Symmetrization of functions and principal eigenvalues of elliptic operators
Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 24, 15 p.

In this paper, we consider shape optimization problems for the principal eigenvalues of second order uniformly elliptic operators in bounded domains of n . We first recall the classical Rayleigh-Faber-Krahn problem, that is the minimization of the principal eigenvalue of the Dirichlet Laplacian in a domain with fixed Lebesgue measure. We then consider the case of the Laplacian with a bounded drift, that is the operator -Δ+v·, for which the minimization problem is still well posed. Next, we deal with more general elliptic operators -div(A)+v·+V , for which the coefficients fulfill various pointwise, integral or geometric constraints. In all cases, some operators with radially symmetric coefficients in an equimeasurable ball are shown to have smaller principal eigenvalues. Whereas the Faber-Krahn proof relies on the classical Schwarz symmetrization, another type of symmetrization is defined to handle the case of general (possibly non-symmetric) operators.

DOI : 10.5802/slsedp.19
Hamel, François 1 ; Nadirashvili, Nikolai 2 ; Russ, Emmanuel 3

1 Aix-Marseille Université & Institut Universitaire de France LATP Avenue Escadrille Normandie-Niemen 13397 Marseille Cedex 20 France
2 CNRS, LATP 39 rue F. Joliot-Curie 13453 Marseille Cedex 13 France
3 Université Joseph Fourier Institut Fourier 100 rue des Maths BP 74 38402 St-Martin d’Hères France
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Hamel, François; Nadirashvili, Nikolai; Russ, Emmanuel. Symmetrization of functions and principal eigenvalues of elliptic operators. Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 24, 15 p. doi : 10.5802/slsedp.19. http://archive.numdam.org/articles/10.5802/slsedp.19/

[1] A. Alvino and G. Trombetti, A lower bound for the first eigenvalue of an elliptic operator, J. Math. Anal. Appl. 94 (1983), 328-337. | MR | Zbl

[2] A. Alvino and G. Trombetti, Isoperimetric inequalities connected with torsion problem and capacity, Boll. Union Mat. Ital. B 4 (1985), 773-787. | MR

[3] A. Alvino, G. Trombetti, P.-L. Lions and S. Matarasso, Comparison results for solutions of elliptic problems via symmetrization, Ann. Inst. Henri Poincaré 16 2 (1999), 167-188. | Numdam | MR | Zbl

[4] M.S. Ashbaugh and R.D. Benguria, A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, Ann. Math. 135 (1992), 601-628. | MR | Zbl

[5] M.S. Ashbaugh and R.D. Benguria, On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions, Duke Math. J. 78 (1995), 1-17. | MR | Zbl

[6] C. Bandle, Isoperimetric Inequalities and Applications, Pitman Monographs and Studies in Math. 7, Boston, 1980. | MR | Zbl

[7] H. Berestycki, L. Nirenberg and S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994), 47-92. | MR | Zbl

[8] F. Brock, An isoperimetric inequality for eigenvalues of the Stekloff problem, Z. Angew. Math. Mech. 81 (2001), 69-71. | MR | Zbl

[9] D. Bucur and A. Henrot, Minimization of the third eigenvalue of the Dirichlet Laplacian, Proc. Royal Soc. London Ser. A 456 (2000), 985-996. | MR | Zbl

[10] S.-Y. Cheng and K. Oden, Isoperimetric inequalities and the gap between the first and second eigenvalues of an Euclidean domain, J. Geom. Anal. 7 (1997), 217-239. | MR | Zbl

[11] D. Daners, A Faber-Krahn inequality for Robin problems in any space dimension, Math. Ann. 335 (2006), 767-785. | MR | Zbl

[12] D. Daners and J. Kennedy, Uniqueness in the Faber-Krahn inequality for Robin problems, SIAM J. Math. Anal. 39 (2007/08), 1191-1207. | MR | Zbl

[13] G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften zu München (1923), 169-172.

[14] F. Hamel, N. Nadirashvili, E. Russ, An isoperimetric inequality for the principal eigenvalue of the Laplacian with drift, C. R. Acad. Sci. Paris Ser. I 340 (2005), 347-352. | MR | Zbl

[15] F. Hamel, N. Nadirashvili, E. Russ, Rearrangement inequalities and applications to isoperimetric problems for eigenvalues, Ann. Math. 174 (2011), 647-755. | MR | Zbl

[16] A. Henrot, Minimization problems for eigenvalues of the Laplacian, J. Evol. Eq. 3 (2003), 443-461. | MR | Zbl

[17] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Birkhäuser, 2006. | MR | Zbl

[18] C.J. Holland, A minimum principle for the principal eigenvalue for second-order linear elliptic equations with natural boundary conditions, Comm. Pure Appl. Math. 31 (1978), 509-519. | MR | Zbl

[19] E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94 (1925), 97-100. | MR

[20] E. Krahn, Über Minimaleigenschaft der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Tartu (Dorpat) A9 (1926), 1-44.

[21] N.S. Nadirashvili, Rayleigh’s conjecture on the principal frequency of the clamped plate, Arch. Ration. Mech. Anal. 129 (1995), 1-10. | MR | Zbl

[22] L.E. Payne, G. Pólya and H.F. Weinberger, On the ratio of consecutive eigenvalues, J. Math. Phys. 35 (1956), 289-298. | MR | Zbl

[23] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies 27, Princeton Univ. Press, Princeton, 1951. | MR | Zbl

[24] J.W.S. Rayleigh, The Theory of Sound, 2 nd ed. revised and enlarged (in 2 vols.), Dover Publications, New York, 1945 (republication of the 1894/1896 edition). | MR | Zbl

[25] G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 343-356. | MR | Zbl

[26] G. Talenti, Linear elliptic P.D.E.’s: level sets, rearrangements and a priori estimates of solutions, Boll. Union Mat. Ital. B 6 (1985), 917-949. | MR | Zbl

[27] G. Trombetti and J. L. Vazquez, A symmetrization result for elliptic equations with lower-order terms, Ann. Fac. Sci. Toulouse 7 (1985), 137-150. | Numdam | MR | Zbl

[28] H.F. Weinberger, An isoperimetric inequality for the N-dimensional free membrane problem, J. Rational Mech. Anal. 5 (1956), 633-636. | MR | Zbl

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