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Examples of classically unsolvable regular scalar variational problems satisfying standard growth conditions

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References

  1. Hilbert's Problems [in Russian], Nauka, Moscow (1969).

  2. M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, Princeton (1983). (Ann. Math. Stud.,105.)

    Google Scholar 

  3. M. A. Sychëv, “A criterion for continuity of an integral functional on a sequence of functions,” Sibirsk. Mat. Zh.,36, No. 1, 203–214 (1995).

    Google Scholar 

  4. L. Tonelli, Fondamenti di Calcolo delle Variazioni. Vol. 1 and 2, Zanichelli, Bologna (1921, 1923).

    Google Scholar 

  5. 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. 1, 325–388 (1985).

    Google Scholar 

  6. F. H. Clarke and R. B. Vinter, “Regularity properties of solutions to the basic problem in 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. De Giorgi, “On differentiability and analyticity of extremals of multiple regular integrals,” Matematika,4, No. 6, 23–38 (1960).

    Google Scholar 

  9. L. Lichtenstein, “Uber den analytischen Charakter der Losungen zweidimesionaler Variationsprobleme,” Bull. de J'Ac. de Sc. de Cracovie (A). Dec., 915–941 (1912).

  10. E. Hopf, “Zum analytischen Character der Losungen zweidimesionaler Variationsprobleme,” Math. Z.,30, 404–413 (1929).

    Article  MathSciNet  Google Scholar 

  11. C. Morrey, “Existence and differentiability theorems for the solutions of variational problems for multiple integrals,” Bull. Amer. Math. Soc.,46, 127–166 (1940).

    Google Scholar 

  12. S. N. Bernstein, Collected Works. Vol. 3 [in Russian], Izdat. Akad. Nauk SSSR, Moscow (1960).

    Google Scholar 

  13. O. A. Ladyzhenskaya and N. N. Ural'tseva, “Quasilinear elliptic equations and variational problems in several independent variables,” Uspekhi Mat. Nauk,16, No. 1, 19–91 (1961).

    Google Scholar 

  14. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  15. M. Giaquinta and G. Modica, “Remarks on the regularity of minimizers of certain degenerate functionals,” Manuscripta Math.,57, No. 1, 55–101 (1986).

    Article  Google Scholar 

  16. J. J. Manfredi, “Regularity for minima of functionals withp-growth,” J. Differential Equations,76, No. 2, 203–212 (1988).

    Article  Google Scholar 

  17. P. Marcellini, “Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions,” Arch. Rational Mech. Anal.,105, No. 3, 267–284 (1989).

    Article  Google Scholar 

  18. N. Fusco and C. Sbordone, “Some remarks on the regularity of minima of anisotropic integrals,” Comm. Partial Differential Equations,18, No. 1–2, 153–169 (1992).

    Google Scholar 

  19. Tang Qi, “Regularity of minimizers of non-isotropic integrals of the calculus of variations,” Ann. Mat. Pura Appl. Ser. 4,164, 77–87 (1993).

    Article  Google Scholar 

  20. E. Mascolo and G. Papi, Local Boundedness of Minimizers of Integrals of the Calculus of Variations [Preprint No. 309/13-92], Univ. Firenze (1992).

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

    Article  Google Scholar 

  22. P. D. Loewen, “On the Lavrentiev phenomenon,” Canad. Math. Bull.,30, No. 1, 102–107 (1987).

    Google Scholar 

  23. 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 

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

    Google Scholar 

  25. F. H. Clarke and R. B. Vinter, “Regularity of solutions to variational problems with polynomial Lagrangians,” Bull. Pol. Acad. Sci. Math.,34, No. 1–2, 73–81 (1986)

    Google Scholar 

  26. J. M. Ball and N. S. Nadirashvili, “Universal singular sets for one-dimensional variational problems,” Calc. Var. Partial Differential Equations,1, No. 4, 417–428 (1993).

    Article  Google Scholar 

  27. M. A. Sychëv, “Lebesgue measure of the universal singular set for the simplest problems in the calculus of variations,” Sibirsk. Mat. Zh.,36, No. 6, 1373–1383 (1994).

    Google Scholar 

  28. M. Giaguinta and E. Giusti, “Quasi-minima,” Ann. Inst. H. Poincaré Anal. Non Linéaire,1, No. 2, 79–101 (1984).

    Google Scholar 

  29. M. Giaguinta and E. Giusti, “GlobalC 1,α-regularity for second order quasilinear elliptic equations in divergence form,” J. Reine Angew. Math.,351, 55–65 (1984).

    Google Scholar 

  30. M. Giaguinta and E. Giusti, “Sharp estimates for the derivatives of local minima of variational integrals,” Boll. Un. Mat. Ital. A (7),6, No. 3, 239–248 (1984).

    Google Scholar 

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The research is financially supported by the Russian Foundation for Basic Research (Grant 94-01-00878).

Translated from Sibirskii Matematicheskii, Vol. 37, No. 6, pp. 1380–1396, November–December, 1996.

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Sychëv, M.A. Examples of classically unsolvable regular scalar variational problems satisfying standard growth conditions. Sib Math J 37, 1212–1227 (1996). https://doi.org/10.1007/BF02106746

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