A condition on the value function both necessary and sufficient for full regularity of minimizers of onedimensional variational problems
Authors:
M. A. Sychev and V. J. Mizel
Journal:
Trans. Amer. Math. Soc. 350 (1998), 119133
MSC (1991):
Primary 49N60, 49L99, 49J45
MathSciNet review:
1357405
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Abstract: We study twopoint 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 EulerLagrange equations is available for guidance. Full regularity results for problems where is nearly autonomous, nearly independent of , or jointly convex in are presented.
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 W. Fleming and M. Soner, Controlled Markov Processes and Viscosity Solutions, SpringerVerlag, New York, 1993. MR 94e:93004
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 M. Lavrentiev, Sur quelques problèmes du calcul des variations, Ann. Pura Mat. Appl. 41 (1926), 107124.
 [Ma]
 M. Mania', Sopra un esempio di Lavrentieff, Boll. Un. Mat. Italiana 13 (1934), 147153.
 [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), 116120.
 [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.
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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:
http://dx.doi.org/10.1090/S0002994798016481
PII:
S 00029947(98)016481
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
