Abstract
The paper is a discussion of Krahn’s proof of the Rayleigh conjecture that amongst all membranes of the same area and the same physical properties, the circular one has the lowest ground frequency. We show how his approach coincides with the modern techniques of geometric measure theory using the co-area formula. We furthermore discuss some issues and generalisations of his proof.
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Ambrosio L., Fusco N., Pallara D.: Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000)
W. Arendt and S. Monniaux, Domain Perturbation for the first Eigenvalue of the Dirichlet Schrödinger Operator, in: M. Demuth and B. Schulze (eds.), Partial Differential Equations and Mathematical Physics, Operator Theory: Advances and Applications, 78, pp. 9–19, Birkhäuser, Basel, Holzau, 1994, 1995.
M. S. Ashbaugh and R. D. Benguria, Isoperimetric inequalities for eigenvalues of the Laplacian, in: Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, Proc. Sympos. Pure Math. 76, pp. 105–139, Amer. Math. Soc., Providence, RI, 2007.
C. Bandle, Isoperimetric inequalities and applications, Monographs and Studies in Mathematics 7, Pitman, Boston, Mass., 1980.
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, 47–50 (1986)
Brothers J.E., Ziemer W.P.: Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384, 153–179 (1988)
D. Bucur and G. Buttazzo, Variational methods in shape optimization problems, Progress in Nonlinear Differential Equations and their Applications 65, Birkhäuser Boston Inc., Boston, MA, 2005.
Bucur D., Daners D.: An alternative approach to the Faber-Krahn inequality for Robin problems. Calc. Var. Partial Differential Equations 37, 75–86 (2010)
Bucur D., Giacomini A.: A variational approach of the isoperimetric inequality for the Robin eigenvalue problem. Arch. Ration. Mech. Anal. 198, 927–961 (2010)
I. Chavel, Isoperimetric inequalities, Cambridge Tracts in Mathematics 145, Cambridge University Press, Cambridge, 2001.
Courant R.: Beweis des Satzes, daß von allen homogenen Membranen gegebenen Umfanges und gegebener Spannung die kreisförmige den tiefsten Grundton besitzt. Math. Z. 1, 321–328 (1918)
Q. Dai and Y. Fu, Faber-Krahn inequality for Robin problem involving p- Laplacian, Preprint (2008), arXiv:0912.0393.
Daners D.: A Faber-Krahn inequality for Robin problems in any space dimension. Math. Ann. 335, 767–785 (2006)
Daners D., Kawohl B.: An isoperimetric inequality related to a Bernoulli problem. Calc. Var. Partial Differential Equations 39, 547–555 (2010)
Daners D., Kennedy J.: Uniqueness in the Faber-Krahn Inequality for Robin Problems. SIAM J. Math. Anal. 39, 1191–1207 (2007)
L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
G. Faber, Beweis dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsbericht der bayerischen Akademie der Wissenschaften, pp. 169–172 (1923).
Federer H.: Curvature measures. Trans. Amer. Math. Soc. 93, 418–491 (1959)
H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften 153, Springer, New York, 1969.
N. Fusco, Geometrical aspects of symmetrization, in: Calculus of variations and nonlinear partial differential equations, Lecture Notes in Math. 1927, pp. 155–181, Springer-Verlag, Berlin, 2008.
Henrot A.: Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006)
B. Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics 1150, Springer-Verlag, Berlin, 1985.
Kennedy J.: An isoperimetric inequality for the second eigenvalue of the Laplacian with Robin boundary conditions. Proc. Amer. Math. Soc. 137, 627–633 (2009)
S. Kesavan, Symmetrization & applications, Series in Analysis 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
Krahn E.: Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann. 94, 97–100 (1925)
E. Krahn, Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Tartu (Dorpat) A 9, 1–44 (1926), English translation in [28, pp 139–175].
G. Leoni, A first course in Sobolev spaces, Graduate Studies in Mathematics 105, American Mathematical Society, Providence, RI, 2009.
Ü. Lumiste and J. Peetre (eds.), Edgar Krahn, 1894–1961, IOS Press, Amsterdam, 1994, a centenary volume.
Maz’ja V.G.: Sobolev spaces, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin (1985)
Osserman R.: The isoperimetric inequality. Bull. Amer. Math. Soc. 84, 1182–1238 (1978)
Payne L.E.: Isoperimetric inequalities and their applications. SIAM Rev. 9, 453–488 (1967)
Pólya G.: Torsional rigidity, principal frequency, electrostatic capacity and symmetrization. Quart. Appl. Math. 6, 267–277 (1948)
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, 27, Princeton University Press, Princeton, N. J., 1951.
Polya G., Weinstein A.: On the torsional rigidity of multiply connected cross-sections. Ann. of Math. (2) 52, 154–163 (1950)
J. W. Rayleigh and B. Strutt, The Theory of Sound, Dover Publications, New York, N. Y., 1945, 2nd ed.
Rayleigh L.: The Theory of Sound, 1st ed. Macmillan, London (1877)
Rudin W.: Real and Complex Analysis, 2nd ed. McGraw-Hill Inc., New York (1974)
Schmidt E.: Über das isoperimetrische Problem im Raum von n Dimensionen. Math. Z. 44, 689–788 (1939)
Sperner E. Jr.: Symmetrisierung für Funktionen mehrerer reeller Variablen. Manuscripta Math. 11, 159–170 (1974)
Talenti G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4) 110, 353–372 (1976)
Tonelli L.: Sur un problème de Lord Rayleigh. Monatsh. Math. Phys. 37, 253–280 (1930)
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Daners, D. Krahn’s proof of the Rayleigh conjecture revisited. Arch. Math. 96, 187–199 (2011). https://doi.org/10.1007/s00013-010-0218-x
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DOI: https://doi.org/10.1007/s00013-010-0218-x