Regularity properties of solutions to the basic problem in the calculus of variations

Abstract:

This paper concerns the basic problem in the calculus of variations: minimize a functional $J$ defined by \[ J(x) = \int _a^b {L(t,x(t),\dot x(t))\;dt} \] over a class of arcs $x$ whose values at $a$ and $b$ have been specified. Existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs, conditions which must be strengthened in order that the standard necessary conditions apply. The question arises: What necessary conditions hold merely under hypotheses of existence theory, say the classical Tonelli conditions? It is shown that, given a solution $x$, there exists a relatively open subset $\Omega$ of $[a,b]$, of full measure, on which $x$ is locally Lipschitz and satisfies a form of the Euler-Lagrange equation. The main theorem, of which this is a corollary, can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions. Three such classes are identified, and results of Bernstein, Tonelli, and Morrey are extended. One of these classes is of a novel nature, and its study implies the new result that when $L$ is independent of $t$, the solution has essentially bounded derivative.