• DOI:10.1007/S00205-017-1112-5
• Corpus ID: 119680063

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}
}

• P. Bates, G. Fusco, Panayotis Smyrnelis
• Published 14 November 2014
• Mathematics
• Archive for Rational Mechanics and Analysis

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 general notion of equivariance with respect to a homomorphism $${f:G\rightarrow\Gamma}$$f:G→Γ ($${G, \Gamma}$$G,Γ reflection groups) and prove two abstract results, concerning the cases of G finite and G discrete, for the existence of equivariant solutions. Our approach is variational and based on a mapping property of the parabolic vector Allen–Cahn equation and on a pointwise estimate for vector minimizers.