Skip to main content
SpringerLink
Log in

Qualitative properties of solutions to the Euler equation and solvability of one-dimensional regular variational problems in the classical sense

  • 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

References

  1. I. M. Gel'fand and S. V. Fomin, Calculus of Variations [in Russian], Fizmatgiz, Moscow (1961).

    Google Scholar 

  2. The Hilbert Problems [in Russian], Nauka, Moscow (1969).

  3. C. Marcelli and A. Salvadori, “Remarks on growth conditions in calculus of variations,” Atti Sem. Mat. Fis. Univ. Modena,40, No. 1, 199–214 (1992).

    Google Scholar 

  4. N. I. Akhiezer, Calculus of Variations [in Russian], Vyshcha Shkola, Khar'kov (1961).

    Google Scholar 

  5. L. Tonelli, Fondamenti di Calcolo Delle Variazioni. Vol. I, II, Zanichelli, Bologna (1921, 1923).

    Google Scholar 

  6. F. H. Clarke and R. B. Vinter, “Regularity properties of solutions to the basic problem of the calculus of variations,” Trans. Amer. Math. Soc,289, No. 1, 73–98 (1985).

    Google Scholar 

  7. M. A. Sychëv, “To the question of regularity of solutions to some variational problems,” Mat. Sb.,183, No. 4, 118–142 (1992).

    Google Scholar 

  8. E. Heil, “Convex minimizers of variational problems. Applied Geometry and Discrete Mathematics,” Amer. Math. Soc. Transl. Ser. 2,4, 325–333 (1991).

    Google Scholar 

  9. B. Botteron and B. Dacorogna, “Existence of solutions for a variational problem associated to models in optimal foraging theory,” J. Math. Anal. Appl.,147, No. 1, 263–276 (1990).

    Article  Google Scholar 

  10. B. Botteron and B. Dacorogna, “Existence and nonexistence results for noncoercive variational problems and applications in ecology,” J. Differential Equations,85, No. 2, 214–235 (1990).

    Article  Google Scholar 

  11. F. Zh. Sadyrbaev, “On regularity of solutions to the basic problem of the calculus of variations,” Mat. Zametki,52, No. 5, 97–101 (1992).

    Google Scholar 

  12. F. H. Clarke and P. D. Loewen, “Variational problems with Lipschitzian minimizers,” Ann. Inst. H. Poincaré Anal. Non Linéaire,6, 185–209 (1989).

    Google Scholar 

  13. F. H. Clarke and P. D. Loewen, “An intermediate existence theory in the calculus of variations,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4),16, No. 4, 487–526 (1989).

    Google Scholar 

  14. F. H. Clarke, “An indirect method in the calculus of variations,” Trans. Amer. Math. Soc,336, No. 2, 655–675 (1993).

    Google Scholar 

  15. B. Botteron and P. Marcellini, “A general approach to the existence of minimizers of onedimensional noncoercive integrals of the calculus of variations,” Ann. Inst. H. Poincaré Anal. Non Linéaire,8, No. 2, 197–223 (1991).

    Google Scholar 

  16. L. Ambrosio, O. Ascenci, and G. Buttazzo, “Lipschitz regularity for the minimizers of integral functions with highly discontinuous integrals,” J. Math. Anal. Appl.,142, No. 2, 301–316 (1985).

    Article  Google Scholar 

  17. M. A. Sychëv, “To the question of regularity of solutions to some variational problems,” Dokl. Akad. Nauk SSSR,316, No. 6, 1326–1330 (1991).

    Google Scholar 

  18. M. A. Sychëv, To the Problem of Classical Solvability of Regular Variational Problems [in Russian], Dis. ... Kand. Fiz.-Mat. Nauk, Novosibirsk (1992).

    Google Scholar 

  19. J. M. Ball and V. J. Mizel, “One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equations,” Arch. Rational Mech. Anal.,90, No. 4, 325–388 (1985).

    Google Scholar 

  20. A. M. Davie, “Singular minimizers in the calculus of variations in one dimension,” Arch. Rational Mech. Anal.,101, No. 2, 161–177 (1988).

    Article  Google Scholar 

  21. F. H. Clarke and R. B. Vinter, “On the conditions under which the Euler equation or the maximum principle hold,” Appl. Math. Optim.,12, No. 1, 73–79 (1984).

    Article  Google Scholar 

  22. M. A. Sychëv, “About continuous dependence on the integrand of solutions to simplest variational problems,” Sibirsk. Mat. Zh.,36, No. 2, 379–388 (1995).

    Google Scholar 

  23. H. Lewy, “Über die Methode der Differenzengleichungen zur Losung von Variations- und Randwertproblemen,” Math. Ann.,98, 107–124 (1927).

    Article  MathSciNet  Google Scholar 

  24. T. I. Zelenyak, “On stabilization of solutions of boundary value problems for second-order parabolic equations in a single space variable,” Differentsial'nye Uravneniya,4, No. 1, 34–45 (1968).

    Google Scholar 

  25. V. S. Belonosov and T. I. Zelenyak, Nonlocal Problems in the Theory of Quasilinear Parabolic Equations [in Russian], Novosibirsk. Univ., Novosibirsk (1975).

    Google Scholar 

  26. L. Cesari, Optimization—Theory and Applications, Springer, New York etc. (1983).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Novosibirsk

    M. A. Sychëv

Authors
  1. M. A. Sychëv
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research is part of the project supported by the Russian Foundation for Fundamental Research (under Grant 94-01-00878).

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 36, No. 4, pp. 873–892, July–August, 1995.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sychëv, M.A. Qualitative properties of solutions to the Euler equation and solvability of one-dimensional regular variational problems in the classical sense. Sib Math J 36, 753–769 (1995). https://doi.org/10.1007/BF02107333

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02107333

Keywords

Search

Navigation