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Transactions of the American Mathematical Society

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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: We study two-point Lagrange problems for integrands $L= L(t,u,v)$: \begin{equation}\tag {P} \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} \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.

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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