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… 

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