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

DOI:
https://doi.org/10.1090/S0002-9947-98-01648-1

MathSciNet review:
1357405

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study two-point Lagrange problems for integrands :

Under very weak regularity hypotheses [ is Hölder continuous and locally elliptic on each compact subset of ] we obtain, when is of superlinear growth in , a characterization of problems in which the minimizers of (P) are -regular for all boundary data. This characterization involves the behavior of the value function : defined by . Namely, all minimizers for (P) are -regular in neighborhoods of and if and only if is Lipschitz continuous at . 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 is nearly autonomous, nearly independent of , or jointly convex in are presented.

**[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), 325-388. MR**86k:49002****[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*, Teubner, Berlin. 2 vols. 1935: English transl., Holden-Day, 1965, 1967. MR**33:597**; MR**38:590****[Ce]**L. Cesari,*Optimization-Theory and Applications*, Springer, New York, 1983. MR**85c:49001****[CV1]**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**(1985), 73-98. MR**86k:49020****[CV2]**-,*Existence and regularity in the small in the calculus of variations*, J. Differential Equations**59**(1985), 336-354. MR**87a:49014****[Da]**A. M. Davie,*Singular minimizers in the Calculus of Variations in One Dimensional*, Arch. Rational Mech. Anal.**101**(1988), 161-177. MR**89c:49002****[ET]**I. Ekeland and R. Temam,*Convex Analysis and Variational Problems*, North-Holland, Amsterdam, 1976. MR**57:3931b****[FS]**W. Fleming and M. Soner,*Controlled Markov Processes and Viscosity Solutions*, Springer-Verlag, New York, 1993. MR**94e:93004****[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.

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

Email:
masychev@math.nsc.ru

**V. J. Mizel**

Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

Email:
vm09@andrew.cmu.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-01648-1

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