The calculus of variations and materials science

Ball, J. M.

doi:10.1090/qam/1668735

Cited by 37 publications

(31 citation statements)

References 93 publications

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“…In Section 7 we use the coercivity condition (H3) to reduce the problem to W 1,p local minima (see e.g. [3,48] for a definition). From that point on the coercivity condition (3.12) (or (3.11), if p = 2) is no longer needed.…”

Section: The Necessary Conditions (41)-(43) and The Uniform Continumentioning

confidence: 99%

“…If in the scalar case the field theory approach yields sufficient conditions for strong local minima that are very close to the necessary conditions, this is not so in the vectorial case. The reason was pointed out by Ball in [3]. The field theory uses translations by null-Lagrangians (see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…The field theory uses translations by null-Lagrangians (see e.g. [9,18,54]), and is thus associated with polyconvexity, while Ball's conjecture [3,Section 6.2] calls for quasiconvexity-based sufficient conditions. Among non-field theory approaches we are aware of only two: Levi's "expansion method" [34] (see also [41]) and Hestenes's normalized or directional convergence method [24].…”

Section: Introductionmentioning

confidence: 99%

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

Grabovsky

Mengesha

2008

Trans. Amer. Math. Soc.

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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 on the functional independently. 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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Section: The Necessary Conditions (41)-(43) and The Uniform Continumentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Grabovsky

Mengesha

2008

Trans. Amer. Math. Soc.

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“…Various numerical methods for detecting singular minimizers have been developed in recent years [2,6,16,17,18,19] (see [7] for a survey and more references), and the corresponding convergence theorems were proved for the case when the integrand f is convex with respect to the deformation gradient Du.…”

Section: 3)mentioning

confidence: 99%

Numerical Solution of Nonlinear Elasticity Problems With Lavrentiev Phenomenon

Bai

2007

Math. Models Methods Appl. Sci.

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Abstract. A convergence theory is established for a truncation method in solving polyconvex elasticity problems involving the Lavrentiev phenomenon. Numerical results on a recent example by Foss et al, which has a polyconvex integrand and admits continuous singular minimizers, not only verify our convergence theorems but also provid a sharper estimate on the upper bound of a perturbation parameter for the existence of the Lavrentiev phenomenon in the example.

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“…Various numerical methods for detecting singular minimizers involving the Lavrentiev phenomenon have been developed in recent years [4,13,14,15] (see [5] for a survey and more references), and corresponding convergence theorems were proved to guarantee that these methods can be successfully used in detecting singular minimizers in case 3 when 1 < p q (see (1.8)). …”

Section: I(u) = Infmentioning

confidence: 99%

A Truncation Method for Detecting Singular Minimizers Involving the Lavrentiev Phenomenon

Bai

2006

Math. Models Methods Appl. Sci.

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Abstract. A numerical method using the truncation technique on the integrand is developed for computing singular minimizers or singular minimizing sequences in variational problems involving the Lavrentiev phenomenon. It is proved that the method can detect absolute minimizers with various singularities whether the Lavrentiev phenomenon is involved or not. It is also proved that, when the absolute infimum is not attainable, the method can produce minimizing sequences. Numerical results on the Manià's example and a 2-dimensional problem involving the Lavrentiev phenomenon with continuous Sobolev exponent dependence, are given to show the efficiency of the method.

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The calculus of variations and materials science

Abstract: Abstract.A review is given of the development and present state of the calculus of variations, starting from the problem of the brachistochrone, and emphasizing the current interaction with problems of materials science.

Cited by 37 publications

References 93 publications

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

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

Numerical Solution of Nonlinear Elasticity Problems With Lavrentiev Phenomenon

A Truncation Method for Detecting Singular Minimizers Involving the Lavrentiev Phenomenon

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