Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

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

Ali Taheria1 p1

a1 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA


Let Ω ⊂ Rn be a bounded domain and let f : Ω × RN × RN ×nR. Consider the functional


over the class of Sobolev functions W1,q(Ω;RN) (1 ≤ q ≤ ∞) 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 ≤ ∞. The case r = ∞ 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 < ∞. Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of ‘directional convergence’.

(Received October 19 1998)

(Accepted November 10 1999)


p1 Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK.