Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 3
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Sychev, M. A. 2011. Another Theorem of Classical Solvability ‘In Small’ for One-Dimensional Variational Problems. Archive for Rational Mechanics and Analysis, Vol. 202, Issue. 1, p. 269.

    Sychev, Mikhail A. 2008. Local minimizers of one-dimensional variational problems and obstacle problems. Comptes Rendus Mathematique, Vol. 346, Issue. 21-22, p. 1213.

    Taheri, Ali 2005. Local Minimizers and Quasiconvexity – the Impact of Topology. Archive for Rational Mechanics and Analysis, Vol. 176, Issue. 3, p. 363.

  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 131, Issue 1
  • February 2001, pp. 155-184

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

  • Ali Taheri (a1)
  • DOI:
  • Published online: 11 July 2007

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’.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *