The Landau-Lifshitz equation with the external field -- a new extension for harmonic maps with values in \(S^ 2\)
We consider critical points of the functional \[ E_H(u; \Omega):= \int_\Omega \textstyle{{1\over 2}} [|\nabla u|^2- H\cdot u] dx \] in the class \(H^{1,2}_\gamma(\Omega, S^2)\), where \(H\) is a constant vector in \(\mathbb{R}^3\) and \(\Omega\) is a domain in \(\mathbb{R}^3\). The Euler-Lagrange equation of \(E_H\), \(\Delta u+ |Du|^2 u- (H\cdot u) u+ H= 0\) in \(\Omega\), is equivalent to a static Landau-Lifshitz equation of physics. We prove non-existence and existence results for nontrivial smooth solutions of the static Landau-Lifshitz equation depending on the boundary conditions. We also show a gap phenomenon for the functional \(E_H\).
scientific article; zbMATH DE number 856558