Abstract
We discuss a Schiffer’s conjecture which is a symmetry problem for an overdetermined spectral p.d.e.. We show the connection between this problem and the critical points of the eigenvalue with a volume constraint as well as the Faber-Krahn inequality. We give two original proofs of these symmetry results in the case of the first eigenvalue.
Chapter PDF
References
Brock, F. (1995) Continuous Steiner Symmetrization. Math. Nachrichten, 172, 25–48.
Brock, F. and Henrot, A. (to appear) A symmetry result for overdetermined boundary value problem using domain derivative and continuous Steiner symmetrization. Chatelain, T. and Henrot, A. (work in progress).
Cox, S.J. (1994) Extremal eigenvalue problems for the Laplacian. M. A. Herrero, E. Zuazua eds, R.A.M. J. Wiley and Masson, 1994.
Kawohl, B. (1985) Rearrangements and Convexity of level sets in Partial Differential Equations. Springer Lecture Notes in Math., 1150.
Mignot, F., Murat, and Puel, J.P. (1979) Variation d’un point de retournement par rapport au domaine. Comm. in p.d.e., 4, 11, 1263–1297.
Rellich, F. (1940) Darstellung der eigenwerk Du + Au durch ein randintegral. Math. Z., 46, 635–646.
Rousselet, B. and Chesnais, D. (1990) Continuité et différentiabilité d’éléments propres: application à l’optimisation de structures. Appl. Math. and Optim., 22, 27–59.
Sanchez-Hubert, J. and Sanchez-Palencia, E. Vibration and coupling of continuous systems: asymptotic methods. Springer-Verlag.
J. Serrin, J. (1971) A symmetry problem in potential theory, Arch. Rat. Mech. Anal., 43, 304–318.
Simon, J. (1980) Differentiation with respect to the domain in boundary value problems. Num. Funct. Anal. Optimiz., 2 (7,8), 649–689.
Sokolowski, J. and Zolesio, J.P. (1992) Introduction to shape optimization: shape sensitivity analysis. Springer Series in Computational Mathematics, vol 10, Springer.
L. Zalcman, L. (1992) A bibliographical survey of the Pompeiu problem, in Approximation by solutions of p.d.e., B. Fuglede et al. eds., Kluwer.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Chatelain, T., Choulli, M., Henrot, A. (1996). Some new ideas for a Schiffer’s conjecture. In: Malanowski, K., Nahorski, Z., Peszyńska, M. (eds) Modelling and Optimization of Distributed Parameter Systems Applications to engineering. IFIP — The International Federation for Information Processing. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-34922-0_7
Download citation
DOI: https://doi.org/10.1007/978-0-387-34922-0_7
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-5864-1
Online ISBN: 978-0-387-34922-0
eBook Packages: Springer Book Archive