$-Laplacian}},
author = {Ababacar Sadikhe Djite and Diaraf Seck},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17978},
eprint = {2609.17978},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17978},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Quadrature Surface Free Boundary Problems for the Riemannian p-Laplacian
AU - Ababacar Sadikhe Djite
AU - Diaraf Seck
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.17978
UR - https://arxiv.org/abs/2609.17978
AB - We study a quadrature-surface free boundary problem driven by the Riemannian p-Laplacian on a smooth compact finite-dimensional Riemannian manifold. We formulate the problem as a shape optimization problem and develop an intrinsic admissible class based on a uniform Riemannian RC-GNP condition~\cite{DS3}. We establish compactness of admissible domains and stability of the associated Dirichlet problems under strong RC-GNP convergence. In particular, the states converge strongly in W^1,p and the associated p-torsional energy is continuous. Under additional boundary regularity, we derive the first variation of the shape functional and obtain the free-boundary Euler--Lagrange condition |∇_g u_Ω|_g^p=σH_∂Ω+k^p. We also formulate the contact optimality inequality, prove the Riemannian comparison of second fundamental forms at tangential contact, and obtain a sufficient condition on a reference domain which rules out contact and yields a solution of the free-boundary problem. Explicit radial examples on geodesic balls of the round sphere are included.
ER -