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 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.
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Acknowledgements
We thank Professor Ying Zhang for his encouragement and support, and are grateful to Professors Daguang Chen and Carlo Nitsch for some helpful discussions and comments. We also would like to thank the anonymous referees for catching some typos, which improved the readability of this paper.
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Part of this work is supported by NSFC No.12171345.
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Cheng, H., Ma, T. & Wang, K. Comparison Results for Poisson Equation with Mixed Boundary Condition on Manifolds. Results Math 78, 16 (2023). https://doi.org/10.1007/s00025-022-01795-1
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DOI: https://doi.org/10.1007/s00025-022-01795-1