Let O ? Rn be a bounded domain and let f : O × RN × RN×n ? R. Consider the functional I(u) := ?Of(x, u, Du) dx, over the class of Sobolev functions W1,q(O;RN)(1 = q = 8) for which the integral on the right is well defined. In this paper we establish sufficient conditions on a given function u0 and f to ensure that u0 provides an Lr local minimizer for I where 1 = r = 8. The case r = 8 is somewhat known and there is a considerable literature on the subject treating the case min(n, N) = 1, mostly based on the field theory of the calculus of variations. The main contribution here is to present a set of sufficient conditions for the case 1 = r = 8. Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of 'directional convergence'.
|Number of pages||30|
|Journal||Proceedings of the Royal Society of Edinburgh, Section A: Mathematics|
|Publication status||Published - 2001|