Ginzburg–Landau functional

Example

Let $\Omega \subseteq \R^2$ be a bounded open set and let $\varepsilon > 0$. The simplified Ginzburg–Landau functional acts on maps $u \colon \Omega \to \C$ in the Sobolev space $H^1(\Omega; \C)$ by $$E_\varepsilon(u) = \int_\Omega \Big( \frac12 \abs{\nabla u}^2 + \frac{1}{4 \varepsilon^2} \big(1 - \abs{u}^2\big)^2 \Big) \, dx.$$ The first term penalizes variation of $u$, and the second term pushes $\abs{u}$ towards $1$.

Check of the zero set: $E_\varepsilon(u) = 0$ holds exactly when $\nabla u = 0$ and $\abs{u} = 1$ almost everywhere. When $\Omega$ is connected, this means that $u$ is a constant of modulus $1$.

Euler–Lagrange equation: the derivative of $E_\varepsilon$ at $u$ in a direction $v$ is $$\int_\Omega \Big( \operatorname{Re}\big(\nabla u \cdot \overline{\nabla v}\big) - \frac{1}{\varepsilon^2} \big(1 - \abs{u}^2\big) \operatorname{Re}\big(u \, \overline{v}\big) \Big) \, dx.$$ So critical points solve $-\Delta u = \varepsilon^{-2} u \big(1 - \abs{u}^2\big)$ weakly, an equation of Ginzburg–Landau type without magnetic field.