Comparison results for Poisson equation with mixed boundary condition on manifolds

Abstract: In this article, we establish a $L^1$ estimate for solutions to Poisson equation with mixed boundary condition, on complete noncompact manifolds with nonnegative Ricci curvature and compact manifolds with positive Ricci curvature respectively. On Riemann surfaces we obtain a Talenti-type comparison. Our results generalize main theorems in [2] to Riemannian setting, and Chen-Li's result [8] to the case of variable Robin parameter.