Orientation-Preserving Young Measures

Koumatos, Konstantinos; Rindler, Filip; Wiedemann, Emil

doi:10.1093/qmath/haw019

Cited by 19 publications

(35 citation statements)

References 42 publications

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“…This question can indeed be solved for our regime 1 < p < d; the following theorem is proved in [2]:…”

Section: Introductionmentioning

confidence: 93%

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Differential inclusions and Young measures involving prescribed Jacobians

Rindler

2014

Proc Appl Math and Mech

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In elasticity theory, one naturally requires that the Jacobian determinant of the deformation is positive or even a-priori prescribed (e.g. for incompressibility). However, such strongly non-linear and non-convex constraints are difficult to deal with in mathematical models. This short note, which is based on joint work with K. Koumatos and E. Wiedemann, presents various recent results on how this constraint can be manipulated in subcritical Sobolev spaces, where the integrability exponent is less than the dimension. In particular, we give a characterization theorem for Young measures under this side constraint. This is in the spirit of the celebrated Kinderlehrer-Pedregal Theorem and based on convex integration and "geometry" in matrix space. Finally, applications to approximation in Sobolev spaces and to the failure of lower semicontinuity for certain integral functionals with "realistic" growth conditions are given. phone +44 (0) 2476 5 -28329 det ∇u = r > 0 a.e.orwhere by J + 1 and J − 2 we denote the positive part of J 1 and the negative part of J 2 , respectively. These assumptions are rather natural (the integrability condition is only relevant in very special cases). Notice that the lower and upper bound can also be non-active for certain x, for example if J 1 (x) = −∞, there is no lower bound at x.A question that is related to the well-known Dacorogna-Moser Theory [8] then is as follows:Question Q3. Given g ∈ W 1−1/p,p (∂ Ω; R d ) (the trace space to W 1,p (Ω; R d ), that is the space of all boundary values of W 1,p (Ω; R d )-functions), does there exist u ∈ W 1,p (Ω; R d ) with u| ∂ Ω = g and det ∇u > 0 ?The same question can also be asked with the even stricter requirement det ∇u = 1 instead of mere positivity of the Jacobian. This expresses incompressibility and is relevant in the theory of fluids. Notice that here we have no "compatibility" assumption on g. Again, one can show with a similar argument to before that this question is unsolvable if p ≥ d. However, for p < d we get a positive answer, this is proved in [9].

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“…This question can indeed be solved for our regime 1 < p < d; the following theorem is proved in [2]:…”

Section: Introductionmentioning

confidence: 93%

“…However, this questions can be positively solved if p < d as is shown in [2]; we will give a more precise statement involving Young measures below. This regime is important for example in the theory of cavitation, see for example [3,4].…”

Section: Introductionmentioning

confidence: 94%

Differential inclusions and Young measures involving prescribed Jacobians

Rindler

2014

Proc Appl Math and Mech

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“…in the space W 1,p . Rindler [29], Koumatos, Rindler, and Wiedemann [19], [20] proved the weak dense-…”

Section: Introductionmentioning

confidence: 93%

Approximation by mappings with singular Hessian minors

2018

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Let Ω ⊂ R n be a Lipschitz domain. Given 1 ≤ p < k ≤ n and any u ∈ W 2,p (Ω) belonging to the little Hölder class c 1,α , we construct a sequence u j in the same space with rank D 2 u j < k almost everywhere such that u j → u in C 1,α and weakly in W 2,p . This result is in strong contrast with known regularity behavior of functions in W 2,p , p ≥ k, satisfying the same rank inequality.

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“…Obviously, the p-growth conditions formulated above cannot hold for energy densities with these physically relevant properties and the general theory of relaxation cannot be applied. An extension of this theory with a suitable definition of W qc and appropriate growth conditions has been recently obtained, see [KRW13,CD14a].…”

Section: Variational Modeling Of Microstructurementioning

confidence: 99%

Variational Modeling of Slip: From Crystal Plasticity to Geological Strata

Conti

Dolzmann

Kreisbeck

2015

Analysis and Computation of Microstructure in Finite Plasticity

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Abstract. Slip processes are soft modes of deformation, characteristic of a variety of layered materials. The layers can be at the atomic scale, as in the plastic deformation of crystalline lattices, or on a macroscopic scale, as in stacks of cards or sheets of paper and geological strata. The characteristic deformation processes involve sliding of the layers over one another, leading to a shear deformation with a specific orientation. If the forcing is not parallel to the layers, complex microstructures may form, which have a remarkable similarity over different systems and often consist of alternating shears of different sign. We review here recent results on the detailed analysis of slip processes in crystal plasticity based on the theory of relaxation, discuss the general variational framework for these microstructures, and compare with available experimental results in different systems. We then address the situation in which slip in several different directions may coexist in the same system, as frequently observed in plastically deformed crystals.

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Orientation-Preserving Young Measures

Cited by 19 publications

References 42 publications

Differential inclusions and Young measures involving prescribed Jacobians

Differential inclusions and Young measures involving prescribed Jacobians

Approximation by mappings with singular Hessian minors

Variational Modeling of Slip: From Crystal Plasticity to Geological Strata

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