On the structure of phase transition maps for three or more coexisting phases

Alikakos, Nicholas D.

doi:10.1007/978-88-7642-473-1_1

Cited by 14 publications

(19 citation statements)

References 43 publications

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“…. Thus by Lemma 7.2 u(·, y k ) − e(· ) > γ C we have the estimate W(u(·, y)) ≥w((γ/2C) 3 2 ) > 0, by the uniqueness and hyperbolicity of e(·), and therefore the lower bound C R k (y k )…”

Section: V(s)|mentioning

confidence: 89%

“…. , a N (Baldo [10], Bronsard and Reitich [12]), (b) the Ginzburg-Landau system ∆u − (|u| 2 − 1)u = 0 (Bethuel, Brezis and Helein [11]) and [10], [1], [12], [13], [26], [5], [3], [4], [21], [22], [23]. It is well known that the phase transition model is linked to minimal surfaces (m = 1), and Plateau Complexes (m ≥ 2), that is, nonorientable minimizers of surface area.…”

mentioning

confidence: 99%

“…It is well known that the phase transition model is linked to minimal surfaces (m = 1), and Plateau Complexes (m ≥ 2), that is, nonorientable minimizers of surface area. In particular in the vector case, entire solutions to (1.2) in a certain class are linked to singular minimal cones which unlike planes have additional structure ( [3]). …”

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confidence: 99%

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Density estimates for vector minimizers and applications

Alikakos¹,

Fusco²

2015

DCDS-A

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We extend the Caffarelli-Córdoba estimates to the vector case (L. Caffarelli and A. Córdoba, Uniform Convergence of a singular perturbation problem, Comm. Pure Appl. Math. 48 (1995)). In particular, we establish lower codimension density estimates. These are useful for studying the hierarchical structure of minimal solutions. We also give applications.2010 Mathematics Subject Classification. 35J20, 35J47, 35J50.(c) the phase separation system ∆u i − j =i u i u j = 0, i = 1, . . . , m (Caffarelli and Lin [15]) and its variants. All these systems are included in the general framework of Rubinstein, Sternberg and Keller [27], [28].We assumewhere · denotes the Euclidean inner product in R m and | · | the associated norm.In the present paper we limit ourselves to uniformly bounded solutions to (1.1) that are minimal with respect to their boundary conditions (called minimizers by some authors),2 0 (Ω; R m ) (1.6) for every open bounded set Ω ⊂ D. Moreover we assume |u − a| < M, |∇u| < M, on R n .(1.7)The basic estimate for minimal solutions (cr. Lemma 2.1) isWe generally omit the center and write it B R . In this paper, we give two different extensions of the Caffarelli-Cordoba density estimates [14] to the vector case, together with a few applications. In the scalar case, among other things, these estimates refine the linking of the phase transition model to minimal surfaces and have played a major role in the resolution of the De Giorgi conjecture in higher dimensions (Savin [29]). Other extensions of the density estimates in different contexts have been provided by Farina and Valdinoci [20], Savin and Valdinoci [30],[31], Sire and Valdinoci [32], and very recently by Cesaroni, Muratov and Novaga [16].We denote L k the k-dimensional Lebesgue measure and let

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Section: V(s)|mentioning

confidence: 89%

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confidence: 99%

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Density estimates for vector minimizers and applications

Alikakos¹,

Fusco²

2015

DCDS-A

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“…If, additionally, W vanishes at least at one point, it is easy to cook up a suitable competitor for the energy and show that each bounded, globally minimizing solution satisfies

\int_{B (x_{0}, R)} ({}, \frac{1}{2} | \nabla u |^{2} + W ((), u)) dx \leq C R^{n - 1}, R > 0, x_{0} \in {double-struckR}^{n},

for some C > 0 (e.g., ). The system with W ≥0 vanishing at a finite number of non‐degenerate, global minima is used to model multi‐phase transitions (see and the references therein). In this case, the system is frequently referred to as the vectorial Allen–Cahn equation .…”

Section: Introductionmentioning

confidence: 99%

“…for some C > 0 (e.g., [17]). The system (1) with W 0 vanishing at a finite number of non-degenerate, global minima is used to model multi-phase transitions (see [18][19][20] and the references therein). In this case, the system (1) is frequently referred to as the vectorial Allen-Cahn equation.…”

Section: Introductionmentioning

confidence: 99%

Optimal potential energy growth lower bounds for minimizing solutions to the vectorial Allen–Cahn equation in two space dimensions

Sourdis

2016

Math Methods in App Sciences

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Allen–Cahn solutions with triple junction structure at infinity

Sandier,

Sternberg

2024

Comm Pure Appl Math

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We construct an entire solution to the elliptic system where is a ‘triple‐well’ potential. This solution is a local minimizer of the associated energy in the sense that minimizes the energy on any compact set among competitors agreeing with outside that set. Furthermore, we show that along subsequences, the ‘blowdowns’ of given by approach a minimal triple junction as . Previous results had assumed various levels of symmetry for the potential and had not established local minimality, but here we make no such symmetry assumptions.

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On the structure of phase transition maps for three or more coexisting phases

Cited by 14 publications

References 43 publications

Density estimates for vector minimizers and applications

Density estimates for vector minimizers and applications

Optimal potential energy growth lower bounds for minimizing solutions to the vectorial Allen–Cahn equation in two space dimensions

Allen–Cahn solutions with triple junction structure at infinity

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