Comparison results for Poisson equation with mixed boundary condition on manifolds

TL;DR

This paper establishes new estimates and comparison results for solutions to the Poisson equation with mixed boundary conditions on various types of manifolds, extending previous theorems to more general Riemannian settings.

Contribution

It generalizes key theorems to Riemannian manifolds and introduces new $L^1$ estimates and comparison results for the Poisson equation with mixed boundary conditions.

Findings

Established $L^1$ estimates for Poisson solutions on manifolds.

Obtained Talenti-type comparison on Riemann surfaces.

Extended Chen-Li's result to variable Robin parameters.

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.