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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Let tex2html_wrap_inline14 be a bounded domain and let tex2html_wrap_inline16. Consider the functional
over the class of Sobolev functions tex2html_wrap_inline20 (tex2html_wrap_inline22) for which the integral on the right is well defined. In this paper we establish sufficient conditions on a given function tex2html_wrap_inline24 and f to ensure that tex2html_wrap_inline24 provides an tex2html_wrap_inline30 local minimizer for I where tex2html_wrap_inline34. The case tex2html_wrap_inline36 is somewhat known and there is a considerable literature on the subject treating the case tex2html_wrap_inline38, 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 tex2html_wrap_inline40. Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of ``directional convergence''.

24.11.2021, 02:11