Skip to main content
SpringerLink
Log in

Anisotropic strict extremum principle for second-order elliptic-parabolic equations

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Literature Cited

  1. E. Hopf, “Elementare Bermekungen über die Lösungen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus,” Sitzungsber. Preus. Akad. Wiss.,19, 147–152 (1927).

    Google Scholar 

  2. A. D. Aleksandrov, “The investigation of the maximum principle. I,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 5, 126–157 (1958).

    Google Scholar 

  3. P. K. Rashevskii, “On the possibility of joining two arbitrary points of a completely nonholonomic space by an admissible line,” Uch. Zap. Pedagog. Inst. im. Libknekhta, Ser. Fiz. Mat.,2, 83–94 (1938).

    Google Scholar 

  4. J.-M. Bony, “Sur la propagation des maximums et l'uncite du probleme de Cauchy pour les operateurs elliptiques degeneres du second ordre,” C. R. Acad. Sci. Paris,266, No. 15, A763-A765 (1968).

    Google Scholar 

  5. J.-M. Bony, “Principle du maximum, inegalite de Harnack et unicite du probleme de Cauchy pour les operateurs elliptiques degeneres,” Ann. Inst. Fourier,19, No. 1, 277–304 (1969).

    Google Scholar 

  6. L. I. Kamynin and B. N. Khimchenko, “On investigations of the maximum principle,” Dokl. Akad. Nauk SSSR,240, No. 4, 774–777 (1978).

    Google Scholar 

  7. L. I. Kamynin and B. N. Khimchenko, “On an isotropic strict extremum principle,” Vestn. Mosk. Univ., Ser. I, Mat. Mekh., No. 6, 58–62 (1979).

    Google Scholar 

  8. L. L. Kamynin and B. N. Khimchenko, “On an isotropic strict extremum principle in a plane domain,” Differents. Uravn.,15, No. 7, 1296–1306 (1979).

    Google Scholar 

  9. L. I. Mamynin and B. N. Khimchenko, “On a strict extremum principle that is isotropic in a plane domain,” Sib. Mat. Zh.,20, No. 2, 278–292 (1979).

    Google Scholar 

  10. L. I. Kamynin and B. N. Khimchenko, “On a strict extremum principle of a weakly elliptically connected second-order operator,” Zh. Vychisl. Mat. Mat. Fiz.,19, No. 1, 129–142 (1979).

    Google Scholar 

  11. L. I. Kamynin and B. N. Khimchenko, “On the strict extremum principle for a D-(Φ, Ω)-elliptically connected second-order operator,” Differents. Uravn.,15, No. 7, 1307–1317 (1979).

    Google Scholar 

  12. L. I. Kamynin and B. N. Khimchenko, “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 

  13. L. I. Kamynin and B. N. Khimchenko, “On the strict extremum principle for a D-(Φ, Ω)elliptically connected second-order operator,” mat. Sb.,112(154), No. 1 (5), 24–55 (1980).

    Google Scholar 

  14. L. I. Kamynin and B. N. Khimchenko, “On theorems of Giraud type for a second-order elliptic operator that is weakly degenerate near the boundary,” Dokl. Akad. Nauk SSSR,224, No. 4, 752–755 (1975).

    Google Scholar 

  15. 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 

  16. L. Nirenberg, “A strong maximum principle for parabolic equations,” Commun. Pure Appl. Math.,6, No. 2, 167–177 (1953).

    Google Scholar 

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

    Google Scholar 

  18. O. A. Oleinik and E. V. Radkevich (Radkevic), Second-Order Equations with Nonnegative Characteristic Form, Amer. Math. Soc. Providence (1973).

    Google Scholar 

  19. L. I. Kamynin and B. N. Khimchenko, “On the strict extremum principle for degenerate second-order parabolic equations,” Dokl. Akad. Nauk SSSR,236, No. 5, 1060–1063 (1977).

    Google Scholar 

  20. 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 

  21. L. I. Kamynin and B. N. Khimchenko, “On a strict extremum principle for weakly parabolically connected second-order operator,” Zh. Vychisl. Mat. Mat. Fiz.,21, No. 4, 907–925 (1981).

    Google Scholar 

  22. L. I. Kamynin and B. N. Khimchenko, “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 

  23. L. I. Kamynin and B. N. Khimchenko, “On the maximum principle for second-order parabolic equations,” Dokl. Akad. Nauk SSSR,200, No. 2, 282–285 (1971).

    Google Scholar 

  24. L. I. Kamynin and B. N. Khimchenko, “On the maximum principle for second-order ellipticparabolic equations,” Sib. Mat. Zh.,13, No. 4, 773–789 (1972).

    Google Scholar 

  25. L. I. Kamynin and B. N. Khimchenko, “On the applications of the maximum principle to second-order parabolic equations,” Dokl. Akad. Nauk SSSR,204, No. 3, 529–532 (1972).

    Google Scholar 

  26. 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 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  31. L. P. Kuptsov, “A mean value theorem and a maximum principle for Kolmogorov's equation,” Mat. Zametki,15, No. 3, 479–489 (1974).

    Google Scholar 

  32. 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 

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

    Google Scholar 

  34. S. Lefschetz, Differential Equations: Geometric Theory, Wiley-Interscience, New York (1963).

    Google Scholar 

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

  36. P. Hartman, Ordinary Differential Equations, Wiley, New York (1964).

    Google Scholar 

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

    Google Scholar 

  38. E. M. Landis, Second-Order Elliptic and Parabolic Equations [in Russian], Nauka, Moscow (1971).

    Google Scholar 

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

    Google Scholar 

Download references

Authors
  1. L. I. Kamynin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. N. Khimchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

M. V. Lomonosov Moscow State University. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 24, No. 2, pp. 26–55, March–April, 1983.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kamynin, L.I., Khimchenko, B.N. Anisotropic strict extremum principle for second-order elliptic-parabolic equations. Sib Math J 24, 173–198 (1983). https://doi.org/10.1007/BF00968735

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00968735

Keywords

Search

Navigation