Talenti comparison results and rigidity results for anisotropic p-Laplacian operator with Robin boundary conditions

In this paper, we first establish the Talenti comparison principle for anisotropic p-Laplacian equation with Robin boundary conditions. This achievement not only extends classical Talenti comparison result for Laplacian equation with Robin boundary conditions in (A. Alvino, C. Nitsch, and C. Trombetti, “A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions,” Commun. Pure Appl. Math., vol. 76, no. 3, pp. 585–603, 2023, doi:10.1002/cpa.22090), but also for the anisotropic Laplacian cases in (R. Sannipoli, “Comparison results for solutions to the anisotropic Laplacian with Robin boundary conditions,” Nonlinear Anal., vol. 214, no. 112615, 2022, doi:10.1016/j.na.2021.112615). Furthermore, we also obtain the rigidity for this kind of Talenti comparison principle. This naturally covers the rigidity for Talenti comparison principle involving anisotropic Laplacian operator which has not been proved before. As an application of rigidity result for anisotropic Talenti comparison principle, we furthermore establish the rigidity of anisotropic Faber–Krahn inequality with Robin boundary conditions and naturally solve the rigidity for classical Faber–Krahn inequality with Robin boundary condition established by Alvino-Nitsch-Trombetti in (A. Alvino, C. Nitsch, and C. Trombetti, “A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions,” Commun. Pure Appl. Math., vol. 76, no. 3, pp. 585–603, 2023, doi:10.1002/cpa.22090). Significantly, our results also answer some open problems contained in (A. L. Masiello and G. Paoli, “A rigidity result for the Robin torsion problem,” J. Geom. Anal., vol. 33, no. 5, p. 149, 2023, doi:10.1007/s12220-023-01202-3) and (A. L. Masiello and G. Paoli, “Rigidity results for the p-Laplacian poisson problem with robin boundary conditions,” J. Optim. Theor. Appl., vol. 202, pp. 628–648, 2024, doi:10.1007/s10957-024-02442-1).