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Sufficient conditions for strong local minima: The case of $ C^{1}$ extremals


Authors: Yury Grabovsky and Tadele Mengesha
Journal: Trans. Amer. Math. Soc. 361 (2009), 1495-1541
MSC (2000): Primary 49K10, 49K20
DOI: https://doi.org/10.1090/S0002-9947-08-04786-7
Published electronically: October 24, 2008
MathSciNet review: 2457407
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Abstract: In this paper we settle a conjecture of Ball that uniform quasiconvexity and uniform positivity of the second variation are sufficient for a $ C^{1}$ extremal to be a strong local minimizer. Our result holds for a class of variational functionals with a power law behavior at infinity. The proof is based on the decomposition of an arbitrary variation of the dependent variable into its purely strong and weak parts. We show that these two parts act independently on the functional. The action of the weak part can be described in terms of the second variation, whose uniform positivity prevents the weak part from decreasing the functional. The strong part ``localizes'', i.e. its action can be represented as a superposition of ``Weierstrass needles'', which cannot decrease the functional either, due to the uniform quasiconvexity conditions.


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Additional Information

Yury Grabovsky
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122-6094
Email: yury@temple.edu

Tadele Mengesha
Affiliation: Department of Mathematics and Statistics, Coastal Carolina University, Conway, South Carolina 29528-6054
Email: mengesha@coastal.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04786-7
Received by editor(s): February 26, 2007
Published electronically: October 24, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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