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On the asymptotic distribution of eigenvalues for semi-elliptic operators


Authors: Akira Tsutsumi and Chung Lie Wang
Journal: Trans. Amer. Math. Soc. 199 (1974), 295-315
MSC: Primary 35P20
DOI: https://doi.org/10.1090/S0002-9947-1974-0355373-4
MathSciNet review: 0355373
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Abstract: This paper is focused on the asymptotic distribution of eigenvalues for semielliptic operators under weaker smoothness assumptions on coefficients of operators than those of F. E. Browder [3] and Y. Kannai [8] by applying the method of Maruo-Tanabe [9].


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  • [1] S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand Math. Studies, no. 2, Van Nostrand, Princeton, N. J., 1965. MR 31 #2504. MR 0178246 (31:2504)
  • [2] -, Asymptotic formulas with remainder estimates for eigenvalues of elliptic operators, Arch. Rational Mech. Anal. 28 (1967/68), 165-183. MR 37 #4430. MR 0228851 (37:4430)
  • [3] F. E. Browder, The asymptotic distribution of eigenfunctions and eigenvalues for semi-elliptic differential operators, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 270-273. MR 19, 862. MR 0090741 (19:862b)
  • [4] A. Cavallucci, Sulle proprietà differenziali delle soluzioni delle equazioni quasi-ellittiche relativamente a domini normali, Boll. Un. Mat. Ital. (3) 19 (1964), 465-477. MR 31 #2501. MR 0178243 (31:2501)
  • [5] J. Friberg, Asymptotic behavior of integrals connected with spectral functions for hypoelliptic operators, Ark. Mat. 7 (1967), 283-298. MR 36 #4143. MR 0221091 (36:4143)
  • [6] E. Giusti, Equazioni quasi ellittiche e spazi $ {L^{p,\theta }}(\Omega ,\delta )$. I, Ann. Mat. Pura Appl. (4) 74 (1967), 313-354. MR 37 #6607. MR 0231050 (37:6607)
  • [7] V. N. Gorčakov, The asymptotic behavior of a spectral function of a hypoelliptic operator of a certain class, Dokl. Akad. Nauk SSSR 152 (1963), 519-522 = Soviet Math. Dokl. 4 (1963), 1328-1332. MR 28 #494.
  • [8] Y. Kannai, On the asymptotic behavior of resolvent kernels, spectral functions and eigenvalues of semielliptic systems, Ann. Scuola Norm. Sup. Pisa 23 (1969), 563-634. MR 0415092 (54:3183)
  • [9] K. Maruo and H. Tanabe, On the asymptotic distribution of eigenvalues of operators associated with strongly elliptic sesquillinear forms, Osaka J. Math. 8 (1971), 323-345. MR 0310467 (46:9565)
  • [10] T. Matsuzawa, On quasi-elliptic boundary problems, Trans. Amer. Math. Soc. 133 (1968), 241-265. MR 37 #600. MR 0225001 (37:600)
  • [11] N. Nilsson, Asymptotic estimates for spectral functions connected with hypoelliptic differential operators, Math. Scand. 9 (1961), 237-251.
  • [12] Å. Pleijel, On a theorem by P. Malliavin, Israel J. Math. 1 (1963), 166-168. MR 29 #5023. MR 0167751 (29:5023)
  • [13] B. Sternin, Quasi-elliptic equations in an infinite cylinder, Dokl. Akad. Nauk SSSR 194 (1970), 1025-1028 = Soviet Math. Dokl. 11 (1970), 1347-1351. MR 42 #6414. MR 0271531 (42:6414)
  • [14] A. Tsutsumi and C. L. Wang, On the asymptotic distribution of eigenvalues for semi-elliptic operators, Proc. Japan Acad. (to appear). MR 0355373 (50:7847)
  • [15] L. R. Volevič, Local properties of solutions of quasi-elliptic systems, Mat. Sb. 59 (101) (1962), supplement, 3-52. (Russian) MR 27 #446. MR 0150448 (27:446)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0355373-4
Keywords: Resolvent kernels, symmetric sesquilinear form, semielliptic operators, norm inequalities, asymptotic estimates
Article copyright: © Copyright 1974 American Mathematical Society

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