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

Taheri, Ali (2001) Sufficiency theorems for local minimizers of multiple integrals of the calculus of variations. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 131 (1). pp. 155-184. ISSN 0308-2105

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Abstract

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'.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Depositing User: Ali Taheri
Date Deposited: 06 Feb 2012 21:19
Last Modified: 10 Jul 2012 14:34
URI: http://sro.sussex.ac.uk/id/eprint/30744
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