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 Free Access

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?)

  • Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
  • Frederick J. Almgren Jr., Plateau’s problem: An invitation to varifold geometry, W. A. Benjamin, Inc., New York-Amsterdam, 1966. MR 0190856
  • Heinrich P. Baltes and Eberhard R. Hilf, Spectra of finite systems, Bibliographisches Institut, Mannheim-Vienna-Zurich, 1976. A review of Weyl’s problem: the eigenvalue distribution of the wave equation for finite domains and its applications on the physics of small systems. MR 0435624
  • Pierre H. Bérard, Remarques sur la conjecture de Weyl, Compositio Math. 48 (1983), no. 1, 35–53 (French). MR 700579
  • M. V. Berry, Distribution of modes in fractal resonators, Structural stability in physics (Proc. Internat. Symposia Appl. Catastrophe Theory and Topological Concepts in Phys., Inst. Inform. Sci., Univ. Tübingen, Tübingen, 1978) Springer Ser. Synergetics, vol. 4, Springer, Berlin, 1979, pp. 51–53. MR 556688, DOI
  • M. V. Berry, Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 13–28. MR 573427
  • 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. G. Bouligand, Ensembles impropres et nombre dimensionnel, Bull. Sci. Math. (2) 52 (1928), 320-344 and 361-376.
  • Louis Boutet de Monvel and Pierre Grisvard, Le comportement asymptotique des valeurs propres d’un opérateur, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A23–A26 (French). MR 275234
  • Jean Brossard and René Carmona, Can one hear the dimension of a fractal?, Comm. Math. Phys. 104 (1986), no. 1, 103–122. MR 834484
  • Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
  • Fractals, Masson, Paris, 1987 (French). Dimensions non entières et applications. [Nonintegral dimensions and applications]; Under the direction of G. Cherbit; With a preface by J. P. Kahane. MR 904064
  • Gustave Choquet, Outils topologiques et métriques de l’analyse mathématique, Centre de Documentation Universitaire, Paris, 1969 (French). Cours rédigé par Claude Mayer. MR 0262426
  • Donald L. Cohn, Measure theory, Birkhäuser Boston, Inc., Boston, MA, 1993. Reprint of the 1980 original. MR 1454121
  • R. Courant, Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik, Math. Z. 7 (1920), no. 1-4, 1–57 (German). MR 1544417, DOI
  • R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
  • Yves Dupain, Michel Mendès France, and Claude Tricot, Dimensions des spirales, Bull. Soc. Math. France 111 (1983), no. 2, 193–201 (French, with English summary). MR 734220
  • D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications. MR 929030
  • Amar El Kolli, $n$ième épaisseur dans les espaces de Sobolev, J. Approximation Theory 10 (1974), 268–294 (French). MR 355575, DOI
  • W. Falconer, The geometry of fractal sets, Cambridge Univ. Press, London, 1985.
  • Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. MR 707957, DOI
  • Jacqueline Fleckinger and Michel L. Lapidus, Eigenvalues of elliptic boundary value problems with an indefinite weight function, Trans. Amer. Math. Soc. 295 (1986), no. 1, 305–324. MR 831201, DOI
  • Jacqueline Fleckinger and Michel L. Lapidus, Remainder estimates for the asymptotics of elliptic eigenvalue problems with indefinite weights, Arch. Rational Mech. Anal. 98 (1987), no. 4, 329–356. MR 872751, DOI
  • Jacqueline Fleckinger and Guy Métivier, Théorie spectrale des opérateurs uniformément elliptiques sur quelques ouverts irréguliers, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A913–A916 (French). MR 320550
  • F. W. Gehring and J. Väisälä, Hausdorff dimension and quasiconformal mappings, J. London Math. Soc. (2) 6 (1973), 504–512. MR 324028, DOI
  • Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. MR 783634
  • Dieter Gromes, Über die asymptotische Verteilung der Eigenwerte des Laplace-Operators für Gebiete auf der Kugeloberfläche, Math. Z. 94 (1966), 110–121 (German). MR 199575, DOI
  • V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved $2$-manifolds, Topology 19 (1980), no. 3, 301–312. MR 579579, DOI
  • John Hawkes, Hausdorff measure, entropy, and the independence of small sets, Proc. London Math. Soc. (3) 28 (1974), 700–724. MR 352412, DOI
  • Lars Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. MR 609014, DOI
  • ---, The analysis of linear partial differential operators, Vols. III and IV, Springer-Verlag, Berlin, 1985.
  • Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
  • John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI
  • V. Ja. Ivriĭ, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary, Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 25–34 (Russian). MR 575202
  • Victor Ivriĭ, Precise spectral asymptotics for elliptic operators acting in fiberings over manifolds with boundary, Lecture Notes in Mathematics, vol. 1100, Springer-Verlag, Berlin, 1984. MR 771297
  • Gerald W. Johnson and Michel L. Lapidus, Generalized Dyson series, generalized Feynman diagrams, the Feynman integral and Feynman’s operational calculus, Mem. Amer. Math. Soc. 62 (1986), no. 351, vi+78. MR 849943, DOI
  • Peter W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), no. 1-2, 71–88. MR 631089, DOI
  • Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), no. 4, 1–23. MR 201237, DOI
  • J.-P. Kahane, Courbes étranges, ensembles minces, Bull. Assoc. Professeurs Enseign. Math. Phys. 275/276 (1970), 325-339.
  • Jean-Pierre Kahane and Raphaël Salem, Ensembles parfaits et séries trigonométriques, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1301, Hermann, Paris, 1963 (French). MR 0160065
  • N. V. Kuznecov, Asymptotic distribution of eigenfrequencies of a plane membrane in the case of separable variables, Differencial′nye Uravnenija 2 (1966), 1385–1402 (Russian). MR 0206524
  • Michel L. Lapidus, Valeurs propres du laplacien avec un poids qui change de signe, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 12, 265–268 (French, with English summary). MR 745319
  • ---, 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. ---, 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.)
  • Michel L. Lapidus, The differential equation for the Feynman-Kac formula with a Lebesgue-Stieltjes measure, Lett. Math. Phys. 11 (1986), no. 1, 1–13. MR 824670, DOI
  • Michel L. Lapidus, The Feynman-Kac formula with a Lebesgue-Stieltjes measure and Feynman’s operational calculus, Stud. Appl. Math. 76 (1987), no. 2, 93–132. MR 965739, DOI
  • Michel L. Lapidus, The Feynman-Kac formula with a Lebesgue-Stieltjes measure: an integral equation in the general case, Integral Equations Operator Theory 12 (1989), no. 2, 163–210. MR 986594, DOI
  • Michel L. Lapidus, Strong product integration of measures and the Feynman-Kac formula with a Lebesgue-Stieltjes measure, Rend. Circ. Mat. Palermo (2) Suppl. 17 (1987), 271–312 (1988). Functional integration with emphasis on the Feynman integral (Sherbrooke, PQ, 1986). MR 950421
  • ---, 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).
  • Michel L. Lapidus, Can one hear the shape of a fractal drum? Partial resolution of the Weyl-Berry conjecture, Geometric analysis and computer graphics (Berkeley, CA, 1988) Math. Sci. Res. Inst. Publ., vol. 17, Springer, New York, 1991, pp. 119–126. MR 1081333, DOI
  • ---, Elliptic differential operators on fractals and the Weyl-Berry conjecture (in preparation).
  • Michel L. Lapidus and Jacqueline 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), no. 4, 171–175 (French, with English summary). MR 930556
  • J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177
  • G. G. Lorentz, Approximation of functions, 2nd ed., Chelsea Publishing Co., New York, 1986. MR 917270
  • H. P. McKean Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), no. 1, 43–69. MR 217739
  • Benoît B. Mandelbrot, La geometría fractal de la naturaleza, Metatemas, vol. 49, Tusquets Editores, S.A., Barcelona, 1997 (Spanish, with Spanish summary). Translated from the English by Josep Llosa. MR 1484414
  • Benoit Mandelbrot, Les objects fractals, 2nd ed., Nouvelle Bibliothèque Scientifique. [New Scientific Library], Flammarion Sciences, Paris, 1984 (French). Forme, hasard et dimension. [Form, chance and dimension]. MR 785362
  • O. Martio and M. Vuorinen, Whitney cubes, $p$-capacity, and Minkowski content, Exposition. Math. 5 (1987), no. 1, 17–40. MR 880256
  • Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985
  • R. B. Melrose, Weyl’s conjecture for manifolds with concave boundary, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 257–274. MR 573438
  • Richard Melrose, The trace of the wave group, Microlocal analysis (Boulder, Colo., 1983) Contemp. Math., vol. 27, Amer. Math. Soc., Providence, RI, 1984, pp. 127–167. MR 741046, DOI
  • G. Métivier, Théorie spectrale d’opérateurs elliptiques sur des ouverts irréguliers, Séminaire Goulaic-Schwartz, No. 21, Ecole Polytechnique, Paris, 1973. ---, 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.
  • Guy Métivier, Valeurs propres de problèmes aux limites elliptiques irrégulières, Bull. Soc. Math. France Suppl. Mém. 51-52 (1977), 125–219 (French). MR 473578
  • Robert Osserman and Alan Weinstein (eds.), Geometry of the Laplace operator, Proceedings of Symposia in Pure Mathematics, XXXVI, American Mathematical Society, Providence, R.I., 1980. MR 573425
  • H.-O. Peitgen and P. H. Richter, The beauty of fractals, Springer-Verlag, Berlin, 1986. Images of complex dynamical systems. MR 852695
  • Phạm The Lại, Meilleures estimations asymptotiques des restes de la fonction spectrale et des valeurs propres relatifs au laplacien, Math. Scand. 48 (1981), no. 1, 5–38 (French). MR 621413, DOI
  • Luciano Pietronero and Erio Tosatti (eds.), Fractals in physics, North-Holland Publishing Co., Amsterdam, 1986. MR 863016
  • Allan Pinkus, $n$-widths in approximation theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 7, Springer-Verlag, Berlin, 1985. MR 774404
  • M. H. Protter, Can one hear the shape of a drum? revisited, SIAM Rev. 29 (1987), no. 2, 185–197. MR 889243, DOI
  • Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
  • C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
  • Benoit Mandelbrot, Les objects fractals, 2nd ed., Nouvelle Bibliothèque Scientifique. [New Scientific Library], Flammarion Sciences, Paris, 1984 (French). Forme, hasard et dimension. [Form, chance and dimension]. MR 785362
  • R. Seeley, A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of ${\bf R}^{3}$, Adv. in Math. 29 (1978), no. 2, 244–269. MR 506893, DOI
  • R. Seeley, An estimate near the boundary for the spectral function of the Laplace operator, Amer. J. Math. 102 (1980), no. 5, 869–902. MR 590638, DOI
  • Barry Simon, Functional integration and quantum physics, Pure and Applied Mathematics, vol. 86, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 544188
  • D’Arcy Wentworth Thompson, On growth and form, Cambridge University Press, New York, 1961. An abridged edition edited by John Tyler Bonner. MR 0128562
  • Claude Tricot, Douze définitions de la densité logarithmique, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 11, 549–552 (French, with English summary). MR 647678
  • Claude Tricot Jr., Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 1, 57–74. MR 633256, DOI
  • ---, 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.
  • Hajime Urakawa, Bounded domains which are isospectral but not congruent, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 441–456. MR 690649
  • S. K. Vodop′janov, V. M. Gol′dšteĭn, and T. G. Latfullin, A criterion for the extension of functions of the class $L_{2}^{1}$ from unbounded plane domains, Sibirsk. Mat. Zh. 20 (1979), no. 2, 416–419, 464 (Russian). MR 530508
  • Hans F. Weinberger, Variational methods for eigenvalue approximation, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. Based on a series of lectures presented at the NSF-CBMS Regional Conference on Approximation of Eigenvalues of Differential Operators, Vanderbilt University, Nashville, Tenn., June 26–30, 1972; Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 15. MR 0400004
  • H. Weyl, Über die asymptotische Verteilung der Eigenwerte, Gott. Nach. (1911), 110-117.
  • Hermann Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), no. 4, 441–479 (German). MR 1511670, DOI
  • ---, Über die Abhängigkeit der Eigenschwingungen einer Membran von deren Begrenzung, J. Angew. Math. 141 (1912), 1-11.
  • Shing-Tung Yau, Nonlinear analysis in geometry, Enseign. Math. (2) 33 (1987), no. 1-2, 109–158. MR 896385

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