Maximal smoothness for solutions to equilibrium equations in 2D nonlinear elasticity

Yan, Xiaodong

doi:10.1090/s0002-9939-06-08645-x

Cited by 10 publications

(6 citation statements)

References 6 publications

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“…The stationarity conditions (3.4) and (3.5) give some information about the regularity of an auxiliary quantity z defined in (4.3) below. We note that the function z also appears naturally in [BOP91] and [Yan07].…”

Section: A Class Of Curves Which Visit the Origin In R 2 At Least Oncementioning

confidence: 79%

On one-homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions

Bevan

2010

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We extend the result [Ph02] of Phillips by showing that one-homogeneous solutions of certain elliptic systems in divergence form either do not exist or must be affine. The result is novel in two ways. Firstly, the system is allowed to depend (in a sufficiently smooth way) on the spatial variable x. Secondly, Phillips's original result is shown to apply to W 1,2 one-homogeneous solutions, from which his treatment of Lipschitz solutions follows as a special case. A singular one-homogeneous solution to an elliptic system violating the hypotheses of the main theorem is constructed using a variational method.

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Section: A Class Of Curves Which Visit the Origin In R 2 At Least Oncementioning

confidence: 79%

On one-homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions

Bevan

2010

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…This drives one to investigate the regularity of the energy-minimal mappings on the basis of the inner-variational equation alone; also known as energy-momentum or equilibrium equations, etc [11,27,30]. Several results in this direction were obtained in [9,12,25,32]. Nevertheless this theory is still in its infancy.…”

Section: Introductionmentioning

confidence: 99%

Lipschitz regularity for inner-variational equations

Iwaniec¹,

Kovalev²,

Onninen³

2013

Duke Math. J.

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We obtain Lipschitz regularity results for a fairly general class of nonlinear first-order PDEs. These equations arise from the inner variation of certain energy integrals. Even in the simplest model case of the Dirichlet energy the inner-stationary solutions need not be differentiable everywhere; the Lipschitz continuity is the best possible. But the proofs, even in the Dirichlet case, turn out to relay on topological arguments. The appeal to the inner-stationary solutions in this context is motivated by the classical problems of existence and regularity of the energy-minimal deformations in the theory of harmonic mappings and certain mathematical models of nonlinear elasticity; specifically, neo-Hookian type problems.2010 Mathematics Subject Classification. Primary 49N60; Secondary 35B65, 73C50.

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“…a = 0 is excluded by the assumption that the solution r 0 is an immediate lift-off function. Then r 0 | [δ,1] solves (33) uniquely and r 0 takes the form (32). Since r 0 ∈ C ∞ ((0, 1]) all derivatives of r 0 need to agree at δ, i.e.…”

Section: Advanced Bop-theorymentioning

confidence: 99%

“…In this paper we follow the method devised initially by P. Bauman, N.C. Owen and D. Phillips (BOP) in two striking papers [4,5] and extended by Yan [32] and Yan and Bevan [8]. Indeed, BOP consider a infinite nonlinear elastic situation, where the integrand W depends on d in such a way, that W (d) → +∞ if d → 0 + or d → +∞ and W (d) = +∞ if d ≤ 0 and they obtain a higher-order regularity result showing that it is enough for an equilibrium solution to be of class C 1,β for some β ∈ (0, 1] for it to be already fully smooth.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, BOP consider a infinite nonlinear elastic situation, where the integrand W depends on d in such a way, that W (d) → +∞ if d → 0 + or d → +∞ and W (d) = +∞ if d ≤ 0 and they obtain a higher-order regularity result showing that it is enough for an equilibrium solution to be of class C 1,β for some β ∈ (0, 1] for it to be already fully smooth. In [32] it is then shown that the previous assumption can be relaxed to C 1 ∩ W 2,2 and that this is optimal in a sense. Additionally, BOP construct a radial symmetric double covering singular equilibrium solution, which is C 1 \ C 1,β for any β ∈ (0, 1].…”

Section: Introductionmentioning

confidence: 99%

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On the assembly of $C^1-$stationary points of a polyconvex functional and finite BOP-theory

Dengler¹

2022

Preprint

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In this work the following energy is consideredwhere B ⊂ R 2 denotes the unit ball, u ∈ W 1,2 (B, R 2 ) and ρ : R → R + 0 smooth and convex with ρ(s) = 0 for all s ≤ 0 and ρ becomes affine when s exceeds some value s0 > 0. Additionally, we may impose (M ∈ N \ {0})−covering maps as boundary conditions in a suitable fashion. For such situations we then construct radially symmetric M −covering stationary points of the energy, which are at least C 1 (in some circumstances even C ∞ ), and verify more refined properties, which these stationary points need to satisfy. We do so by following the strategy first and foremost developed by P. Bauman, N. C. Owen, and D. Phillips (BOP) confirming and generalising that the method remains valid beyond the M = 2−case for an arbitrary M. Furthermore, as far as we know, this is the first treatise of BOP-theory in finite elasticity. The finiteness, not imposing such strict conditions, allows for a richer class of possible behaviours of the stationary points, making it more difficult to completely determine them.

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Maximal smoothness for solutions to equilibrium equations in 2D nonlinear elasticity

Abstract: For a class of variational integrals from 2D nonlinear elasticity, we prove that any W 2,2 ∩ C 1 weak solution for the equilibrium equations is smooth. Moreover, we present an example showing that the assumption u ∈ W 2,2 is optimal.

Cited by 10 publications

References 6 publications

On one-homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions

On one-homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions

Lipschitz regularity for inner-variational equations

On the assembly of $C^1-$stationary points of a polyconvex functional and finite BOP-theory

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