

Preprint 86/2000
Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations
Ali Taheri
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Submission date: 08. Jan. 2001
Pages: 28
published in: Proceedings of the Royal Society of Edinburgh / A, 131 (2001) 1, p. 155-184
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Abstract:
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''.