Null Lagrangians, Weak Continuity, and Variational Problems of Arbitrary Order

where 0 c R* is a bounded open set, u: R --* W, and Vtklu denotes the set of all partial derivatives nf of u of ail orders 11) < k. The admissible functions u are required to satisfy suitable boundary conditions or other constraints. It is well known that if I is sequentially weakly lower semicontinuous in the Sobolev space wk,“(.Q), 1 < a < co, and if F satisfies certain growth conditions, then the direct method of the calculus,of variations can be used to prove the existence of a minimizer for I. The lower semicontinuity of integrals of the form (1.1) has been studied by Morrey [28,29] in the case k = 1, and by Meyers [27] for arbitrary k. These authors showed that if F is continuous then a necessary and sufficient condition for I to be sequentially 135 0022.1236/81/050135-40$02.00/0