A simple proof of the characterization of functions of low Aviles Giga energy on a ball<i>via</i>regularity

Lorent, Andrew

doi:10.1051/cocv/2010102

Cited by 7 publications

(7 citation statements)

References 16 publications

(27 reference statements)

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“…which is precisely the well-studied Aviles-Giga model, see e.g. [6,4,9,10,17,19,22,23] and the references therein. Singular structures for that model emerging in the ε → 0 limit take the form of domain wallsgenerically curves-across which the normal component of ∇ψ jumps.…”

Section: Introductionmentioning

confidence: 87%

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A Ginzburg--Landau-Type Problem for Highly Anisotropic Nematic Liquid Crystals

Golovaty¹,

Sternberg²,

Venkatraman³

2019

SIAM J. Math. Anal.

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We carry out an asymptotic analysis of a variational problem relevant in the studies of nematic liquid crystalline films when one elastic constant dominates over the others, namelyHere u : Ω → R 2 is a vector field, 0 < ε 1 is a small parameter, and L > 0 is a fixed constant, independent of ε. We identify a candidate for the Γ-limit E 0 , which is a sum of a bulk term penalizing divergence and an Aviles-Giga type wall energy involving the cube of the jump in the tangential component of the S 1 -valued nematic director. We establish the lower bound and provide the recovery sequence for this candidate within a restricted class. Then we consider a set of variational problems for E 0 arising for a choice of domain geometries and boundary conditions. We demonstrate that the criticality conditions for E 0 can be expressed as a pair of scalar conservation laws that share characteristics. We use the method of characteristics to analytically construct critical points of E 0 that we observe numerically.

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Section: Introductionmentioning

confidence: 87%

“…We point out that for the Aviles-Giga energy, the authors in [18] classify zero energy states of the Aviles-Giga energy. More recently, [23] provides a quantitative version of the result in [18].…”

Section: Tangential Boundary Conditionsmentioning

confidence: 99%

A Ginzburg--Landau-Type Problem for Highly Anisotropic Nematic Liquid Crystals

Golovaty¹,

Sternberg²,

Venkatraman³

2019

SIAM J. Math. Anal.

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“…All the sets admitting sequences with vanishing energy were characterized in [17] and with the appropriate boundary conditions the limit function is in these casesū = dist(•, ∂ ). A quantitative version of this result is proven in [20] (see also [19]). In a different direction, it was shown in [21] that the vanishing of the two entropy defect measures div e 1 ,e 2 (m) and div ε 1 ,ε 2 (m) is sufficient to establish div (m) = 0 for every ∈ E. Here we denoted by (e 1 , e 2 ) the standard orthonormal system in R 2 and by…”

Section: Zero-energy Statesmentioning

confidence: 75%

Characterization of Minimizers of Aviles–Giga Functionals in Special Domains

Marconi

2021

Arch Rational Mech Anal

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We consider the singularly perturbed problem $$F_\varepsilon (u,\Omega ):=\int _\Omega \varepsilon |\nabla ^2u|^2 + \varepsilon ^{-1}|1-|\nabla u|^2|^2$$ F ε ( u , Ω ) : = ∫ Ω ε | ∇ 2 u | 2 + ε - 1 | 1 - | ∇ u | 2 | 2 on bounded domains $$\Omega \subset {\mathbb {R}}^2$$ Ω ⊂ R 2 . Under appropriate boundary conditions, we prove that if $$\Omega $$ Ω is an ellipse, then the minimizers of $$F_\varepsilon (\cdot ,\Omega )$$ F ε ( · , Ω ) converge to the viscosity solution of the eikonal equation $$|\nabla u|=1$$ | ∇ u | = 1 as $$\varepsilon \rightarrow 0$$ ε → 0 .

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“…All the sets Ω admitting sequences with vanishing energy were characterized in [JOP02] and with the appropriate boundary conditions the limit function is in these cases ū = dist(•, ∂Ω). A quantitative version of this result is proven in [Lor14] (see also [Lor12]). In a different direction, it was shown in [LP18] that the vanishing of the two entropy defect measures divΣ e1,e2 (m) and divΣ ε1,ε2 (m) is sufficient to establish div Φ(m) = 0 for every Φ ∈ E. Here we denoted by (e 1 , e 2 ) the standard orthonormal system in R 2 and by…”

mentioning

confidence: 75%