Comparison Results for Poisson Equation with Mixed Boundary Condition on Manifolds

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 Alvino et al. (J Math Pures Appl 9(152):251-261, 2021) to Riemannian setting, and Chen-Li's result (Talenti's comparison theorem for poisson equation and applications on Riemannian manifold with nonnegative Ricci curvature, 2021. arXiv:2104.05568) to the case of variable Robin parameter.