Comparison results for solutions of Poisson equations with Robin boundary on complete Riemannian manifolds. (English) Zbl 1525.53046
Authors’ abstract: In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove
comparison results for the solutions of Poisson equations on complete Riemannian manifolds with
Ric\(\ge(n-1)\kappa\), \(\kappa =0\) or \(1\), which extends the results in [A. Alvino et al., Commun. Pure Appl. Math. 76, No. 3, 585–603 (2023; Zbl 1525.35076)]. Furthermore, as applications of our comparison results, we
obtain the Saint-Venant inequality and Bossel-Daners inequality for Robin Laplacian
Reviewer: Eleutherius Symeonidis (Bucureşti)
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35R01 | PDEs on manifolds |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
Keywords:
Poisson equations; Robin Laplacian; Bossel-Daners inequality; isoperimetric inequality; Talenti comparisonCitations:
Zbl 1525.35076References:
| [1] | Agostiniani, V., Fogagnolo, M. and Mazzieri, L., Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature, Invent. Math.222(3) (2020) 1033-1101. · Zbl 1467.53062 |
| [2] | Alvino, A., Lions, P.-L. and Trombetti, G., Comparison results for elliptic and parabolic equations via Schwarz symmetrization, Ann. Inst. H. Poincaré C Anal. Non Linéaire7(2) (1990) 37-65. · Zbl 0703.35007 |
| [3] | Alvino, A., Nitsch, C. and Trombetti, C., A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions, Comm. Pure Appl. Math.76(3) (2023) 585-603. · Zbl 1525.35076 |
| [4] | Amato, V., Gentile, A. and Masiello, A. L., Comparison results for solutions to \(p\)-Laplace equations with Robin boundary conditions, Ann. Mat. Pura Appl. (4)201(3) (2022) 1189-1212. · Zbl 1491.35247 |
| [5] | Baernstein, A., Drasin, D. and Laugesen, R. S., Symmetrization in Analysis, With a foreword by Walter Hayman, , Vol. 36 (Cambridge University Press, Cambridge, 2019). · Zbl 1509.32001 |
| [6] | Balogh, Z. M. and Kristály, A., Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature, Math. Ann.385(3-4) (2023) 1747-1773. · Zbl 1514.53079 |
| [7] | Bossel, M. H., Membranes élastiquement liées: Extension du théorème de Rayleigh-Faber-Krahn et de l’inégalité de Cheeger, C. R. Acad. Sci. Paris Sér. I Math.302(1) (1986) 47-50. · Zbl 0606.73018 |
| [8] | Brendle, S., Sobolev inequalities in manifolds with nonnegative curvature, Comm. Pure. Appl. Math. (2022), https://doi.org/10.1002/cpa.22070 · Zbl 1527.53030 |
| [9] | Bucur, D. and Daners, D., An alternative approach to the Faber-Krahn inequality for Robin problems, Calc. Var. Partial Differential Equations37(1-2) (2010) 75-86. · Zbl 1186.35118 |
| [10] | Bucur, D. and Giacomini, A., The Saint-Venant inequality for the Laplace operator with Robin boundary conditions, Milan J. Math.83(2) (2015) 327-343. · Zbl 1334.35183 |
| [11] | Chen, D. G., Cheng, Q.-M and Li, H., Faber-Krahn inequalities for the Robin Laplacian on bounded domain in Riemannian manifolds, J. Differential Equations336 (2022) 374-386. · Zbl 1502.53060 |
| [12] | Chen, D. G. and Li, H., Talenti’s comparison theorem for Poisson equation and applications on Riemannian manifold with nonnegative Ricci curvature, J. Geom. Anal.33(4) (2023) 123. · Zbl 1519.53028 |
| [13] | Chiacchio, F., Gavitone, N., Nitsch, C. and Trombetti, C., Sharp estimates for the Gaussian torsional rigidity with Robin boundary conditions, Potential Anal. (2022), https://doi.org/10.1007/s11118-022-09995-8 · Zbl 1525.35092 |
| [14] | Colladay, D., Langford, J. and McDonald, P., Comparison results, exit time moments, and eigenvalues on Riemannian manifolds with a lower Ricci curvature bound, J. Geom. Anal.28(4) (2018) 3906-3927. · Zbl 1410.58016 |
| [15] | Dai, Q. Y. and Fu, Y. X., Faber-Krahn inequality for Robin problems involving \(p\)-Laplacian, Acta Math. Appl. Sin. Engl. Ser.27(1) (2011) 13-28. · Zbl 1209.35093 |
| [16] | Daners, D., A Faber-Krahn inequality for Robin problems in any space dimension, Math. Ann.333 (2006) 767-785. · Zbl 1220.35103 |
| [17] | Fogagnolo, M. and Mazzieri, L., Minimising hulls, \(p\)-capacity and isoperimetric inequality on complete Riemannian manifolds, J. Funct. Anal.283(9) (2022) 109638. · Zbl 1495.49025 |
| [18] | Gamara, N., Hasnaoui, A. and Makni, A., Torsional rigidity on compact Riemannian manifolds with lower Ricci curvature bounds, Open Math.13(1) (2015) 557-570. · Zbl 1515.35168 |
| [19] | M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Modern Birkhäuser Classics, Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates, Reprint of the 2001 English edition (Birkhäuser Boston, Boston, MA, 2007). · Zbl 1113.53001 |
| [20] | Kawohl, B., Rearrangements and Convexity of Level Sets in PDE, , Vol. 1150 (Springer-Verlag, Berlin, 1985). · Zbl 0593.35002 |
| [21] | Kesavan, S., Some remarks on a result of Talenti, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)15(3) (1988)453-465 (1989). · Zbl 0703.35047 |
| [22] | Kesavan, S., Symmetrization & Applications, , Vol. 3 (World Scientific Publishing, Hackensack, NJ, 2006). · Zbl 1110.35002 |
| [23] | Mondino, A. and Vedovato, M., A Talenti-type comparison theorem for \(RCD(K,N)\) spaces and applications, Calc. Var. Partial Differential Equations60(4) (2021) 157. · Zbl 1480.53060 |
| [24] | Talenti, G., Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)3(4) (1976) 697-718. · Zbl 0341.35031 |
| [25] | Talenti, G., Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. (4)120 (1979) 160-184. · Zbl 0419.35041 |
| [26] | Talenti, G., Inequalities in rearrangement invariant function spaces, in Nonlinear Analysis, Function Spaces and Applications, Vol. 5 (Prometheus, Prague, 1994), pp. 177-230. · Zbl 0872.46020 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
