On the question of regularity of the solutions of variational problems

M. A. Sychev

Sbornik. Mathematics, Tome 75 (1993) no. 2, pp. 535-556

Résumé

Under the assumptions that $L(t,u,v)\in C(\mathbf R^3)$, $L_{vv}>\mu>0$, and $L>\mu v^2$ a study is made of the problem of minimizing the functional $\mathcal F(u(t))=\int_a^bL(t,u(t),\dot u(t))\,dt$ in the class of absolutely continuous functions $u(t)$ with $u(a)=A$ and $u(b)=B$. A direct method is presented for investigating the regularity of solutions and their dependence on the parameters of the problem. An example is given of a problem in which $L$ is analytic, $L_{vv}>\mu>0$, $L>\mu v^2$, and all the sequences minimizing the functional in the class of admissible smooth functions converge to a nonsmooth function $u_0(t)$ that is not a generalized solution of the Euler equation. An analogous example is given for the two-dimensional problem in the disk.

Bibliographie

[1] Problemy Gilberta, Nauka, M., 1969 | MR

[2] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[3] Giaquinta M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. Math. Studies, no. 105, Princeton Univ. Press, 1983 | MR | Zbl

[4] Tonelli L., Fondamenti di Calcolo delle Variazioni, v. II, Zanichelli, Bologna, 1921–1923

[5] Ball J. M., Mizel V. J., “One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation”, Arch. ration. mech. and anal., 90:1 (1985), 325–388 | DOI | MR | Zbl

[6] Clarke F. H., Vinter R. B., “Regularity properties of solutions to the basic problem in the calculus of variations”, TAMS, 289:1 (1985), 73–98 | DOI | MR | Zbl

[7] Lavrentiev M., “Sur quelques problems du calcul des variations”, Ann. mat. pure ed appl., 1926, no. 4, 7–28

[8] Ball J. M., Mizel V. J., “Singular minimizers for regular one dimensional problems in the calculus of variations”, Bull. Amer. Math. Soc., 2:1 (1984), 143–146 | DOI | MR

[9] Davie A. M., “Singular Minimizers in the Calculus of variations”, Arch. ration. mech. and anal., 101:2 (1988), 161–177 | MR | Zbl

[10] Clarke F. H., Vinter R. B., “On the conditions under which the Euler Equation or the maximum principle hold”, Appl. math. and optim., 1984, no. 12, 73–79 | DOI | MR | Zbl

[11] Bernshtein S. N., Sobr. soch., t. 3, Izd. AN SSSR, M., 1960

[12] Giaquinta M., Modica G., “Remarks on the Regularity of Minimizers of Certain Denerate Functionals”, Manuscr. math., 57:1 (1986), 55–101 | DOI | MR

[13] Manfredi J., “Regularity for Minima of Functionals with $p$-Growth”, J. different. equat., 76:2 (1988), 203–213 | DOI | MR

[14] Giaquinta M., “Growth conditions and regularity, a countrexample”, Manuscr. math., 59:2 (1987), 245–249 | DOI | MR

[15] Marcellini P., “Regularity of Minimizers of Integrals of the Calculus of Variations with Non Standard Growth Conditions”, Arch. ration. mech. and anal., 105:3 (1989), 267–284 | MR | Zbl

[16] Boccardo L., Marcellini P., Sbordone C., “$L^\infty$-regularity for variational problems with sharp non standard growth conditions”, Boll. Unione mat. ital., VII Ser., IV-A:2 (1990), 219–227 | MR

[17] Fusco N., Sbordone C., “Local boundeness of minimizers in a Limit Case”, Manuscr. math., 69:1 (1990), 19–25 | DOI | MR | Zbl

[18] Zhikov V. V., “Effekt Lavrenteva i usrednenie nelineinykh variatsionnykh zadach”, Differents. uravneniya, 27:1 (1991), 142–151 | MR

[19] Clarke F. H., Vinter R. B., “Existence and regularity in the small in the calculus of variations”, J. different. equat., 59:3 (1985), 336–355 | DOI | MR

[20] Loewen P. D., “On the Lavrentiev phenomenon”, Can. math. bull., 30:1 (1987), 102–107 | MR