Echoes and glimpses of a distant drum

• Autorzy: Turner John Wm.

Identyfikatory

DOI

10.7494/OpMath.2020.40.6.737

• Języki publikacji: EN

Abstrakty

EN

To what extent does the spectrum of the Laplacian operator on a domain D with prescribed boundary conditions determine its shape? This paper first retraces the history of this problem, then Kac’s approach in terms of a diffusion process with absorbing boundary conditions. It is shown how the restriction to a polygonal boundary for D in this method, which required taking the limit of an infinite number of sides to obtain a smooth one, can be avoided by using the Duhamel method.

Słowa kluczowe

EN

Kac drum problem; inverse methods; diffusion process

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2020
• Tom: Vol. 40, no. 6
• Strony: 737--–750
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Turner John Wm.

• Universite Libre de Bruxelles Brussels, Belgium

Bibliografia

• 1] M. Berger, P. Gauduchon, E. Mazet, Le spectre d’une variete riemanienne, Lecture Notes in Mathematics, vol. 194, Springer-Verlag, Berlin-New York, 1971 [in French].
• [2] O. Giraud, K. Thas, Hearing shapes of drums: Mathematical and physical aspects of isospectrality, Rev. Mod. Phys. 82 (2010), 2213-2255.
• [3] C. Gordon, D.L. Webb, S. Wolpert, One cannot hear the shape of a drum, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 134-138.
• [4] F. John, Partial Differential Equations, 4th edition, Applied Mathematical Sciences, Springer-Verlag, New York, 1982.
• [5] M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), 1-23.
• [6] H.P. McKean Jr., I.M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), 43-69.
• [7] S. Minakshisundaram, A. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canad. J. Math. 1 (1949), 242-256.
• [8] F.W.J. Olver, The asymptotic expansion of Bessel functions of large order, Philos. Trans. Roy. Soc. London Ser. A 247 (1954), 328-368. REFERENCES
• [1] M. Berger, P. Gauduchon, E. Mazet, Le spectre d’une variete riemanienne, Lecture Notes in Mathematics, vol. 194, Springer-Verlag, Berlin-New York, 1971 [in French].
• [2] O. Giraud, K. Thas, Hearing shapes of drums: Mathematical and physical aspects of isospectrality, Rev. Mod. Phys. 82 (2010), 2213-2255.
• [3] C. Gordon, D.L. Webb, S. Wolpert, One cannot hear the shape of a drum, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 134-138.
• [4] F. John, Partial Differential Equations, 4th edition, Applied Mathematical Sciences, Springer-Verlag, New York, 1982.
• [5] M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), 1-23.
• [6] H.P. McKean Jr., I.M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), 43-69.
• [7] S. Minakshisundaram, A. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canad. J. Math. 1 (1949), 242-256.
• [8] F.W.J. Olver, The asymptotic expansion of Bessel functions of large order, Philos. Trans. Roy. Soc. London Ser. A 247 (1954), 328-368.
• [9] A. Pleijel, A study of certain Green’s functions with applications in the theory of vibrating membranes, Ark. Mat. 2 (1954), 553-569.
• [10] L. Smith, The asymptotics of the heat equation for a boundary value problem, Invent. Math. 63 (1981), 467-493.
• [11] M. van den Berg, S. Srisatkunarajah, Heat equation for a region in R2 with a polygonal boundary, J. London Math. Soc. (2) 37 (1988), 119-127.
• [12] S. Zelditch, Inverse spectral problem for analytic domains. II, Z2 -symmetric domains, Ann. of Math. (2) 170 (2009), 205-269.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020)