Differential Inclusions and Young Measures Involving Prescribed Jacobians

Koumatos, Konstantinos; Rindler, Filip; Wiedemann, Emil

doi:10.1137/140968860

Cited by 20 publications

(26 citation statements)

References 45 publications

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“…(iii) A different picture arises if one instead uses a constraint on the pointwise determinant, with the material becoming substantially softer, see [4,8] and [28,27,38].…”

Section: Relaxation Of Orientation-preserving Modelsmentioning

confidence: 99%

“…The case p < n is substantially different, since the deformations are not continuous and can develop holes, as was first shown by Ball [4]. Correspondingly, the constraint of having positive determinant does not pass to the limit and the relaxed problem has a substantially different structure, see for example [8,28,27] for further developments. Relaxation in a related situation in which the constraints are lost after rank-one convexification was discussed in [10].…”

Section: Introductionmentioning

confidence: 99%

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On the Theory of Relaxation in Nonlinear Elasticity with Constraints on the Determinant

Conti

Dolzmann

2014

Arch Rational Mech Anal

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We consider vectorial variational problems in nonlinear elasticity of the form I [u] =´W (Du) dx, where W is continuous on matrices with a positive determinant and diverges to infinity along sequences of matrices whose determinant is positive and tends to zero. We show that, under suitable growth assumptions, the functional´W qc (Du) dx is an upper bound on the relaxation of I , and coincides with the relaxation if the quasiconvex envelope W qc of W is polyconvex and has p-growth from below with p n. This includes several physically relevant examples. We also show how a constraint of incompressibility can be incorporated in our results.

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“…(iii) A different picture arises if one instead uses a constraint on the pointwise determinant, with the material becoming substantially softer, see [4,8] and [28,27,38].…”

Section: Relaxation Of Orientation-preserving Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Theory of Relaxation in Nonlinear Elasticity with Constraints on the Determinant

Conti

Dolzmann

2014

Arch Rational Mech Anal

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“…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].…”

mentioning

confidence: 55%

“…Then the following two statements are shown in [9]: W 1,p -orientation-preserving quasiconvex is necessary for W 1,p -weak lower semicontinuity, but on the other hand, the elastic growth assumptions (4) are incompatible with this quasiconvexity notion. Hence: (4).…”

Section: Lack Of Lower Semicontinuity For a Class Of Functionalsmentioning

confidence: 99%

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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“…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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Differential Inclusions and Young Measures Involving Prescribed Jacobians

Cited by 20 publications

References 45 publications

On the Theory of Relaxation in Nonlinear Elasticity with Constraints on the Determinant

On the Theory of Relaxation in Nonlinear Elasticity with Constraints on the Determinant

Differential inclusions and Young measures involving prescribed Jacobians

Approximation by mappings with singular Hessian minors

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