Bo Su
Department of Mathematics
Iowa State University

Abstract: A well known problem in calculus of variation is to identify the $\Gamma -$ limit as $\epsilon \rightarrow 0$ of the following energy functional in the convex domain $$E_{\epsilon}(u)=\inf _{u\in {\mathcal A}}\int _{\Omega }\frac 1{\epsilon} (|\nabla u|^2-1)^2+\epsilon |\nabla ^2u|^2\,dx,$$ where ${\mathcal A}=\{u\in H^2(\Omega)\mid u|_{\partial \Omega}=0, \frac {\partial u}{\partial n}|_{\partial \Omega }=-1\}$. Twenty years ago, Aviles- Giga conjectured that the limit is $$\frac 13\int _{\Gamma _{\varphi}}|[\nabla \varphi ]^3|\,d{\mathcal H}= \inf _{u\in {\mathcal B}}\frac 13\int _{\Gamma _{u}}|[\nabla u ]^3|\,d {\mathcal H}\equiv E(u),$$ where $\varphi =\hbox {dist}(x,\partial \Omega)$, ${\mathcal B}=\{u\in W^ {1,\infty }(\Omega)\mid |\nabla u|=1, u|_{\partial \Omega}=0, \frac {\partial u}{\partial n}|_{\partial \Omega }=-1\}$, and $\Gamma _u=\{x\in \Omega \mid \lim _{y\rightarrow x}\nabla u(y)$ does not exist $\}$. Around 2000, Jin-Kohn proved it in the case that $\Omega $ is an ellipse. In the general case, DeSimone-Kohn-M\"uller-Otto, Ambrosio-De Lellis-Mentagazza showed independently that any sequence of $u^{\epsilon }$ with $E_{\epsilon}(u^ {\epsilon)$ uniform bound converge to a solution of $\nabla u|=1$ in $\Omega $. Here we prove the conjecture is true in the general case.