Sufficiency theorems for local minimizers of multiple integrals of the calculus of variations

Let Omega subset of R-n be a bounded domain and let f : Omega x R-N X R-NXn --> R. Consider the functional I(u) := f(Omega)f(x,u,Du)dx, over the class of Sobolev functions W-1,W-q(Omega ;R-N) (1 less than or equal to q less than or equal to infinity) for which the integral on the right is well defined. In this paper we establish sufficient conditions on a given function u(0) and f to ensure that u(0) provides an L-r local minimizer for I where 1 less than or equal to r less than or equal to infinity. The case r = infinity 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 less than or equal to r less than or equal to infinity. Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of 'directional convergence'.