Sufficient conditions for strong local minima: The case of $C^{1}$ extremals
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- by Yury Grabovsky and Tadele Mengesha PDF
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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 âlocalizesâ, i.e. its action can be represented as a superposition of âWeierstrass needlesâ, which cannot decrease the functional either, due to the uniform quasiconvexity conditions.References
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Additional Information
- Yury Grabovsky
- Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122-6094
- MR Author ID: 338656
- Email: yury@temple.edu
- Tadele Mengesha
- Affiliation: Department of Mathematics and Statistics, Coastal Carolina University, Conway, South Carolina 29528-6054
- Email: mengesha@coastal.edu
- Received by editor(s): February 26, 2007
- Published electronically: October 24, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2457407