Minimizing under relaxed symmetry constraints: triple and N-junctions (Q6609497)

The author proves the existence of solutions \(u:\mathbb R^n\to\mathbb R^m\) of the elliptic system \N\[\N\Delta u=W(u),\tag{1} \N\]\Nwhere \(W:\mathbb R^m\to\mathbb R\) is a smooth non-negative potential such that \(0=W(a)<W(u)\), \(a\in A\), \(u\in\mathbb R^m\setminus A\), and \(A=\{a_1\dotsc,a_N\}\subset\mathbb R^m\) is a set of \(N\) distinct points. The following conditions are assumed. \N\N\(H_1\): \(W\) is invariant under the action of the rotation group \(C_N\) with \(N\ge3\), \(W(\omega u)=W(u)\), where \(u\in\mathbb R^2\), \(\omega=\big( \begin{smallmatrix}\cos2 \pi/N &-\sin2\pi/N \\\N\sin2\pi/N &\cos2\pi/N\end{smallmatrix}\big)\). \N\N\(H_2\): \(W\ge0\) and \(A=\{a,\omega a,\dotsc, \omega^{N-1}a\}\) for some \(a\in\mathbb R^2\setminus\{0\}\) and the Hessian matrix \(W_{uu}(a)\) is positive definite and \(W_{u}(u)\cdot u\ge0\) for \(|u|\ge M>0\). \N\NThe main Theorem 1.1 is proved based on several lemmas and propositions: Assume that \(H_1\) and \(H_2\) hold. Then there exists a \(C_N\)-equivariant classical solution \(U:\mathbb R^2\to \mathbb R^2\) of (1), \(U(\omega x)=\omega U(x)\), \(x\in\mathbb R^2\). Moreover there are positive constants such that \(|U(x)-a|\le Ke^{-kd(x,\partial Q)}\), where \[Q=\Big\{x=x(r,\theta)\,:\,r>r_0,\;c/\sqrt{r}<\theta< 2\pi/N-C/\sqrt{r}\Big\}.\] In addition, two more very similar Theorems 1.2 and 1.3 are proved.