MiS Preprint Repository

We examine the singularly perturbed variational problem $E_{\epsilon }(\psi )=\int {\epsilon }^{-1}(1-|\mathrm{\nabla }\psi {|}^2{)}^2+\epsilon |\mathrm{\nabla }\mathrm{\nabla }\psi {|}^2$ in the plane. As $\epsilon \to 0$ this functional favors $|\mathrm{\nabla }\psi |=1$ and penalizes singularities where $|\mathrm{\nabla }\mathrm{\nabla }\psi |$ concentrates. Our main result is a compactness theorem: if $E_{\epsilon }({\psi }_{\epsilon })$ is uniformly bounded then $\mathrm{\nabla }{\psi }_{\epsilon }$ is compact in $L^2$. Thus, in the limit $\epsilon \to 0$ $\psi$ solves the eikonal equation $|\mathrm{\nabla }\psi |=1$ almost everywhere. Our analysis uses "entropy relations" and the "div-curl lemma," adopting Tartar\'s approach to the interaction of linear differential equations and nonlinear algebraic relations.