Abstract
Let (M, g) be a Riemannian manifold with a distinguished point O and assume that the geodesic distance d from O is an isoparametric function. Let \(\varOmega \subset M\) be a bounded domain, with \(O \in \varOmega \), and consider the problem \(\varDelta _p u = -1\ \mathrm{in}\ \varOmega \) with \(u=0\ \mathrm{on}\ \partial \varOmega \), where \(\varDelta _p\) is the p-Laplacian of g. We prove that if the normal derivative \(\partial _{\nu }u\) of u along the boundary of \(\varOmega \) is a function of d satisfying suitable conditions, then \(\varOmega \) must be a geodesic ball. In particular, our result applies to open balls of \(\mathbb {R}^n\) equipped with a rotationally symmetric metric of the form \(g=dt^2+\rho ^2(t)\,g_S\), where \(g_S\) is the standard metric of the sphere.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Babaoglu, C., Shahgholian, H.: Symmetry in multi-phase overdetermined problems. J. Convex Anal. 18, 1013–1024 (2011)
Brandolini, B., Nitsch, C., Salani, P., Trombetti, C.: Serrin type overdetermined problems: an alternative proof. Arch. Rational Mech. Anal. 190, 267–280 (2008)
Choulli, M., Henrot, A.: Use of the domain derivative to prove symmetry results in partial differential equations. Math. Nachr. 192, 91–103 (1998)
Ciraolo, G., Vezzoni, L.: A rigidity problem on the round sphere. arXiv:1512.07749
Di Benedetto, E.: \(C^{1,\alpha }\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827–850 (1983)
Enciso, A., Peralta-Salas, D.: A symmetry result for the \(p\)-Laplacian in a punctured manifold. J. Math. Anal. Appl. 354, 619–624 (2009)
Farina, A., Mari, L., Valdinoci, E.: Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds. Commun. P.D.E. 38 1818–1862 (2013)
Farina, Alberto, Sire, Yannick, Valdinoci, Enrico: Stable solutions of elliptic equations on Riemannian manifolds. J. Geom. Anal. 23, 1158–1172 (2013)
Farina, A., Valdinoci, E.: A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete Contin. Dyn. Syst. 30, 1139–1144 (2011)
Garofalo, N., Lewis, J.L.: A symmetry result related to some overdetermined boundary value problems. Am. J. Math. 111, 9–33 (1989)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1998)
Greco, A.: Symmetry around the origin for some overdetermined problems. Adv. Math. Sci. Appl. 13, 383–395 (2003)
Greco, A.: Constrained radial symmetry for monotone elliptic quasilinear operators. J. Anal. Math. 121, 223–234 (2013)
Heinonen, J., Kilpeläinen, T., Martio, O.: NonlineaR Potential Theory of Degenerate Elliptic Equations. Dover Publications Inc, Mineola (2006)
Hopf, E.: A remark on linear elliptic differential equations of second order. Proc. Am. Math. Soc. 3, 791–793 (1952)
Lewis, J.L.: Regularity of the derivatives of solutions to certain degenerate elliptic equations. Indiana Univ. Math. J. 32, 849–858 (1983)
Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)
Pucci, P., Serrin, J.: The strong maximum principle revisited. J. Differ. Equ. 196, 1–66 (2004)
Rosenberg, S.: The Laplacian on a Riemannian manifold. An introduction to analysis on manifolds. Lond. Math. Soc. Stud. Texts 31, X+172 (1997). Cambridge University Press, Cambridge
Serrin, J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304–318 (1971)
Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)
Vogel, A.L.: Symmetry and regularity for general regions having a solution to certain overdetermined boundary value problems. Atti Semin. Mat. Fis. Univ. Modena 50, 443–484 (1992)
Wang, Q.M.: Isoparametric functions on Riemannian manifolds. Math. Ann. 277, 639–646 (1987)
Weinberger, H.F.: Remark on the preceding paper of Serrin. Arch. Rational Mech. Anal. 43, 319–320 (1971)
Acknowledgments
The second author is grateful to the organizers of “Geometric Properties for Parabolic and Elliptic PDE’s 4th Italian-Japanese Workshop” for the invitation and the very kind hospitality during the workshop.
This work was partially supported by the project PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica”, the projects FIRB “Differential Geometry and Geometric functions theory” and “Geometrical and Qualitative aspects of PDE”, and GNSAGA and GNAMPA (INdAM) of Italy.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Ciraolo, G., Vezzoni, L. (2016). A Remark on an Overdetermined Problem in Riemannian Geometry. In: Gazzola, F., Ishige, K., Nitsch, C., Salani, P. (eds) Geometric Properties for Parabolic and Elliptic PDE's. GPPEPDEs 2015. Springer Proceedings in Mathematics & Statistics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-41538-3_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-41538-3_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-41536-9
Online ISBN: 978-3-319-41538-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)