Minimizers with topological singularities in two dimensional elasticity

Bevan, Jonathan J.; Yan, Xiaodong

doi:10.1051/cocv:2007043

Cited by 5 publications

(9 citation statements)

References 21 publications

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“…We also note that the logarithm plays a similarly important role in the analysis of the elastic stored energy functionals considered in [5].…”

Section: A Global Results For the Double-covering Mapmentioning

confidence: 91%

“…Ball remarks in [3] that no member of A is C 1 ; see [5] for a proof of this fact. In the absence of the doubling boundary condition, Evans and Gariepy examine in [10] the effect of the constraint det ∇v = 1 a.e.…”

Section: The Class a Of Admissible Functionsmentioning

confidence: 99%

“…2 + 3 ln R det ∇ϕ dx, and then apply ideas of [5] to G. See Section 3 for details, and Theorem 3.1 in particular.…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, the effect of the volume constraint and double-covering boundary conditions on the symmetry, or otherwise, of the global minimizer of stored-energy functions remains an interesting open question. This question is studied in [4,5] in the compressible case, that is, where the pointwise a.e. constraint det ∇u = 1 is replaced by det ∇u > 0 a.e.…”

Section: Introductionmentioning

confidence: 99%

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On double-covering stationary points of a constrained Dirichlet energy

Bevan

2014

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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The double-covering map u dc : R 2 → R 2 is given byin cartesian coordinates. This paper examines the conjecture that u dc is the global minimizer of the Dirichlet energy I(u) = B |∇u| 2 dx among all W 1,2 mappings u of the unit ball B ⊂ R 2 satisfying (i) u = u dc on ∂B, and (ii) det ∇u = 1 almost everywhere. Let the class of such admissible maps be A. The chief innovation is to express I(u) in terms of an auxiliary functional G(u − u dc ), using which we show that u dc is a stationary point of I in A, and that u dc is a global minimizer of the Dirichlet energy among members of A whose Fourier decomposition can be controlled in a way made precise in the paper. By constructing variations about u dc in A using ODE techniques, we also show that u dc is a local minimizer among variations whose tangent ψ to A at u dc obeys G(ψ o ) > 0, where ψ o is the odd part of ψ. In addition, a Lagrange multiplier corresponding to the constraint det ∇u = 1 is identified by an analysis which exploits the well-known Fefferman-Stein duality. AbstractLe double-revêtement fonction u dc : R 2 → R 2 est donn paren coordonnes cartesian. Cet article examine la conjecture que u dc est le minimiseur global de l'énergie de Dirichlet I(u) = B |∇u| 2 dx pour les fonctions satisfaisant (i) u ∈ W 1,2 (B), où B est la boule unité de R 2 , (ii) u = u dc sur ∂B, et (iii) det ∇u = 1 presque partout. Soit A la classe admissible de telles fonctions. La principale innovation est ici * 2010 AMS Classification 35A15, 49J40, 49N60 † Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK. tel: +44 (0)1483 682620. email: j.bevan@surrey.ac.uk 1 Double-covering stationary points of a constrained Dirichlet Energy 2 d'exprimer I(u) sous forme d'une fonction auxiliaire G(u − u dc ), avec laquelle nous montrons que u dc est un point stationnaire de I en A, et que u dc est un minimiseur global de l'énergie de Dirichlet parmi les membres de A dont la décomposition de Fourier peutêtre contrôlée d'une manire détaillée dans l'article. En construisant des variations autour de u dc en A par des techniques variationnelles, nous montronségalement que u dc est un minimiseur local parmi les variations dont la tangente ψ de u dc vers A obéissentà G(ψ o ) > 0, où ψ o est la partie impaire de ψ. Additionnellement, un multiplicateur de Lagrange correspondantà la contrainte det ∇u = 1 est identifiée par une analyse qui exploite la dualité de Fefferman-Stein.

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“…We also note that the logarithm plays a similarly important role in the analysis of the elastic stored energy functionals considered in [5].…”

Section: A Global Results For the Double-covering Mapmentioning

confidence: 91%

Section: The Class a Of Admissible Functionsmentioning

confidence: 99%

“…2 + 3 ln R det ∇ϕ dx, and then apply ideas of [5] to G. See Section 3 for details, and Theorem 3.1 in particular.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On double-covering stationary points of a constrained Dirichlet energy

Bevan

2014

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…We also present an example showing that the above result fails if we only assume u ∈ W 2,r ∩ C 1 for some r < 2. We use the same example u 0 constructed by BOP in [4] together with some new estimates obtained in [5]. Based on those estimates, we show u 0 belongs to W 2,r ∩ C 1 for any r < 2 but not W 2,2 .…”

Section: Introductionmentioning

confidence: 98%