Sufficient conditions for regularity of area-preserving deformations

Karakhanyan, Aram

doi:10.1007/s00229-011-0500-7

Cited by 9 publications

(13 citation statements)

References 9 publications

(10 reference statements)

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“…From Theorem A we can conclude that the first equation in (1.3) is well defined in Ω. Moreover applying the duality argument from [6] we infer that there is a function P : Ω ⋆ → R such that the pair (u −1 , P ) is a solution the corresponding Euler-Lagrange equations in Ω ⋆ , see Theorem 2 [6]. Combining Theorem A with this observation we obtain Theorem B.…”

Section: Introductionmentioning

confidence: 67%

“…In fact the term p(∇u) −t is not welldefined unless ∇u is better than L 2 integrable, see [2]. The lack of higher integrability of ∇u produces a number of technical difficulties, see [6]. To circumvent them author and N. Chaudhuri succeeded to compute the first variation of the energy (1.6) in the image domain Ω ⋆ = u(Ω) under very weak assumptions (note that u is open map [10]).…”

Section: Introductionmentioning

confidence: 99%

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Regularity for Energy-Minimizing Area-Preserving Deformations

Karakhanyan

2013

J Elast

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In this paper we establish the square integrability of the nonnegative hydrostatic pressure p, that emerges in the minimization problem inf KˆΩ |∇v| 2 , Ω ⊂ R 2 as the Lagrange multiplier corresponding to the incompressibility constraint det ∇v = 1 a.e. in Ω. Our method employs the Euler-Lagrange equation for the mollified Cauchy stress C satisfied in the image domain Ω ⋆ = u(Ω). This allows to construct a convex function ψ, defined in the image domain, such that the measure of the normal mapping of ψ controls the L 2 norm of the pressure. As a by-product we conclude that u ∈ C 1 2 loc (Ω) if the dual pressure (introduced in [6]) is nonnegative.

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Section: Introductionmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Regularity for Energy-Minimizing Area-Preserving Deformations

Karakhanyan

2013

J Elast

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“…They show that if v minimizes a suitably quasiconvex energy, which includes the case of the Dirichlet energy, then ∇v must be Hölder continuous on a dense subset of B ′ . Interesting results in this vein, although under different assumptions involving the dual pressure, have recently been obtained by Karakhanyan in [13]. In the degree two case considered in this paper, the regularity of a general stationary point of the energy would still seem to be open.…”

Section: The Class a Of Admissible Functionsmentioning

confidence: 90%

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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“…[Bal77], [LO81], [BOP92], [Le Dre85], [CK09], [Kar12] and the references contained therein. The setting of [CK09] and [Kar12] is mathematically particularly close to ours.…”

Section: Local Study Of Energy-minimal Solutions and Lagrange Multiplmentioning

confidence: 99%

On the Jacobian equation and the Hardy space H^1(C)

Lindberg¹

2015

Ann. Acad. Sci. Fenn. Math. Diss.

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Sufficient conditions for regularity of area-preserving deformations

Cited by 9 publications

References 9 publications

Regularity for Energy-Minimizing Area-Preserving Deformations

Regularity for Energy-Minimizing Area-Preserving Deformations

On double-covering stationary points of a constrained Dirichlet energy

On the Jacobian equation and the Hardy space H^1(C)

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