Semi-continuity

Abstract: Here x = (x', , x'), u = (u .., utm), Q is a bounded domain and the integrand f(x, p, **, p) is a continuous function of its arguments. In 1952 Morrey studied the case I = 1 and introduced the concept of quasiconvexity (see [3]). Extending this concept to the cases I > 1, we say that an integrand f(pl) is quasi-convex if each polynomial of degree ? I minimizes the integral, fnf(D'u(x))dx, among all functions whose derivatives of order I 1 satisfy a Lipschitz condition on Q (we denote this function space by 4' (Q)) and assume the same Dirichlet data on a0 as the polynomial. The reason for the term quasi-convexity becomes clear when one sees that convexity implies quasiconvexity and quasi-convexity in turn implies the Legendre condition (at least for smooth integrands) which contains within it various convexities. Hence, quasi-convexity is a condition which falls between convexity and a weaker kind of convexity. In Theorems 1 and 2 of ?2,1 extend Morrey's results by showing that the necessary and sufficient condition for lower semi-continuity of I(u; Q) in f under uniform convergence of derivatives of order ? I 1 and uniform boundedness of derivatives of order 1, is that f(x, ps,.,p') be quasi-convex in p1 for each fixed value of the variables (x, p, ... p1). The proof is a straightforward extension of Morrey's for the case 1 = 1. However, it contains the added feature that the necessity is derived asssuming only that the admissible functions satisfy fixed Dirichlet boundary conditions. I then go on to consider lower semi-continuity under weak convergence in a space 4/ r(Q) (1 < r < oo ), the space of functions with strong derivatives up to the order I which are in Yr(Q). Two cases are considered, though they do not require separate treatment: first, the case where the admissible functions satisfy a fixed Dirichlet boundary condition, and second, the case of no boundary condition.