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

We study two-point Lagrange problems for integrands L = L(t,u,v): (P) F[u] = integral(a)(b) L(t,u(t),u(t))dt --> inf, u is an element of A = {v is an element of W-1,W-1([a,b];R-n)\v(a)= A, v(b) = B}. Under very weak regularity hypotheses [L is Holder continuous and locally elliptic on each compact subset of R x R-n x R-n] we obtain, when L is of superlinear growth in u, 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: R x R-n x R x R-n --> R defined by S(a, A, b, B) = inf(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.