Variational Properties of Unbounded Order Parameters
Bo Li · SIAM Journal on Mathematical Analysis · 2006
Order parameters in physical and biological systems can sometimes become unbounded as the size of an underlying system increases. It is proposed that such a quantity be modeled as a minimizer of the energy functional \[ I_\varepsilon(u) = \dashint \left[ \frac{\varepsilon^2}{2} | abla u|^2 - \frac12 \log (1 + |u|^2 )\right]dx, \] where u is constrained by a side condition, and $\varepsilon > 0$ is a parameter that is inversely proportional to the linear size of the system. It is shown that a minimizer of $I_\varepsilon $ exists; the minimum value of $I_\varepsilon $ scales as $\log \varepsilon$; and both the $L^2$ and $H^1$ norms of any minimizer of $I_\varepsilon $ are of the order $O(1/\varepsilon)$, indicating the unboundedness of the order parameter. It is also shown that the renormalized energy functionals \[ J_\varepsilon(v) = I_\varepsilon \left( \frac{v}{ \varepsilon} \right) - \log \varepsilon \] $\Gamma$-converge to the functional \[ J(v) = \dashint \left( \frac12 | abla v|^2 - \log |v|\right) dx. \] Minimizers of this $\Gamma$-limit for scalar order parameters with the Dirichlet boundary condition are well characterized.