# Multiphase Solutions to the Vector Allen–Cahn Equation: Crystalline and Other Complex Symmetric Structures

@article{Bates2014MultiphaseST, title={Multiphase Solutions to the Vector Allen–Cahn Equation: Crystalline and Other Complex Symmetric Structures}, author={Peter W. Bates and Giorgio Fusco and Panayotis Smyrnelis}, journal={Archive for Rational Mechanics and Analysis}, year={2014}, volume={225}, pages={685-715} }

AbstractWe present a systematic study of entire symmetric solutions $${u : \mathbb{R}^n \rightarrow\mathbb{R}^m}$$u:Rn→Rm of the vector Allen–Cahn equation
$$\Delta u - W_u(u) = 0 \quad\text{for all}\quad x \in \mathbb{R}^n,$$Δu-Wu(u)=0for allx∈Rn,where $${W:\mathbb{R}^m \rightarrow \mathbb{R}}$$W:Rm→R is smooth, symmetric, nonnegative with a finite number of zeros, and where $${ W_u= (\partial W / \partial u_1,\dots,\partial W / \partial u_m)^{\top}}$$Wu=(∂W/∂u1,⋯,∂W/∂um)⊤. We introduce a…

## 13 Citations

### Symmetry and the Vector Allen–Cahn Equation: Crystalline and Other Complex Structures

- Mathematics
- 2018

We present a systematic study of entire symmetric solutions \(u:{\mathbb R}^n \to {\mathbb R}^m\) of the vector Allen–Cahn equation Δu − Wu(u) = 0, \(x \in {\mathbb R}^n\), where \(W:{\mathbb R}^m…

### Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections

- Computer Science
- 2017

It is proved that, under a nondegeneracy condition, the existence of a solution to the existence problem is given and a new proof of the existence is given.

### On the Existence of N-Junctions for a Symmetric Nonnegative Potential with $$N+1$$ N + 1 Zeros

- MathematicsJournal of Dynamics and Differential Equations
- 2022

We consider a nonnegative potential $$W:{\mathbb {R}}^2\rightarrow {\mathbb {R}}$$ W : R 2 → R which is invariant under $$C_N$$ C N , the rotation group of the regular polygon with N sides:…

### Entire Vortex Solutions of Negative Degree for the Anisotropic Ginzburg–Landau System

- MathematicsArchive for Rational Mechanics and Analysis
- 2022

The anisotropic Ginzburg–Landau system Δu+δ∇(divu)+δcurl∗(curlu)=(|u|2-1)u,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}…

### On the growth of the energy of entire solutions to the vector Allen-Cahn equation

- Mathematics
- 2014

We prove that the energy over balls of entire, nonconstant, bounded solutions to the vector Allen-Cahn equation grows faster than $(\ln R)^k R^{n-2}$, for any $k>0$, as the volume $R^n$ of the ball…

### LAYERED SOLUTIONS TO THE VECTOR ALLEN-CAHN EQUATION IN R 2 . MINIMIZERS AND HETEROCLINIC CONNECTIONS

- Mathematics
- 2017

. Let W : R m → R be a nonnegative potential with exactly two nondegenerate zeros a − (cid:54) = a + ∈ R m . We assume that there are N ≥ 1 distinct heteroclinic orbits connecting a − to a +…

### On the structure of minimizers of the Allen-Cahn energy for $$Z_N$$ symmetric nonnegative potentials with $$N+1$$ zeros

- MathematicsCalculus of Variations and Partial Differential Equations
- 2022

Let W : R 2 → R a smooth nonnegative potential invariant under Z N , the symmetry group of the regular polygon with N sides. We assume that W has exactly N + 1 zeros:

### Entire Minimizers of Allen–Cahn Systems with Sub-Quadratic Potentials

- MathematicsJournal of Dynamics and Differential Equations
- 2021

We study entire minimizers of the Allen-Cahn systems. The specific feature of our systems are potentials having a finite number of global minima, with sub-quadratic behaviour locally near their…

### Uniform estimates for positive solutions of semilinear elliptic equations and related Liouville and one-dimensional symmetry results

- Mathematics
- 2012

We consider a semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce a condition on the size of the domain that…

### AN ASYMPTOTIC MONOTONICITY FORMULA FOR MINIMIZERS TO A CLASS OF ELLIPTIC SYSTEMS OF ALLEN-CAHN TYPE AND THE LIOUVILLE PROPERTY

- Mathematics
- 2015

We prove an asymptotic monotonicity formula for bounded, globally minimizing solutions to a class of semilinear elliptic systems of the form ∆u = Wu(u), x ∈ R, n ≥ 2, with W : R → R, m ≥ 1,…

## References

SHOWING 1-10 OF 23 REFERENCES

### Entire Solutions to Equivariant Elliptic Systems with Variational Structure

- Mathematics
- 2008

We consider the system Δu − Wu(u) = 0, where $${u : \mathbb{R}^n \to \mathbb{R}^n}$$ , for a class of potentials $${W : \mathbb{R}^n \to \mathbb{R}}$$ that possess several global minima and are…

### Equivariant entire solutions to the elliptic system $$\Delta u=W_u(u)$$ for general $$G$$-invariant potentials

- Mathematics
- 2014

We consider the system $$\Delta u - W_u(u) = 0$$, where $$u: \mathbb R ^n \rightarrow \mathbb R ^m$$, for potentials $$W: \mathbb R ^m \rightarrow \mathbb R $$ that possess $$N$$ global minima and…

### On some elementary properties of vector minimizers of the Allen-Cahn energy

- Mathematics
- 2013

We derive a point-wise estimate for a map $u: \Omega \subset R^n \rightarrow R^m$ that minimizes $J_A(v): \int_A \frac{1}{2}|\nabla v|^2+U(v)$ subjected to the Dirichlet condition $v=u$ on…

### EXISTENCE OF LATTICE SOLUTIONS TO SEMILINEAR ELLIPTIC SYSTEMS WITH PERIODIC POTENTIAL

- Mathematics
- 2012

Under the assumption that the potential W is invariant under a general discrete reflection group G′ = TG acting on Rn, we establish existence of G′-equivariant solutions to ∆u − Wu(u) = 0, and find…

### New solutions of equations on $\mathbb {R}^n$

- Mathematics
- 2001

We consider some weakly nonlinear elliptic equations on the whole
space and use local and global bifurcations methods to construct solutions periodic
in one variable and decaying in the other…

### A New Proof for the Existence of an Equivariant Entire Solution Connecting the Minima of the Potential for the System Δu − W u (u) = 0

- Mathematics
- 2011

Recently, Giorgio Fusco and the author in [2] studied the system Δu − W u (u) = 0 for a class of potentials that possess several global minima and are invariant under a general finite reflection…

### On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation

- Mathematics
- 1993

We present a formal asymptotic analysis which suggests a model for three-phase boundary motion as a singular limit of a vector-valued Ginzburg-Landau equation. We prove short-time existence and…

### The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces

- Mathematics
- 1976

In this paper we provide a complete classification of the local structure of singularities in a wide class of two-dimensional surfaces in R3 collected under the adjective (M, i, a) minimal by Almgren…

### The Number of Group Homomorphisms from D[subscript m] into D[subscript n].

- Mathematics
- 2013

Counting homomorphisms between cyclic groups is a common exercise in a first course in abstract algebra. A similar problem, accessible at the same level, is to count the number of group homomorphisms…