Necessary and Sufficient Condition of Existence for the $p$-Laplacian Quadrature Surfaces Free Boundary Problem in Riemannian Geometry

We generalize the necessary and sufficient condition of existence for the quadrature surfaces free boundary problem to the case of the $p$-Laplacian operator in Riemannian geometry. Following the shape optimization approach of Barkatou for the Euclidean case and the Riemannian framework recently developed by Djité and Seck, we formulate the $p$-Laplacian quadrature surface problem as a shape optimization problem on a compact Riemannian manifold. Using the Riemannian $RC$-GNP condition, the stability results, and the $p$-Laplacian shape derivative formula from \cite{DjiteSeck2026b}, we prove that the $p$-Laplacian quadrature surface problem $Q{S}_p(f,k)$ admits a solution strictly containing the totally convex hull $C$ of the support of $f$ if and only if $${\int }_{C}f(x)dv(g)>k^{p-1}\mathrm{\mid }\mathrm{\partial }C{\mathrm{\mid }}_g,$$ for $1<p<\mathrm{\infty }$, under suitable regularity and stability assumptions. This extends the Euclidean result of Barkatou for $p=2$ and the Riemannian result of Djité and Seck to the nonlinear $p$-Laplacian setting. Explicit examples on the round sphere are provided, including the computation of the radial $p$-harmonic solution and the boundary flux.