transactions of the american mathematical society Volume 289, Number 1, May 1985 REGULARITY PROPERTIES OF SOLUTIONS TO THE BASIC PROBLEM IN THE CALCULUS OF VARIATIONS BY F. H. CLARKE1 AND R. B. VINTER ABSTRACT. This paper concerns the basic problem in the calculus of varia- tions: minimize a functional J defined by J(x) = / L(t,x(t),x(t))dt Jo 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 f! 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 con- junction with various classes of additional hypotheses to deduce the global smoothness of solutions. Three such classes are identified, and results of Bern- stein, 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. 1. Introduction. The basic problem in the calculus of variations, which we denote by (P), is that of minimizing the functional J defined by J(x) := / L{t,x{t),x(t))dt Ja over a given class of functions x, assuming given values at a and b: x(a) = A, x(b) — B. (Here L is a function from [a, b] x Rn x Rn to R, and x signifies the derivative of x.) It has been studied now for almost three hundred years. One of the fundamental issues that was broached relatively late is that of exis- tence: under what hypotheses on L, and within what class of functions x, can one be assured that a solution exists? It was Tonelli, in work that also had great signif- icance in several areas of functional analyis, who was able to develop a satisfactory existence theory. Let us suppose that L is C2, the function v —> L(t,x,v) satisfies globally Lvv > 0 for all (t,x), and the following coercivity condition holds: L{t,x,v) > ß\v\2 + A for all (í,x,u), Received by the editors January 29, 1984 and, in revised form, July 5, 1984. 1980 Mathematics Subject Classification. Primary 49B05. 'The support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. ©1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page 73 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use