Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture

Author: Michel L. Lapidus
Journal: Trans. Amer. Math. Soc. 325 (1991), 465-529
MSC: Primary 58G25; Secondary 28A75, 35J25, 35P20
MathSciNet review: 994168
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Omega $ be a bounded open set of $ {\mathbb{R}^n}\;(n \geq 1)$ with "fractal" boundary $ \Gamma $. We extend Hermann Weyl's classical theorem by establishing a precise remainder estimate for the asymptotics of the eigenvalues of positive elliptic operators of order $ 2m\;(m \geq 1)$ on $ \Omega $. We consider both Dirichlet and Neumann boundary conditions. Our estimate--which is expressed in terms of the Minkowski rather than the Hausdorff dimension of $ \Gamma $--specifies and partially solves the Weyl-Berry conjecture for the eigenvalues of the Laplacian. Berry's conjecture--which extends to "fractals" Weyl's conjecture--is closely related to Kac's question "Can one hear the shape of a drum?"; further, it has significant physical applications, for example to the scattering of waves by "fractal" surfaces or the study of porous media. We also deduce from our results new remainder estimates for the asymptotics of the associated "partition function" (or trace of the heat semigroup). In addition, we provide examples showing that our remainder estimates are sharp in every possible "fractal" (i.e., Minkowski) dimension.

The techniques used in this paper belong to the theory of partial differential equations, the calculus of variations, approximation theory and--to a lesser extent--geometric measure theory. An interesting aspect of this work is that it establishes new connections between spectral and "fractal" geometry.

References [Enhancements On Off] (What's this?)

  • [Ad] R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
  • [Ag] S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand, Princeton, N. J., 1965. MR 0178246 (31:2504)
  • [Al] F. J. Almgren, Jr., Plateau's problem: an invitation to varifold geometry, Benjamin, New York, 1966. MR 0190856 (32:8266)
  • [BaHi] H. P. Baltes and E. R. Hilf, Spectra of finite systems, B. I.-Wissenschaftsverlag, Vienna, 1976. MR 0435624 (55:8582)
  • [Bd] P. H. Bérard, Remarques sur la conjecture de Weyl, Compositio Math. 48 (1983), 35-53. MR 700579 (85d:58082)
  • [Be1] M. V. Berry, Distribution of modes in fractal resonators, Structural Stability in Physics (W. Güttinger and H. Eikemeier, eds.), Springer-Verlag, Berlin, 1979, pp. 51-53. MR 556688
  • [Be2] -, Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals, Geometry of the Laplace Operator, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc., Providence, R. I., 1980, pp. 13-38. MR 573427 (81f:58012)
  • [BiSo] M. S. Birman and M. Z. Solomjak, Piecewise polynomial approximation of functions of the class $ W_p^\alpha $, Math. USSR-Sb. 2 (1967), 295-317.
  • [Bo] G. Bouligand, Ensembles impropres et nombre dimensionnel, Bull. Sci. Math. (2) 52 (1928), 320-344 and 361-376.
  • [BG] L. Boutet de Monvel and P. Grisvard, Le comportement asymptotique des valeurs propres d'un opérateur, C.R. Acad. Sci. Paris Sér. A 272 (1971), 23-25. MR 0275234 (43:991)
  • [BrCa] J. Brossard and R. Carmona, Can one hear the dimension of a fractal?, Comm. Math. Phys. 104 (1986), 103-122. MR 834484 (87h:58218)
  • [C] I. Chavel, Eigenvalues in Riemannian geometry, Academic Press, Orlando, Fla., 1984. MR 768584 (86g:58140)
  • [Ce] G. Cherbit (ed.), Fractals, dimensions non entières et applications, Masson, Paris, 1987. MR 904064 (89f:00046)
  • [Ch] G. Choquet, Outils topologiques et métriques de l'analyse mathématique, Centre de Documentation Universitaire, Paris, 1969. MR 0262426 (41:7033)
  • [Cn] D. L. Cohn, Measure theory, Birkhäuser, Boston, Mass., 1980. MR 1454121 (98b:28001)
  • [Co] R. Courant, Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik, Math. Z. 7 (1920), 1-57. MR 1544417
  • [CoHi] R. Courant and D. Hilbert, Methods of mathematical physics, Vol. I, English transl., Interscience, New York, 1953. MR 0065391 (16:426a)
  • [DMT] Y. Dupain, M. Mendès-France, and C. Tricot, Dimension des spirales, Bull. Soc. Math. France 111 (1983), 193-201. MR 734220 (86i:28011)
  • [EdEv] D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford Univ. Press, Oxford, 1987. MR 929030 (89b:47001)
  • [Ek] A. El Kolli, Nième épaisseur dans les espaces de Sobolev, J. Approx. Theory 10 (1974), 268-294. MR 0355575 (50:8049)
  • [Fa] W. Falconer, The geometry of fractal sets, Cambridge Univ. Press, London, 1985.
  • [Fe] H. Federer, Geometric measure theory, Springer, New York, 1969. MR 0257325 (41:1976)
  • [Ff] C. L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), 129-206. MR 707957 (85f:35001)
  • [FlLa1] J. Fleckinger and M. L. Lapidus, Eigenvalues of elliptic boundary value problems with an indefinite weight function, Trans. Amer. Math. Soc. 295 (1986), 305-324. MR 831201 (87j:35282)
  • [FlLa2] -, Remainder estimates for the asymptotics of elliptic eigenvalue problems with indefinite weights, Arch. Rational Mech. Anal. 98 (1987), 329-356. MR 872751 (88b:35149)
  • [FlMt] J. Fleckinger and G. Métivier, Théorie spectrale des opérateurs uniformément elliptiques sur quelques ouverts irréguliers, C.R. Acad. Sci. Paris Sér. A 276 (1973), 913-916. MR 0320550 (47:9087)
  • [GeVa] F. W. Gehring and J. Väisälä, Hausdorff dimension and quasiconformal mappings, J. London Math. Soc. 6 (1973), 504-512. MR 0324028 (48:2380)
  • [Gi] P. B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, 2nd ed., Publish or Perish, Wilmington, Del., 1984. MR 783634 (86j:58144)
  • [Gr] D. Gromes, Über die asymptotische Verteilung der Eigenwerte des Laplace Operators für Gebiete auf der Kugeloberfläche, Math. Z. 94 (1966), 110-121. MR 0199575 (33:7718)
  • [GuKz] V. W. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved $ 2$-manifolds, Topology 19 (1980), 301-312. MR 579579 (81j:58082)
  • [Ha] J. Hawkes, Hausdorff measure, entropy, and the independence of small sets, Proc. London Math. Soc. (3) 28 (1974), 700-724. MR 0352412 (50:4899)
  • [Ho1] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193-218. MR 0609014 (58:29418)
  • [Ho2] -, The analysis of linear partial differential operators, Vols. III and IV, Springer-Verlag, Berlin, 1985.
  • [HrWa] W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, N. J., 1941. MR 0006493 (3:312b)
  • [Hu] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 625600 (82h:49026)
  • [Iv1] V. Ja. Ivrii, Second term of the spectral asymptotic expansion of the Laplace-Beltrami operator on manifolds with boundary, Functional Anal. Appl. 14 (1980), 98-106. MR 575202 (82m:58057)
  • [Iv2] -, Precise spectral asymptotics for elliptic operators acting in fiberings over manifolds with boundary, Lecture Notes in Math., vol. 1100, Springer-Verlag, Berlin, 1984. MR 771297 (86h:58139)
  • [JhLa] G. W. Johnson and M. L. Lapidus, Generalized Dyson series, generalized Feynman diagrams, the Feynman integral and Feynman's operational calculus, Mem. Amer. Math. Soc. No. 351, 62 (1986), 1-78. MR 849943 (88f:81034)
  • [Jn] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), 71-88. MR 631089 (83i:30014)
  • [Ka] M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly (Slaught Memorial Papers, No. 11) (4) 73 (1966), 1-23. MR 0201237 (34:1121)
  • [Kh] J.-P. Kahane, Courbes étranges, ensembles minces, Bull. Assoc. Professeurs Enseign. Math. Phys. 275/276 (1970), 325-339.
  • [KhSa] J.-P. Kahane and R. Salem, Ensembles parfaits et séries trigonométriques, Hermann, Paris, 1963. MR 0160065 (28:3279)
  • [Ku] N. V. Kuznetsov, Asymptotic distribution of the eigenfrequencies of a plane membrane in the case when the variables can be separated, Differential Equations 2 (1966), 715-723. MR 0206524 (34:6342)
  • [La1] M. L. Lapidus, Valeurs propres du laplacien avec un poids qui change de signe, C.R. Acad. Sci. Paris Sér. I Math. 298 (1984), 265-268. MR 745319 (85j:35139)
  • [La2] -, Spectral theory of elliptic problems with indefinite weights, Spectral Theory of Sturm-Liouville Differential Operators (Hans G. Kaper and A. Zettle, eds.), ANL-84-73, Argonne National Laboratory, Argonne, 1984, pp. 159-168.
  • [La3] -, Formules de Trotter et calcul opérationnel de Feynman, Thèse de Doctorat d'Etat ès Sciences, Mathématiques, Université Pierre et Marie Curie (Paris VI), France, 1986. (Part II: Problèmes aux valeurs propres elliptiques avec un poids non défini. Part III: Calcul opérationnel de Feynman.)
  • [La4] -, The differential equation for the Feynman-Kac formula with a Lebesgue-Stieltjes measure, Lett. Math. Phys. 11 (1986), 1-13. (Dedicated to the memory of Mark Kac.) MR 824670 (87d:58033)
  • [La5] -, The Feynman-Kac formula with a Lebesgue-Stieltjes measure and Feynman's operational calculus, Stud. Appl. Math. 76 (1987), 93-132. MR 965739 (89j:81057)
  • [La6] -, The Feynman-Kac formula with a Lebesgue-Stieltjes measure: an integral equation in the general case, Integral Equations Operator Theory 12 (1989), 163-210. MR 986594 (92e:47064)
  • [La7] -, Strong product integration of measures and the Feynman-Kac formula with a Lebesgue-Stieltjes measure, Suppl. Rend. Circ. Mat. Palermo Ser. II 17 (1987), 271-312. MR 950421 (90c:28021)
  • [La8] -, Asymptotic distribution of the eigenvalues of elliptic boundary value problems and Schrödinger operators with indefinite weights, abridged version of a talk given at the VIIIth Latin American School of Mathematics on "Partial Differential Equations" (IMPA, Rio de Janeiro, Brazil, July 1986).
  • [La9] -, Can one hear the shape of a fractal drum? Partial resolution of the Weyl-Berry conjecture, Geometric Analysis and Computer Graphics (P. Concus et al., eds.), Proc. Workshop Differential Geometry, Calculus of Variations, and Computer Graphics (MSRI, Berkeley, May 1988), Mathematical Sciences Research Institute Publications, vol. 17, Springer-Verlag, New York, 1990, pp. 119-126. MR 1081333 (92d:58211)
  • [La10] -, Elliptic differential operators on fractals and the Weyl-Berry conjecture (in preparation).
  • [LaF1] M. L. Lapidus and J. Fleckinger-Pellé, Tambour fractal: vers une résolution de la conjecture de Weyl-Berry pour les valeurs propres du laplacien, C.R. Acad. Sci. Paris Sér. I Math. 306 (1988), 171-175. MR 930556 (89d:35133)
  • [LiMa] J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I, English transl., Springer-Verlag, Berlin, 1972. MR 0350177 (50:2670)
  • [Lo] G. G. Lorentz, Approximation of functions, 2nd ed., Chelsea, New York, 1986. MR 917270 (88j:41001)
  • [McSn] H. P. McKean and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 1 (1967), 43-69. MR 0217739 (36:828)
  • [Md1] B. B. Mandelbrot, The fractal geometry of nature, rev. and enlarged ed., Freeman, New York, 1983. MR 1484414 (98g:00011)
  • [Md2] -, Les objets fractals, 2nd ed., Flammarion, Paris, 1984. MR 785362 (86j:00021)
  • [MrVu] O. Martio and M. Vuorinen, Whitney cubes, $ p$-capacity, and Minkowski content, Exposition. Math. 5 (1987), 17-40. MR 880256 (88e:28004)
  • [M] V. G. Maz'ja, Sobolev spaces, Springer-Verlag, Berlin, 1985. MR 817985 (87g:46056)
  • [Ms1] R. B. Melrose, Weyl's conjecture for manifolds with concave boundary, Geometry of the Laplace Operator, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc., Providence, R. I., 1980, pp. 254-274. MR 573438 (82b:58101)
  • [Ms2] -, The trace of the wave group, Contemp. Math., vol. 27, Amer. Math. Soc., Providence, R. I., 1984, pp. 127-167. MR 741046 (86f:35144)
  • [Mt1] G. Métivier, Théorie spectrale d'opérateurs elliptiques sur des ouverts irréguliers, Séminaire Goulaic-Schwartz, No. 21, Ecole Polytechnique, Paris, 1973.
  • [Mt2] -, Etude asymptotique des valeurs propres et de la fonction spectrale de problèmes aux limites, Thèse de Doctorat d'Etat, Mathématiques, Université de Nice, France, 1976.
  • [Mt3] -, Valeurs propres de problèmes aux limites elliptiques irréguliers, Bull. Soc. Math. France Mém. 51-52 (1977), 125-219. MR 0473578 (57:13244)
  • [OsWi] R. Osserman and A. Weinstein (eds.), Geometry of the Laplace operator, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc., Providence, R. I., 1980. MR 573425 (81c:58002)
  • [PeRi] H. O. Peitgen and P. H. Richter, The beauty of fractals, Springer-Verlag, Berlin, 1986. MR 852695 (88e:00019)
  • [Ph] Pham The Lai, Meilleures estimations asymptotiques des restes de la fonction spectrale et des valeurs propres relatifs au laplacien, Math. Scand. 48 (1981), 5-38. MR 621413 (83b:35127a)
  • [PiTo] L. Pietronero and M. Tozzati (eds.), Fractals in physics, North-Holland, Amsterdam, 1986. MR 863016 (87i:00022)
  • [Pn] A. Pinkus, $ n$-widths in approximation theory, Springer-Verlag, New York, 1985. MR 774404 (86k:41001)
  • [Pr] M. H. Protter, Can one hear the shape of a drum? Revisited, SIAM Rev. 29 (1987), 185-197. MR 889243 (88g:58185)
  • [ReSi] M. Reed and B. Simon, Methods of modern mathematical physics, Vol. IV, Analysis of Operators, Academic Press, New York, 1978. MR 0493421 (58:12429c)
  • [Ro] C. A. Rogers, Hausdorff measures, Cambridge Univ. Press, Cambridge, 1970. MR 0281862 (43:7576)
  • [SMR] M. F. Schlesinger, B. B. Mandelbrot, and R. J. Rubin (eds.), Fractals in the physical sciences, J. Statist. Phys. 36 Nos. 5/6, (1984). MR 785362 (86j:00021)
  • [Se1] R. T. Seeley, A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of $ {\mathbb{R}^3}$, Adv. in Math. 29 (1978), 244-269. MR 506893 (80a:35096)
  • [Se2] -, An estimate near the boundary for the spectral function of the Laplace operator, Amer. J. Math. 102 (1980), 869-902. MR 590638 (82k:58097)
  • [Si] B. Simon, Functional integration and quantum physics, Academic Press, New York, 1979. MR 544188 (84m:81066)
  • [Th] W. Thompson, d'Arcy, On growth and form, abridged ed. (J. T. Bonner, ed.), Cambridge Univ. Press, Cambridge, 1966. MR 0128562 (23:B1601)
  • [Tr1] C. Tricot, Jr., Douze définitions de la densité logarithmique, C.R. Acad. Sci. Paris Sér. I Math. 293 (1981), 549-552. MR 647678 (83i:28007)
  • [Tr2] -, Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 57-74. MR 633256 (84d:28013)
  • [Tr3] -, Metric properties of compact sets of measure zero in $ {\mathbb{R}^2}$, preprint; in Mesures et Dimensions, Thèse de Doctorat d'Etat, Mathématiques, Université Paris-Sud (Orsay), France, 1983.
  • [Ur] H. Urakawa, Bounded domains which are isospectral but not congruent, Ann. Sci. Ecole Norm. Sup. (4) 15 (1982), 441-456. MR 690649 (84g:58106)
  • [VGL] S. K. Vodopjanov, V. M. Gol'dstein, and T. G. Latfullin, Criteria for extension of functions of the class $ L_2^1$ from unbounded plane domains, Siberian Math. J. 20 (1979), 298-301. MR 530508 (80j:46061)
  • [Wb] H. F. Weinberger, Variational methods for eigenvalue approximation, CBMS Regional Conf. Ser. Appl. Math., vol. 15, SIAM, Philadelphia, Pa., 1974. MR 0400004 (53:3842)
  • [We1] H. Weyl, Über die asymptotische Verteilung der Eigenwerte, Gott. Nach. (1911), 110-117.
  • [We2] -, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann. 71 (1912), 441-479. MR 1511670
  • [We3] -, Über die Abhängigkeit der Eigenschwingungen einer Membran von deren Begrenzung, J. Angew. Math. 141 (1912), 1-11.
  • [Ya] S.-T. Yau, Nonlinear analysis in geometry, Enseign. Math. (2) 33 (1987), 109-158. MR 896385 (88g:58003)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58G25, 28A75, 35J25, 35P20

Retrieve articles in all journals with MSC: 58G25, 28A75, 35J25, 35P20

Additional Information

Keywords: Dirichlet and Neumann Laplacians, higher order elliptic operators, variational boundary value problems, inverse problems, spectral theory, approximation theory, asymptotics of eigenvalues, remainder estimate, Weyl-Berry conjecture, fractals, Minkowski and Hausdorff dimensions, vibrations of fractal drums, spectral and fractal geometry
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society