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

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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.

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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:
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