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

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.