On Quadrature Surface Free Boundary Problems for the Riemannian $p$-Laplacian

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 $$\mathrm{\mid }{\mathrm{\nabla }}_g{u}_{\mathrm{\Omega }}{\mathrm{\mid }}_g^p=\sigma H_{\mathrm{\partial }\mathrm{\Omega }}+k^p\mathrm{.}$$ 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.