Isoperimetric inequalities for eigenvalues of the Laplacian and the Schrödinger operator
Tóm tắt
The purpose of this manuscript is to present a series of lecture notes on isoperimetric inequalities for the Laplacian, for the Schrödinger operator, and related problems.
Tài liệu tham khảo
Andrews B., Clutterbuck J.: Proof of the fundamental gap conjecture. J. Am. Math. Soc. 24, 899–916 (2011)
Antunes, P.R.S.: On the buckling eigenvalue Problem. J. Phys. A Math. Theor. 44, 215205(13) (2011)
Ashbaugh, M.S.: Isoperimetric and universal inequalities for eigenvalues. In: Davies B., Safarov Yu. (eds.) Spectral Theory and Geometry (Edinburgh, 1998), E. London Math. Soc. Lecture Notes, vol. 273, pp. 95–139. Cambridge University Press, Cambridge (1999)
Ashbaugh, M.S.: Open problems on eigenvalues of the Laplacian. In: Rassias, Th.M., Srivastava, H.M. (eds.) Analytic and Geometric Inequalities and Applications, Mathematics and Its Applications, 478, pp. 13–28. Kluwer Academic Publishers, Dordrecht (1999) [see also M. S. Ashbaugh contribution (pp. 2–10) to ESI-Workshop on Geometrical Aspects of Spectral Theory (Matrei, Austria, 1999), L. Friedlander and T. HoffmannOstenhof, editors, 33 pp., available electronically at http://www.esi.ac.at/ESIPreprints.html, preprint no. 768]
Ashbaugh M.S.: The universal eigenvalue bounds of Payne–Pólya–Weinberger, Hile–Protter, and H. C. Yang, In: Spectral and Inverse Spectral Theory (Goa, 2000). Proc. Indian Acad. Sci. Math. Sci., vol. 112, 3–30 (2002)
Ashbaugh M.S.: The Fundamental Gap (2006) http://www.aimath.org/WWN/loweigenvalues/
Ashbaugh M.S., Benguria R.D.: Log-concavity of the ground state of Schrödinger operators: a new proof of the Baumgartner–Grosse–Martin inequality. Phys. Lett. A 131, 273–276 (1988)
Ashbaugh M.S., Benguria R.D.: Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single well potentials. Proc. Am. Math. Soc. 105, 419–424 (1989)
Ashbaugh M.S., Benguria R.D.: Proof of the Payne–Pólya–Weinberger conjecture. Bull. Am. Math. Soc. 25, 19–29 (1991)
Ashbaugh M.S., Benguria R.D.: A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions. Ann. Math. 135, 601–628 (1992)
Ashbaugh M.S., Benguria R.D.: A second proof of the Payne–Pólya–Weinberger conjecture. Commun. Math. Phys. 147, 181–190 (1992)
Ashbaugh M.S., Benguria R.D.: Sharp upper bound to the first nonzero Neumann eigenvalue for bounded domains in spaces of constant curvature. J. Lond. Math. Soc. 52(2), 402–416 (1995)
Ashbaugh M.S., Benguria R.D.: On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions. Duke Math. J. 78, 1–17 (1995)
Ashbaugh M.S., Benguria R.D.: A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of . Trans. Am. Math. Soc. 353, 1055–1087 (2001)
Ashbaugh, M.S., Benguria, R.D.: Isoperimetric inequalities for eigenvalues of the Laplacian. In: Gesztesy, F., Deift, P., Galvez, C., Perry, P., Schlag, W. (eds.) Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, Proceedings of Symposia in Pure Mathematics, vol. 76, Part 1, pp. 105–139. Amer. Math. Soc., Providence, RI (2007)
Ashbaugh, M.S., Benguria, R.D.: El problema de la Reina Dido: Panorama sobre los problemas de la isoperimetri a, Joven Matemático, vol. 1, 3–8 (2010). [The original English version of this essay, Dido and Isoperimetry, can be found on the web page of the International Conference on the isoperimetric problem of Queen Dido and its mathematical ramifications, that was held in Carthage, Tunisia, May 24–29, 2010; see, http://math.arizona.edu/~dido/didon.html a french translation, by Jacqueline Fleckinger–Pellé, can be found in the same web page]
Ashbaugh, M.S., Benguria, R.D., Laugesen R.S.: Inequalities for the first eigenvalues of the clamped plate and buckling problems. in General inequalities 7, (Oberwolfach, 1995), pp. 95–110. Internat. Ser. Numer. Math., vol. 123. Birkhäuser, Basel (1997)
Ashbaugh M.S., Laugesen R.S.: Fundamental tones and buckling loads of clamped plates. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23, 383–402 (1996)
Baernstein II, A.: A unified approach to symmetrization, In: Partial Differential Equations of Elliptic Type, Cortona, 1992, Sympos. Math., vol. XXXV, pp. 47–91. Cambridge University Press, Cambridge, UK (1994)
Bandle C.: Isoperimetric inequalities and applications, Pitman monographs and studies in mathematics, vol. 7. Pitman, Boston (1980)
Bañuelos R., Kröger P.: Gradient estimates for the ground state Schrödinger eigenfunction and applications. Commun. Math. Phys. 224, 545–550 (2001)
Baumgartner B., Grosse H., Martin A.: The Laplacian of the potential and the order of energy levels. Phys. Lett. B 146, 363–366 (1984)
Benguria R.D., Linde H.: A second eigenvalue bound for the Dirichlet Schrödinger operator. Commun. Math. Phys. 267, 741–755 (2006)
Benguria R.D., Linde H.: A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space. Duke Math. J. 140, 245–279 (2007)
Benguria, R.D., Linde, H.: Isoperimetric inequalities for eigenvalues of the Laplace operator. In: Villegas-Blas, C. (ed.) Fourth summer school in analysis and mathematical physics: topics in spectral theory and quantum mechanics. Contemporary Mathematics (AMS), vol. 476, pp. 1–40 (2008)
Benguria, R.D., Loss, M.: Connection between the Lieb–Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane. In: Conca, C., Manásevich, R., Uhlmann, G., Vogelius, M.S. (eds.) Partial Differential Equations and Inverse Problems. Contemp. Math., vol. 362, pp. 53–61. Amer. Math. Soc., Providence, R.I. (2004)
Bérard P.: Transplantation et isospectralité I. Math. Ann. 292, 547–559 (1992)
Bérard P.: Transplantation et isospectralité II. J. Lond. Math. Soc. 48, 565–576 (1993)
Bérard P.: Domaines plans isospectraux a la Gordon–Web–Wolpert: une preuve elementaire. Afr. Math. 1, 135–146 (1993)
van den Berg M.: On condensation in the free-boson gas and the spectrum of the Laplacian. J. Stat. Phys. 31, 623–637 (1983)
Bernstein, J., Breiner, C.: A variational characterization of the catenoid. (2010) (preprint)
Birman M.S., Solomjak M.Z.: Spectral theory of self-adjoint operators in Hilbert Space. D. Reidel Publishing Company, Dordrecht (1987)
Borell C.: The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30, 207–211 (1975)
Brascamp J., Lieb E.H.: On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22, 366–389 (1976)
Brock F., Solynin A.Y.: An approach to symmetrization via polarization. Trans. Am. Math. Soc. 352, 1759–1796 (2000)
Brooks R.: Constructing isospectral manifolds. Am. Math. Mon. 95, 823–839 (1988)
Burchard A., Thomas L.E.: On the Cauchy problem for a dynamical Euler’s elastica. Commun. Partial Differ. Equ. 28, 271–300 (2003)
Burchard A., Thomas L.E.: On an isoperimetric inequality for a Schrödinger operator depending on the curvature of a loop. J. Geom. Anal. 15, 543–563 (2005)
Chasman, L.M.: Isoperimetric problem for eigenvalues of free plates. Ph. D thesis, University of Illinois at Urbana–Champaign (2009)
Chasman L.M.: An isoperimetric inequality for fundamental tones of free plates. Commun. Math. Phys. 303, 421–429 (2011)
Chasman L.M.: Vibrational modes of circular free plates under tension. Appl. Anal. 90, 1877–1895 (2011)
Chavel I.: Eigenvalues in Riemannian geometry. Academic Press, NY (1984)
Courant R., Hilbert D.: Methods of Mathematical Physics, vol. 1. Interscience Publishers, New York (1953)
Daners D.: A Faber–Krahn inequality for Robin problems in any space dimension. Math. Ann. 335, 767–785 (2006)
Davies E.B.: Heat kernels and spectral theory, paperback edn. Cambridge University Press, Cambridge (1990)
Davies E.B.: Spectral theory and differential operators. Cambridge University Press, Cambridge (1996)
Davis B.: On the spectral gap for fixed membranes. Ark. Mat. 39, 65–74 (2001)
Duclos P., Exner P.: Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7, 73–102 (1995)
Exner, P., Harrell, E.M., Loss, M.: Optimal eigenvalues for some Laplacians and Schrödinger operators depending on curvature. In: Dittrich, J., Exner, P., Tater, M. (eds.) Mathematical Results in Quantum Mechanics (Prague, 1998). Oper. Theory Adv. Appl., vol. 108, pp. 47–58 (1999)
Faber, G.: Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt. In: Sitzungberichte der mathematisch- physikalischen Klasse der Bayerischen Akademie der Wissenschaften zu München Jahrgang, pp. 169–172 (1923)
Federer H.: Geometric Measure Theory. Springer Verlag, New York (1969)
Geisinger L., Weidl T.: Universal bounds for traces of the Dirichlet Laplace operator. J. Lond. Math. Soc. 82, 395–419 (2010)
Giraud O., Thas K.: Hearing shapes of drums: Mathematical and physical aspects of isospectrality. Rev. Mod. Phys. 82, 2213–2255 (2010)
Gordon C., Webb D., Wolpert S.: Isospectral plane domains and surfaces via Riemannian orbifolds. Invent. Math. 110, 1–22 (1992)
Hardy G.H., Littlewood J.E., Pólya G.: Inequalities. Cambridge Univ. Press, Cambridge, UK (1964)
Harrell, E.M.: Gap estimates for Schrödinger operators depending on curvature talk delivered at the 2002 UAB International Conference on Differential Equations and Mathematical Physics. Available electronically at http://www.math.gatech.edu/~harrell/
Harrell E.M., Loss M.: On the Laplace operator penalized by mean curvature. Commun. Math. Phys. 195, 643–650 (1998)
Harrell E.M., Hermi L.: Differential inequalities for Riesz means and Weyl type bounds for eigenvalues. J. Funct. Anal. 254, 3173–3191 (2008)
Heath, T.L.: A History of Greek Mathematics. vol. 2. The Clarendon Press, Oxford (1921)
Henrot A.: Extremum Problems for Eigenvalues of Elliptic Operators. Collection Frontiers in Mathematics, Birkhauser (2006)
Kac M.: On some connections between probability theory and differential and integral equations. In: Neyman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley, CA, pp. 189–215 (1951)
Kac M.: Can one hear the shape of a drum?. Am. Math. Mon. 73, 1–23 (1966)
Kornhauser E.T., Stakgold I.: A variational theorem for and its applications. J. Math. Phys. 31, 45–54 (1952)
Krahn E.: Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann. 94, 97–100 (1925)
Krahn, E. :Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Tartu (Dorpat), vol. A9, 1–44 (1926) [English translation: Minimal properties of the sphere in three and more dimensions. In: Edgar Krahn 1894–1961: A Centenary Volume, Ü. Lumiste and J. Peetre, editors, IOS Press, Amsterdam, The Netherlands, pp. 139–174 (1994)]
Lavine R.: The eignevalue gap for one-dimensional convex potentials. Proc. Am. Math. Soc. 121, 815–821 (1994)
Lieb E.H.: The stability of matter Rev. Mod. Phys. 48, 553–569 (1976)
Lieb, E.H.: Lieb–Thirring inequalities. In: Encyclopaedia of Mathematics, Suppl. II, Kluwer, Dordrecht, pp. 311–312 (2000)
Lieb, E.H., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, RI (1997)
Lieb, E.H., Thirring, W.: Bounds for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35, 687–689 (1975) (Errata: PRL 35, 1116 (1975))
Lieb E.H., Thirring W.: Inequalities for the moments of the eigenvalues of the Schrödinger hamiltonian and their relation to Sobolev inequalities. In: Lieb, E.H., Simon, B., Wightman, A.S. (eds) Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann, Princeton University Press, Princeton (1986)
Linde H.: A lower bound for the ground state energy of a Schrödinger operator on a loop. Proc. Am. Math. Soc. 134, 3629–3635 (2006)
McHale, K.P.: Eigenvalues of the Laplacian, “Can you Hear the Shape of a Drum?”, Master’s Project. Mathematics Department University of Missouri, Columbia, MO (1994)
McKean H.P., Singer I.M.: Curvature and the eigenvalues of the Laplacian. J. Differ. Geom. 1, 662–670 (1967)
Maz’ja V.G. : Sobolev spaces, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin (1985) (Translated from the Russian by T.O. Shaposhnikova)
Melrose, R.B.: The inverse spectral problem for planar domains. In: Proceedings of the Centre for Mathematics and its Applications, Australian National University, vol. 34 (1996)
Milnor J.: Eigenvalues of the Laplace operator on certain manifolds. Proc. Nat. Acad. Sci. 51, 542 (1964)
Munkres J.R.: Topology, A first course. Englewood Cliffs, Prentice-Hall (1975)
Nadirashvili N.: Rayleigh’s conjecture on the principal frequency of the clamped plate. Arch. Ration. Mech. Anal. 129, 1–10 (1995)
Osserman, R.: Isoperimetric inequalities and eigenvalues of the Laplacian. In: Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 435–442, Acad. Sci. Fennica, Helsinki (1980)
Payne L.E.: A note on inequalities for plate eigenvalues. J. Math. Phys. 39, 155–159 (1960/1961)
Payne L.E.: Isoperimetric inequalities and their applications. SIAM Rev. 9, 453–488 (1967)
Payne L.E., Pólya G., Weinberger H.F.: Sur le quotient de deux fréquences propres consécutives. Comptes Rendus Acad. Sci. Paris 241, 917–919 (1955)
Payne L.E., Pólya G., Weinberger H.F.: On the ratio of consecutive eigenvalues. J. Math. Phys. 35, 289–298 (1956)
Payne L.E., Weinberger H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)
Pleijel Å.: A study of certain Green’s functions with applications in the theory of vibrating membranes. Ark. Math. 2, 553–569 (1954)
Pólya G., Szegö G.: Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, Princeton, NJ (1951)
Rayleigh, J.W.S.: The Theory of Sound, 2nd edn. revised and enlarged (in 2 vols.), Dover Publications, New York (1945) (republication of the 1894/1896 edition)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 3. Academic Press, NY (1979)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics Vol. 4. Analysis of Operators. Academic Press, NY (1978)
Riemann, B.: Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie, pp. 671–680 (1859)
Singer I.M., Wong B., Yau S.-T., Yau S.S.-T.: An estimate of the gap of the first two eigenvalues in the Schrödinger operator. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12(4), 319–333 (1985)
Sridhar S., Kudrolli A.: Experiments on not “Hearing the Shape” of drums. Phys. Rev. Lett. 72, 2175–2178 (1994)
Sunada T.: Riemannian coverings and isospectral manifolds. Ann. Math. 121, 169–186 (1985)
Szegö G.: On membranes and plates. Proc. Nat. Acad. Sci. USA 36, 210–216 (1950)
Szegö, G. : On the vibrations of a clamped plate. In: Atti del Quarto Congresso dell’Unione Matematica Italiana, Taormina, 1951, vol. II, pp. 573–577. Casa Editrice Perrella, Roma (1953)
Szegö G.: Inequalities for certain eigenvalues of a membrane of given area. J. Ration. Mech. Anal. 3, 343–356 (1954)
Szegö G.: Note to my paper “On membranes and plates”. Proc. Nat. Acad. Sci. USA 44, 314–316 (1958)
Talenti G.: On the first eigenvalue of the clamped plate. Ann. Mat. Pura Appl. 129(4), 265–280 (1981)
Talenti G.: Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa 3(4), 697–718 (1976)
Thomas, I.: Greek Mathematics. In: Newman J.R. (ed.) The World of Mathematics, vol. 1, pp. 189–209, Dover, NY (2000)
Vergil: The Aeneid, [English translation by Sarah Ruden], Yale University Press, New Haven, CT (2008)
Weinberger H.F.: An isoperimetric inequality for the n-dimensional free membrane problem. J. Ration. Mech. Anal. 5, 633–636 (1956)
Weyl H.: Über die asymptotische Verteilung der Eigenwerte. Nachr. Akad. Wiss. Göttingen Math. Phys., Kl. II, 110–117 (1911)
Weyl H.: Ramifications, old and new, of the eigenvalue problem. Bull. Am. Math. Soc. 56, 115–139 (1950)
Yau, S.-T.: Nonlinear analysis in geometry, Monographies de LEnseignement Mathématique, vol. 33. Série des Conférences de lUnion Mathématique Internationale, vol. 8, Geneva (1986)
Yu Q.-H., Zhong J.-Q.: Lower bounds of the gap between the first and second eigenvalues of the Schrödinger operator. Trans. Am. Math. Soc. 294, 341–349 (1986)
Zelditch S.: Spectral determination of analytic bi-axisymmetric plane domains. Geom. Funct. Anal. 10, 628–677 (2000)
Zelditch S.: Inverse spectral problems for analytic domains II: \({\mathbb{Z}^2}\) -symmetric domains. Ann. Math. 170, 205–269 (2009)