06/09/2016, 09:40 — 10:20 — Amphitheatre Pa2, Mathematics Building
Giorgio Fusco, Universitá degli Studi dell’Aquila

On the existence of an heteroclinic connection between two global minimizers of the the Ginzburg-Landau functional

We assume that $W: \mathbb{R}^m\rightarrow \mathbb{R}$ is a nonnegative potential with exactly two nondegenerate zeros $\pm a \in \mathbb{R}^m$ and that there exist two (unique modulo translation) heteroclinic orbits $\overline{u}_- \ne \overline{u}_+$ connecting $-a$ to $a$, that is two minimizers of the Ginzburg-Landau functional $J(\varphi)= \int_ {\mathbb{R}} (W(\varphi)+\frac12|\varphi’|^2) \ ds$:

\[ J(\overline u_\pm) = \min_A J(\varphi) , \quad A\equiv \{\varphi \in W^{1,2}_{loc}(\mathbb{R};\mathbb{R}^m): \lim_{s\rightarrow\pm\infty} \varphi(s)=\pm a\} . \]

M. Schatzman in her remarkable paper Asymmetric heteroclinic double layers considered the PDE \[ \Delta u=W_u(u) , \quad W_u=(D_{u_1}W,\dots, D_{u_m}W)^T \] and, under a nondegeneracy condition on $\overline u_\pm$, proved the existence of a solution $u^S: \mathbb{R}^2\rightarrow\mathbb{R}^m$ that connects $\overline u_-$ to $\overline u_+$ in the sense that

\begin{align*}\lim_{y\rightarrow\pm\infty} u^S(x,y) & = \pm a , \\\lim_{x\rightarrow\pm\infty} u^S(x,y) & = \overline u_\pm(y-\eta_\pm)\end{align*} for some $\eta_\pm \in \mathbb{R}$.

Assuming that $W$ is a $C^\infty$ function, we give an alternative elementary proof of this result. Our approach is based on the idea of regarding $u^S$ as an Heteroclinic map $\mathbb{R} \ni x \rightarrow u^S(x, \cdot) \in A$ that connects the minimizers $\overline u_\pm(\cdot-\eta_\pm)$ of the Effettive Potential $J(\varphi)$.