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Integrability of reciprocals of the Green's function for elliptic operators: counterexamples


Author: M. Cristina Cerutti
Journal: Proc. Amer. Math. Soc. 119 (1993), 125-134
MSC: Primary 35J25; Secondary 35B65, 35C15
DOI: https://doi.org/10.1090/S0002-9939-1993-1145941-1
MathSciNet review: 1145941
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Abstract: We construct examples of elliptic operators for which the set of points where the reciprocal $ 1/g(x, \cdot )$ of the Green's function is not locally integrable in a dense set of points.


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  • [1] A. D. Aleksandrov, Uniqueness conditions and bounds for the solution of the Dirichlet problem, Vestnik Leningrad. Univ. Math. 18 (1963), 5-29. MR 0164135 (29:1434)
  • [2] P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints, Ark. Mat. 22 (1984), 153-173. MR 765409 (86m:35008)
  • [3] -, A Wiener test for nondivergence structure, second-order elliptic equations, Indiana Univ. Math. J. 4 (1985), 825-844. MR 808829 (87b:35047)
  • [4] L. Caffarelli, Elliptic second order equations, Rend. Sem. Mat. Fis. Milano 57 (1988). MR 1069735 (91h:35070)
  • [5] M. C. Cerutti, L. Escauriaza, and E. B. Fabes, Uniqueness for the Dirichlet problem for some elliptic operators with discontinuous coefficients, Ann. Mat. Pura Appl. (to appear).
  • [6] C. Evans, Some estimates for nondivergence structure second order elliptic equations, Trans. Amer. Math. Soc. 287 (1985), 701-712. MR 768735 (86g:35056)
  • [7] E. Fabes and D. Strook, The $ {L^p}$-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J. 51 (1984), 977-1016.
  • [8] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, New York and Berlin, 1983. MR 737190 (86c:35035)
  • [9] F.-H. Lin, Second derivative $ {L^p}$-estimates for elliptic equations of nondivergence form, Proc. Amer. Math. Soc. 96 (1986), 447-451. MR 822437 (88b:35058)
  • [10] B. Oksendhal, Dirichlet forms, quasiregular functions and Brownian motion, Invent. Math. 91 (1988), 273-297. MR 922802 (89d:60143)
  • [11] C. Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. 74 (1966), 15-30. MR 0214905 (35:5752)
  • [12] M. V. Safanov, Harnack's inequality for elliptic equations and the Hölder property of their solutions, Zap. Nauchn. Sem. Leningrad. Odtel. Mat. Inst. Steklov. (LOMI) 96 (1980), 272-287; English transl. in J. Soviet Math. 21 (1983). MR 579490 (82b:35045)

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DOI: https://doi.org/10.1090/S0002-9939-1993-1145941-1
Article copyright: © Copyright 1993 American Mathematical Society

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