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Sufficiency theorems for local minimizers of multiple integrals of the calculus of variations
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'.
History
Publication status
- Published
Journal
Proceedings of the Royal Society of Edinburgh: Section A MathematicsISSN
0308-2105Publisher
Cambridge University PressExternal DOI
Issue
1Volume
131Page range
155-184Department affiliated with
- Mathematics Publications
Full text available
- No
Peer reviewed?
- Yes