Ginzburg-Landau vortices with pinning functions and self-similar solutions in harmonic maps
The authors deal with the following boundary value problem for \(V_\varepsilon= (V^1_\varepsilon,\dots, V^m_\varepsilon):\Omega\to \mathbb{R}^m\), \[ \begin{cases} -\text{div}(A(x)\nabla V)= {B(x)V\over \varepsilon^2} (1-|V|^2)+ P(x)V\quad &\text{in }\Omega,\\ V(x)= g(x)\quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(\Omega\subset \mathbb{R}^m\) is a smooth, bounded and simply connected domain. They prove that, under some suitable assumptions, all the vortices of (1) are pinned together to the minimum points of \(A(x)\) in \(\Omega\) as \(\varepsilon\to 0\). Moreover, the authors construct a self-similar solution in the evolution of harmonic maps. To this end they use so-called Pokhozhaev identity.
scientific article; zbMATH DE number 1782005