Semicontinuity problems in the calculus of variations

Abstract

Two meaningful cases in which (1.2b) is satisfied are the following: (i)f = f(x, 5) is measurable in x and upper semicontinuous in t; (ii)f is a Caratheodory function, i.e. measurable in x and continuous in (s, 5). Several authors have studied (and proved, under suitable hypothesis) the sequential lower semicontinuity (s.1.s.) of F(Q, . ) in the weak topology of HlsP(Q). A well known theorem of Serrin [l] assures the s.1.s. of F under the assumption that f is a non negative continuous function, convex in r. More recently some improvements of Serrin’s theorem have been given by De Giorgi [2], Berkowitz [3], Cesari [4], Ioffe [S], Olech [6] by considering the s.1.s. of the functional