Abstract
It is proven that among all regions with given value for the maximum of the torsion function, the first eigenvalue of the Laplacian under Robin boundary condition is a minimum for an infinite slab. This holds provided the mean curvature of the boundary is nonnegative.
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
Preview
Unable to display preview. Download preview PDF.
References
M. H. Bossel, Longeurs extrémales et fonctionelles de domaine. Complex Variables 6 (1986), 203–234.
M.H. Bossel, Membranes élastiquement liées inhomogènes ou sur une surface: une nouvelle extension du théorème isopérimetrique de Rayleigh-FaberKrahn. ZAMP 39 (1988), 733–742.
G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton hat. Sitz.ber. bayer. Akad. Wiss. (1923), 169–172.
J. Hersch, Sur la fréquence fondamental d’une membrane vibrante; évaluation par défaut et principe de maximum. J. Math. Phys. Appl. 11 (1960), 387–413.
E. Krahn, Uber Minimaleigenschaften der Kugel in drei und mehr Dimensionen. Acta. Comm. Univ. Tartu (Dorpat) A9 (1926), 1–144.
L.E. Payne, Isoperimetric inequalities and their applications. SIAM Rev. 9 (1967), 453–488.
L.E. Payne, Bounds for the maximum stress in the Saint Venant torsion problem. Indian J. Mech. Math., special issue (1968), 51–59.
L.E. Payne, Bounds for solutions of a class of quasilinear elliptic boundary value problems in terms of the torsion function. Proc. Roy. Soc. Edinborough 88A (1981), 251–265.
L.E. Payne and I. Stakgold, On the mean value of fundamental mode in the fixed membrane problem. Applicable Anal. 3 (1971), 295–303.
M. H. Protter, Lower bounds for the first eigenvalue of elliptic equations. Ann. of Math. 71 (1960), 423–444.
R.P. Sperb, Maximum principles and their applications. Mathematics in Science and Engineering 157, Academic Press (1981.)
R. P. Sperb, Optimal bounds for the critical value in a semilinear boundary value problem on a surface. General ineq. 5, ISNM 80 (1987), 391–400.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Basel AG
About this chapter
Cite this chapter
Sperb, R.P. (1992). An isoperimetric inequality for the first eigenvalue of the Laplacian under Robin boundary conditions. In: Walter, W. (eds) General Inequalities 6. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 103. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7565-3_28
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7565-3_28
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7567-7
Online ISBN: 978-3-0348-7565-3
eBook Packages: Springer Book Archive