Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A condition on the value function
both necessary and sufficient
for full regularity of minimizers
of one-dimensional variational problems

Authors: M. A. Sychev and V. J. Mizel
Journal: Trans. Amer. Math. Soc. 350 (1998), 119-133
MSC (1991): Primary 49N60, 49L99, 49J45
MathSciNet review: 1357405
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study two-point Lagrange problems for integrands $L=\break L(t,u,v)$:

 \begin{equation}\begin{split} F[u]=\int _a^b L(t,u(t),\dot u(t))&\,dt \to \inf,\\ & u\in\mathcal A=\{v\in W^{1,1} ([a,b];\mathbb R^n)|v(a)=A,v(b)=B\}. \end{split}\tag{P}\label{tagp} \end{equation}

Under very weak regularity hypotheses [$L$ is Hölder continuous and locally elliptic on each compact subset of $\mathbb R\times\mathbb R^n\times\mathbb R^n$] we obtain, when $L$ is of superlinear growth in $v$, a characterization of problems in which the minimizers of (P) are $C^1$-regular for all boundary data. This characterization involves the behavior of the value function $S$: $\mathbb R\times\mathbb R^n\times\mathbb R\times\mathbb R^n\to\mathbb R$ defined by $S(a,A,b,B)=\inf _{\mathcal A} F$. Namely, all minimizers for (P) are $C^1$-regular in neighborhoods of $a$ and $b$ if and only if $S$ is Lipschitz continuous at $(a,A,b,B)$. Consequently problems (P) possessing no singular minimizers are characterized in cases where not even a weak form of the Euler-Lagrange equations is available for guidance. Full regularity results for problems where $L$ is nearly autonomous, nearly independent of $u$, or jointly convex in $(u,v)$ are presented.

References [Enhancements On Off] (What's this?)

  • [BM] J. M. Ball and V. J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation, Arch. Rational Mech. Anal. 90 (1985), no. 4, 325–388. MR 801585,
  • [Bo] O. Bolza, Vorlesungen über Variationsrechnung, Teubner 1909 (Koehler and Amelang 1949).
  • [Ca] C. Carathéodory, Calculus of variations and partial differential equations of the first order. Part I: Partial differential equations of the first order, Translated by Robert B. Dean and Julius J. Brandstatter, Holden-Day, Inc., San Francisco-London-Amsterdam, 1965. MR 0192372
    C. Carathéodory, Calculus of variations and partial differential equations of the first order. Part II: Calculus of variations, Translated from the German by Robert B. Dean, Julius J. Brandstatter, translating editor, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1967. MR 0232264
  • [Ce] Lamberto Cesari, Optimization—theory and applications, Applications of Mathematics (New York), vol. 17, Springer-Verlag, New York, 1983. Problems with ordinary differential equations. MR 688142
  • [CV1] Jerzy Sztajnic, Transversality conditions for controls with bounded variation, Acta Univ. Lodz. Folia Math. 1 (1984), 131–149 (English, with Polish summary). MR 809022
  • [CV2] Frank H. Clarke and R. B. Vinter, Existence and regularity in the small in the calculus of variations, J. Differential Equations 59 (1985), no. 3, 336–354. MR 807852,
  • [Da] A. M. Davie, Singular minimisers in the calculus of variations in one dimension, Arch. Rational Mech. Anal. 101 (1988), no. 2, 161–177. MR 921937,
  • [ET] Ivar Ekeland and Roger Temam, Analyse convexe et problèmes variationnels, Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French). Collection Études Mathématiques. MR 0463993
    Ivar Ekeland and Roger Temam, Convex analysis and variational problems, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. Translated from the French; Studies in Mathematics and its Applications, Vol. 1. MR 0463994
  • [FS] Wendell H. Fleming and H. Mete Soner, Controlled Markov processes and viscosity solutions, Applications of Mathematics (New York), vol. 25, Springer-Verlag, New York, 1993. MR 1199811
  • [L] M. Lavrentiev, Sur quelques problèmes du calcul des variations, Ann. Pura Mat. Appl. 41 (1926), 107-124.
  • [Ma] M. Mania', Sopra un esempio di Lavrentieff, Boll. Un. Mat. Italiana 13 (1934), 147-153.
  • [S1] M. A. Sychev, On the question of regularity of the solutions of variational problems, Russian Acad. Sci. Sb. Math. 75 (1993), No 2.
  • [S2] -, On a classical problem of the calculus of variations, Soviet Math. Dokl. 44 (1992), 116-120.
  • [S3] -, Lebesgue measure of the universal singular set for the simplest problems in the calculus of variations, Siberian Math. J. 35 (1994).
  • [T] T. Tonelli, Fondamenti di calcolo delle variazíoni, vol. II, Zanichelli, Bologna, 1921.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 49N60, 49L99, 49J45

Retrieve articles in all journals with MSC (1991): 49N60, 49L99, 49J45

Additional Information

M. A. Sychev
Affiliation: Institute of Mathematics of the Russian Academy of Science (Siberian Department), 630090 Novosibirsk, Russia

V. J. Mizel
Affiliation: Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

Keywords: Tonelli partial regularity result, conditionally equicontinuous family, value function
Received by editor(s): August 17, 1995
Received by editor(s) in revised form: December 28, 1995
Additional Notes: Research of the first author partially supported by the NSF under Grant DMS9320104 and by the grant “Lavrentiev’s effect and applications” of the Siberian Division of the Russian Academy of Science
Research of the second author partially supported by the NSF under Grant DMS9201221
Article copyright: © Copyright 1998 American Mathematical Society