Abstract
In this paper, we provide comparisons of solutions to nonlinear parabolic equations on complete manifolds with \(\mathrm Ric\ge (n-1)\kappa \), where \(\kappa =0\) or 1, which extend the results of Cheng et al. (Schwarz symmetrizations in parabolic equations on complete manifolds, 2021) and Vázquez (C R Acad Sci Paris Ser I Math 295(2):71–74, 1982). As an application, we give an alternate proof to Faber–Krahn inequality for Dirichlet Laplacian on these manifolds, which was proved in Balogh and Kristály (Math Ann https://doi.org/10.1007/s00208-022-02380-1, 2022) [see also Chen and Li (J Geom Anal 33(4):123, 2023) and Fogagnolo and Mazzieri (J Funct Anal 283:109638, 2022)]. Additionally, we also obtain corresponding comparison results on minimal submanifolds in Riemannian manifolds with nonnegative sectional curvature.
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19 March 2023
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Funding
This work was partially supported by the National Natural Science Foundation of China (NSFC grant No. 11831005) and the Cooperative Programme developed by the National Natural Science Foundation of China (NSFC) and the Research Foundation - Flanders (FWO) (NSFC-FWO 11961131001).
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Chen, D., Wei, Y. Comparison Results for Filtration Equations on Manifolds via Schwarz Rearrangements. Results Math 78, 81 (2023). https://doi.org/10.1007/s00025-023-01841-6
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DOI: https://doi.org/10.1007/s00025-023-01841-6