Sufficient conditions for strong local minima: The case of 𝐶¹ extremals

Grabovsky, Yury; Mengesha, Tadele

doi:10.1090/S0002-9947-08-04786-7

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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