On the Quadrature Surface Free Boundary Problem on Riemannian Manifolds: Sufficient Condition of Existence

Abstract

We study the quadrature surface free boundary problem on a compact Riemannian manifold $(\M,g)$ of dimension $N \geq 2$. Given a positive source $f \in L^2_g(\M)$ with compact support $K$, and $k>0$, one seeks a domain $\Omega \supset K$ on which the overdetermined problem $-\Delta_g u = f$ in $\Omega$, $u=0$ and $|\nabla_g u|_g = k$ on $\partial\Omega$ admits a solution. Following the geometric and variational framework of Djit\'e--Seck, we prove that the integral condition \[ (NS)_g \qquad \int_C f\,dv(g)>k\,|\partial C|_g, \] where $C$ is the totally convex hull of $K$, is \emph{sufficient} for the existence of a solution strictly containing $C$. We then show, by an explicit counterexample on the round sphere $S^n_R$, that $(NS)_g$ is \emph{not necessary} in general under nonnegative curvature: the phenomenon arises because the perimeter is not monotone with respect to inclusion in positive curvature. Finally, we identify an additional geometric hypothesis---perimeter monotonicity within a normal convex neighborhood of $C$---under which necessity is restored, and we discuss the critical equality case $\int_C f = k|\partial C|_g$.