Null Lagrangians, weak continuity, and variational problems of arbitrary order

Ball, J. M.; Currie, J. C.; Olver, Peter J.

doi:10.1016/0022-1236(81)90085-9

Cited by 295 publications

(258 citation statements)

References 29 publications

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“…The underlying idea was to view the integrand as convex function of null Lagrangians. The term null Lagrangian pertains to a nonlinear differential expression whose integral over any open region depends only on the boundary values of the mapping, see [10,16,18]. In this paper the mappings of our interest are homeomorphisms h : X onto − − → Y that are given only on a part of the boundary (possibly empty part, in which case we are dealing with a traction free problem).…”

Section: Some Free Lagrangians Normal and Tangential Distortionsmentioning

confidence: 99%

Mappings of Least Dirichlet Energy and their Hopf Differentials

Iwaniec

Onninen

2013

Arch Rational Mech Anal

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Abstract. The paper is concerned with mappings h : X onto − − → Y between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in X ) of the energyminimal mappings is established within the class H 2(X, Y) of strong limits of homeomorphisms in the Sobolev space W 1,2 (X, Y) , a result of considerable interest in the mathematical models of Nonlinear Elasticity. The inner variation (of the independent variable in X) leads to the Hopf differential hzhz dz ⊗ dz and its trajectories. For a pair of doubly connected domains, in which X has finite conformal modulus, we establish the following principle:A mapping h ∈ H 2(X, Y) is energy-minimal if and only if its Hopf-differential is analytic in X and real along ∂X. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of cracks in X . Nevertheless, cracks are triggered only by the points in ∂Y where Y fails to be convex. The general law of formation of cracks reads as follows:Cracks propagate along vertical trajectories of the Hopf differential from ∂X toward the interior of X where they eventually terminate before making a crosscut.

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Section: Some Free Lagrangians Normal and Tangential Distortionsmentioning

confidence: 99%

Mappings of Least Dirichlet Energy and their Hopf Differentials

Iwaniec

Onninen

2013

Arch Rational Mech Anal

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“…for all balls B and all ∈ C ∞ 0 (B; R n ): (We refer to [3] for more on null-Lagrangians.) Assume also that N is homogeneous of degree k ≥ 2.…”

Section: Some Examples and Applicationsmentioning

confidence: 99%

Lp-Mean coercivity, regularity and relaxation in the calculus of variations

Yan

Zhou

2001

Nonlinear Analysis: Theory, Methods & Applications

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“…We shall frequently use the following well-known result that a subdeterminant of the gradient is a null-Lagrangian [4,5,8,18,19]. Lemma 3.1.…”

Section: The Uniqueness Of the Young Measurementioning

confidence: 99%

The Simply Laminated Microstructure in Martensitic Crystals that Undergo a Cubic-to-Orthorhombic Phase Transformation

Bhattacharya

Luskin

1999

Archive for Rational Mechanics and Analysis

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Abstract. We study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type P (432) → P (222) . The free energy density modeling such a crystal is minimized on six energy wells that are pairwise rank-one connected. We consider the energy minimization problem with Dirichlet boundary data compatible with an arbitrary but fixed simple laminate. We first show that for all but a few isolated values of transformation strains, this problem has a unique Young measure solution solely characterized by the boundary data that represents the simply laminated microstructure. We then present a theory of stability for such a microstructure, and apply it to the conforming finite element approximation to obtain the corresponding error estimates for the finite element energy minimizers.

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Null Lagrangians, weak continuity, and variational problems of arbitrary order

Cited by 295 publications

References 29 publications

Mappings of Least Dirichlet Energy and their Hopf Differentials

Mappings of Least Dirichlet Energy and their Hopf Differentials

Lp-Mean coercivity, regularity and relaxation in the calculus of variations

The Simply Laminated Microstructure in Martensitic Crystals that Undergo a Cubic-to-Orthorhombic Phase Transformation

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