### Abstract

Let O ? R^{n} be a bounded domain and let f : O × R^{N} × R^{N×n} ? R. Consider the functional I(u) := ?_{O}f(x, u, Du) dx, over the class of Sobolev functions W^{1,q}(O;R^{N})(1 = q = 8) 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 = r = 8. The case r = 8 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 = 8. Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of 'directional convergence'.

Original language | English |
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Pages (from-to) | 155-184 |

Number of pages | 30 |

Journal | Proceedings of the Royal Society of Edinburgh, Section A: Mathematics |

Volume | 131 |

Issue number | 1 |

Publication status | Published - 2001 |