Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands

Abstract

The study of regularity properties for minimizers of problems of the form min , u, u’) dt: u E IV’-‘(Q; R”), u(a) = a, u(b) = B (1.1) has been undertaken for a long time in Calculus of Variations, and a number of results are by now available on this subject. When f(t, s, z) is a smooth coercive function, problem ( 1.1) was considered by Tonelli (see [ 18, 19]), who proved that every minimizer u(t) of the problem (1.1) is locally Lipschitz on a relatively open set of full measure in [a, b]. Under some additional assumptions on f, it is also possible to prove global regularity theorems (see for instance Cesari [3, Chap 2, Sect. 61). In recent years, much attention has been devoted to problems of the form (l.l), on the one hand in weakening the hypotheses of Tonelli’s regularity results, and on the other in finding counterexamples of nonregular minimizers for coercive functionals. With respect to this last subject, we recall that it has been shown (see Ball and Mizel [l] and Clarke and Vinter [6]) that there exist polynomial functions f(t, s, z) such that, for suitable C > 0 and p,