Harmonic Maps in Connection of Phase Transitions with Higher Dimensional Potential Wells

Lin, Fanghua; Wang, Changyou

doi:10.1007/s11401-019-0160-6

Cited by 7 publications

(10 citation statements)

References 22 publications

(40 reference statements)

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“…The general case of the Keller-Rubinstein-Sternberg problem is fairly sophisticated and remains open. We refer the interested readers to a recent work of Lin-Wang [21] for the well-posedness of the limiting system. On the other hand, the static problem has been investigated by Lin et al [19].…”

Section: Resultsmentioning

confidence: 99%

Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions

Laux

Liu

2021

Arch Rational Mech Anal

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In this work, we study the nematic–isotropic phase transition based on the dynamics of the Landau–De Gennes theory of liquid crystals. At the critical temperature, the Landau–De Gennes bulk potential favors the isotropic phase and nematic phase equally. When the elastic coefficient is much smaller than that of the bulk potential, a scaling limit can be derived by formal asymptotic expansions: the solution gradient concentrates on a closed surface evolving by mean curvature flow. Moreover, on one side of the surface the solution tends to the nematic phase which is governed by the harmonic map heat flow into the sphere while on the other side, it tends to the isotropic phase. To rigorously justify such a scaling limit, we prove a convergence result by combining weak convergence methods and the modulated energy method. Our proof applies as long as the limiting mean curvature flow remains smooth.

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Section: Resultsmentioning

confidence: 99%

Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions

Laux

Liu

2021

Arch Rational Mech Anal

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“…It is something never exists in the scalar case or several vector-valued cases studied by various authors before. It also illustrate why there are different types of boundary conditions along the interface in earlier works [16,17,8,19,18], for examples. Let us describe it roughly below.…”

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confidence: 78%

“…There is also a mixed Dirichlet-Neumann boundary condition for the phase field along the interface. A regularity theory for minimizing maps of the limit problem was obtained in [17].…”

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confidence: 99%

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Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-Sternberg problem

Fei¹,

Lin²,

Wang³

et al. 2021

Preprint

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In this paper, we consider the sharp interface limit of a matrix-valued Allen-Cahn equation, which takes the form:We show that the sharp interface system is a two-phases flow system: the interface evolves according to the motion by mean curvature; in the two bulk phase regions, the solution obeys the heat flow of harmonic maps with values in O `pnq and O ´pnq (represent the sets of n ˆn orthogonal matrices with determinant `1 and ´1 respectively); on the interface, the phase matrices in two sides satisfy a novel mixed boundary condition.The above result provides a solution to the Keller-Rubinstein-Sternberg's (conjecture) problem in this Opnq setting. Our proof relies on two key ingredients. First, in order to construct the approximate solution by matched asymptotic expansions as the standard approach does not seem to work, we introduce the notion of quasi-minimal connecting orbits. They satisfy the usual leading order equations up to some small higher order terms. In addition, the linearized systems around these quasi-minimal orbits needs to be solvable up to some good remainders. These flexibilities are needed for the possible "degenerations" or higher dimensional kernels for the linearized operators due to intriguing boundary conditions at the sharp interface. The second key point is to establish a spectral uniform lower bound estimate for the linearized operator around approximate solutions. To this end, we introduce additional decompositions to reduce the problem into the coercive estimates of several linearized operators for scalar functions and some singular product estimates which is accomplished by exploring special cancellation structures between eigenfunctions of these linearized operators.

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“…While Fonseca-Tartar [14], Sternberg [46], and Andre-Shafrir [4] studied the gradient theory of phase transitions involving Allen-Cahn type energy functionals with potential wells of points or curves in R 2 . More recently, partly motivated by the Keller-Rubinstein-Sternberg problem [28], Lin-Pan-Wang [31] have made a systematic study of the vectorial singular perturbation problem of general high dimensional wells, see also [32].…”

Section: Isotropic and Nematic Phase Transitionsmentioning

confidence: 99%

Isotropic-Nematic Phase Transition and Liquid Crystal Droplets

Lin¹,

Wang²

2020

Preprint

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Liquid crystal droplets are of great interest from physics and applications. Rigorous mathematical analysis is challenging as the problem involves harmonic maps (and in general the Oseen-Frank model), free interfaces and topological defects which could be either inside the droplet or on its surface along with some intriguing boundary anchoring conditions for the orientation configurations. In this paper, through a study of the phase transition between the isotropic and nematic states of liquid crystal based on the Ericksen model, we can show, when the size of droplet is much larger in comparison with the ratio of the Frank constants to the surface tension, a Γ-convergence theorem for minimizers. This Γ-limit is in fact the sharp interface limit for the phase transition between the isotropic and nematic regions when the small parameter ε, corresponding to the transition layer width, goes to zero. This limiting process not only provides a geometric description of the shape of the droplet as one would expect, and surprisingly it also gives the anchoring conditions for the orientations of liquid crystals on the surface of the droplet depending on material constants. In particular, homeotropic, tangential, and even free boundary conditions as assumed in earlier phenomenological modelings arise naturally provided that the surface tension, Frank and Ericksen constants are in suitable ranges.

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Harmonic Maps in Connection of Phase Transitions with Higher Dimensional Potential Wells

Cited by 7 publications

References 22 publications

Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions

Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions

Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-Sternberg problem

Isotropic-Nematic Phase Transition and Liquid Crystal Droplets

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