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A linear boundary-value problem for a second-order elliptic-parabolic equation

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Literature Cited

  1. M. V. Keldysh, “On certain cases of degeneration of equations of elliptic type on the boundary of a domain,” Dokl. Akad. Nauk SSSR,77, No. 2, 181–183 (1951).

    Google Scholar 

  2. G. Fichera, “Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine,” Atti Accad. Naz. Lincei, Mem. Cl. Sci. Fiz. Mat. Natur., Sez. I,5, 1–30 (1956).

    Google Scholar 

  3. G. Fichera, “On a unified theory of boundary-value problems for elliptic-parabolic equations of second order,” in: Boundary Problems in Differential Equations, Univ. of Wisconsin Press, Madison, Wisconsin (1960), pp. 97–120.

    Google Scholar 

  4. A. N. Kolmogorov (A. Kolmogoroff), “Zufällige Bewegungen,” Ann. Math.,35, No. 1, 116–117 (1934).

    Google Scholar 

  5. A. D. Aleksandrov, “The investigation of the maximum principle. II,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 3–15 (1959).

    Google Scholar 

  6. L. I. Kamynin, “On the uniqueness of the solution of a boundary-value problem with A. A. Samarskii's boundary conditions for a second-order parabolic equation,” Zh. Vychisl. Mat. Mat. Fiz.,16, No. 6, 1480–1488 (1976).

    Google Scholar 

  7. L. I. Kamynin, “Uniqueness of boundary-value problems for a second-order degenerate elliptic equation,” Differents. Uravn.,14, No. 1, 39–49 (1978).

    Google Scholar 

  8. J.-M. Bony, “Principe du maximum. inégalité de Harnack et unicité de problème de Cauchy pour les opérateurs elliptiques dégénérés,” Ann. Inst. Fourier,19, No. 1, 277–304 (1969).

    Google Scholar 

  9. C. D. Hill, “A sharp maximum principle for degenerate elliptic-parabolic equations,” Indiana Univ. Math. J.,20, No. 3, 213–229 (1970).

    Google Scholar 

  10. R. M. Redheffer, “The sharp maximum principle for nonlinear inequalities,” Indiana Univ. Math. J.,21, No. 3, 227–248 (1971).

    Google Scholar 

  11. K. Amano, “Maximum principles for degenerate elliptic-parabolic operators,” Indiana Univ. Math. J.,28, No. 4, 545–557 (1979).

    Google Scholar 

  12. L. I. Kamynin and B. N. Khimchenko, “On the anisotropic strict extremum principle for second-order elliptic-parabolic equations,” Dokl. Akad. Nauk SSSR,258, No. 2, 288–293 (1981).

    Google Scholar 

  13. L. I. Kamynin and B. N. Khimchenko, “In vestigations on the anisotropic strict extremum principle for second-order elliptic-parabolic equations,” Sib. Mat. Zh.,24, No. 2, 26–55 (1983).

    Google Scholar 

  14. S. E. Myers, “A boundary maximum principle for degenerate elliptic-parabolic inequalities, for characteristic boundary points,” Bull. Am. Math. Soc.,80, No. 3, 527–530 (1974).

    Google Scholar 

  15. S. Zaremba, “Sur un problème mixte relatif a l'équation de Laplace,” Bull. Acad. Sci. Math. Natur., A, 313–344 (1910).

    Google Scholar 

  16. G. Giraud, “Généralisation des problèmes dur les opérations du type elliptique,” Bull. Sci. Math.,56, 316–352 (1932).

    Google Scholar 

  17. L. I. Kamynin and B. N. Khimchenko, “On the maximum principle for a second-order elliptic-parabolic equation,” Sib. Mat. Zh.,13, No. 4, 773–789 (1972).

    Google Scholar 

  18. L. I. Kamynin and B. N. Khimchenko, “On analogues of Giraud's theorem for second-order parabolic equations,” Sib. Mat. Zh.,14, No. 1, 86–110 (1973).

    Google Scholar 

  19. L. I. Kamynin and B. N. Khimchenio, “On an aspect of the development of the Theory of the isotropic strict extremum principle of A. D. Aleksandrov,” Differents. Uravn.,16, No. 2, 280–292 (1980).

    Google Scholar 

  20. L. I. Kamynin and B. N. Khimchenko, “Theorems of Giraud type for second-order equations with a weakly degenerate nonnegative characteristic part,” Sib. Mat. Zh.,18, No. 1, 103–121 (1977).

    Google Scholar 

  21. L. I. Kamynin and B. N. Khimchenko, “Theorems of Giraud type for second-order parabolic equations that admit degeneration,” Sib. Mat. Zh.,21, No. 4, 72–94 (1980).

    Google Scholar 

  22. L. I. Kamynin and B. N. Khimchenko, “Giraud's theorem on the time derivative for a second-order parabolic operator,” Vestn. Mosk. Univ., Ser. I, Mat. Mekh., No. 5, 45–53 (1977).

    Google Scholar 

  23. L. I. Kamynin, “On the uniqueness of the solution of a linear boundary-value problem for a second-order elliptic-parabolic equation,” Dokl. Akad. Nauk SSSR,262, No. 4, 791–794 (1982).

    Google Scholar 

  24. S. M. Nikol'skii, A Course in Mathematical Analysis [in Russian], Vol. I, Nauka, Moscow (1973).

    Google Scholar 

  25. V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton Univ. Press (1960).

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M. V. Lomonosov Moscow State University. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 24, No. 4, pp. 38–63, July–August, 1983.

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Kamynin, L.I. A linear boundary-value problem for a second-order elliptic-parabolic equation. Sib Math J 24, 521–543 (1983). https://doi.org/10.1007/BF00969551

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  • DOI: https://doi.org/10.1007/BF00969551

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