Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations
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Submission date: 08. Jan. 2001
published in: Proceedings of the Royal Society of Edinburgh / A, 131 (2001) 1, p. 155-184
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Let be a bounded domain and let . Consider the functional
over the class of Sobolev functions () for which the integral on the right is well defined. In this paper we establish sufficient conditions on a given function and f to ensure that provides an local minimizer for I where . The case is somewhat known and there is a considerable literature on the subject treating the case , 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 . Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of ``directional convergence''.