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

Abstract

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 Djit\'{e} 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 $QS_p(f,k)$ admits a solution strictly containing the totally convex hull $C$ of the support of $f$ if and only if \[ \int_C f(x)\,dv(g) > k^{p-1} |\partial C|_g, \] for $1 < p < \infty$, under suitable regularity and stability assumptions. This extends the Euclidean result of Barkatou for $p=2$ and the Riemannian result of Djit\'{e} 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.