Multiphase solutions to the vector Allen-Cahn equation: crystalline and other complex symmetric structures (Q2014377)

The authors consider the vector Allen-Cahn equation \(\Delta u-W_{u}(u)=0\) in \(\mathbb{R}^{n}\) where \(W:\mathbb{R}^{m}\rightarrow \mathbb{R}\) is smooth, symmetric, nonnegative and with a finite number of zeros and \(W_{u}=(\frac{\partial W}{\partial u_{1}} ,\ldots ,\frac{\partial W}{\partial u_{m}})^{\top }\). The authors assume there exists a finite or discrete (infinite) reflection group \(G\) acting on \( \mathbb{R}^{n}\), a finite reflection group \(\Gamma \) acting on \(\mathbb{R}^{m}\) and a homomorphism \(f:G\rightarrow \Gamma \). They look for \(f\)-equivariant solutions to the above Allen-Cahn equation that is solutions which satisfy \( u(gx)=f(g)u(x)\) for every \(g\in G\) and \(x\in \mathbb{R}^{n}\). In the case of a finite reflection group, the authors prove the existence of an \(f\)-equivariant classical solution \(u\) to the above Allen-Cahn equation. In the case of a discrete reflection group, the authors prove that for every \(R>R_{0}\) the Allen-Cahn equation \(\Delta u_{R}-R^{2}W_{u}(u_{R})=0\) in \(\mathbb{R}^{n}\) has an \(f\)-equivariant classical solution. Both existence results require hypotheses on the homomorphism \(f\) between \(G\) and \(\Gamma \) (notion of positive homomorphism which is defined) and on \(W\) (smoothness, invariance with respect to \(\Gamma \) and number and properties of minima). For the proof, the authors introduce the functional associated to the Allen-Cahn equation and defined on \(W^{1,2}(\Omega ;\mathbb{R}^{m})\) as \(J_{\Omega }(u)=\int_{\Omega }( \frac{1}{2}\left| \nabla u\right| ^{2}+W(u))dx\) for every bounded subset \(\Omega \subset \mathbb{R}^{n}\) and they consider in the case of a finite reflection group \(G\) the parabolic equation \(\frac{\partial u}{\partial t} =\Delta u-W_{u}(u)\) in \(B_{R}\times (0,\infty )\) with the boundary condition \(\frac{\partial u}{\partial n}=0\) on \(\partial B_{R}\) and the initial condition \(u(x,0)=u_{0}(x)\). In the case of a discrete reflection group \(G\), they consider instead the parabolic equation \(\frac{\partial u}{\partial t} =\Delta u-R^{2}W_{u}(u)\) in \(\mathbb{R}^{n}\times (0,\infty )\) with the initial condition \(u(x,0)=u_{0}(x)\). They prove that if \(u_{0}\) is smooth and \(f\)-equivariant then the solution to these parabolic equations is also \(f\)-equivariant. The paper ends with deeper analyses in particular cases.