Saddle solutions to Allen-Cahn equations in doubly periodic media (Q2806003)

The authors consider a class of semilinear periodic Allen-Cahn equations NEWLINE\[CARRIAGE_RETURNNEWLINE- \Delta u(x,y) + a(x,y)W'(u(x,y)) = 0, \quad (x,y)\in \mathbb{R}^2, \eqno{(\mathrm{E})} CARRIAGE_RETURNNEWLINE\]NEWLINE where \(a\) is a function representing a doubly periodic media and \(W : \mathbb{R} \to \mathbb{R}\) is a classical double well potential satisfyingNEWLINE{\parindent=9mmNEWLINE\begin{itemize}NEWLINE\item[(\(H_1\))] \(a\in C(\mathbb{R}^2)\) is strictly positive, and the following hold: NEWLINE{\parindent=14mmNEWLINE\begin{itemize}\item[(i)] \(a(x+1,y)= a(x,y)=a(x,y+1)\) for all \((x,y)\in \mathbb{R}^2\)NEWLINE\item[(ii)] \(a(x,y)= a(x,-y)=a(-x,y)\), \(a(x,y)=a(y,x)\) for each \((x,y)\in \mathbb{R}^2\)NEWLINE\end{itemize}}NEWLINE\item[(\(H_2\))] \(W\in C(\mathbb{R}^2)\) satisfies \(W(\pm 1)=0, W''(\pm 1)>0\) and \(W(s)>0\) for any \(s\in(-1,1)\), and \(W(s)=W(-s)\) for \(s\in \mathbb{R}\). NEWLINE\end{itemize}}NEWLINENEWLINEThe authors use variational methods, and letting \(S_0=\mathbb{R}\times [0,1]\), they look for minima of the action potential NEWLINE\[CARRIAGE_RETURNNEWLINE \varphi(u)= \int\!\!\!\int_{S_0} \frac{1}{2} |\nabla u(x,y)|^2 +a(x,y) W(u(x,y)) \, dxdy CARRIAGE_RETURNNEWLINE\]NEWLINE on the class NEWLINE\[CARRIAGE_RETURNNEWLINE E_0=\{ u\in H_{\mathrm{loc}}^1(\mathbb{R}\times[0,1]) \mid u(x,y)=-u(-x,y) \text{ for } x\in \mathbb{R},\text{ and } 1\geq u(x,y)\geq 0\text{ for } x>0 \} CARRIAGE_RETURNNEWLINE\]NEWLINE and denote by \({\mathcal K}\) the set of minima \(\varphi\) on \(E_0\).NEWLINENEWLINENEWLINENEWLINEThe main result of this paper is the following theorem.NEWLINENEWLINENEWLINENEWLINETheorem 1.1. There exists \(v\in C^2(\mathbb{R}^2)\), a solution of (E) on \(\mathbb{R}^2\) that verifies of the following:NEWLINE{\parindent=6mmNEWLINE\begin{itemize}\item[(a)] \(v(x,y) >0\) on the first quadrant in \(\mathbb{R}^2\);NEWLINE\item[(b)] \(v(x,y)= -v(-x,y)=-v(x,-y)=-v(x,-y)\) and \(v(x,y)=v(y,x)\) on \(\mathbb{R}^2\), and such that NEWLINE\[CARRIAGE_RETURNNEWLINE \operatorname{dist}_{L^\infty(\mathbb{R}\times[j,j+1])}( v,{\mathcal K})\to 0 \quad\text{as } j \to +\infty. CARRIAGE_RETURNNEWLINE\]NEWLINENEWLINENEWLINE\end{itemize}}