Upper bounds for singular perturbation problems involving gradient fields

Abstract

We prove an upper bound for the Aviles–Giga problem, which involves the minimization of the energy $E_{\epsilon }(v)=\epsilon {\int }_{\Omega }|{\nabla }^{2}v{|}^{2}dx+\frac{1}{\epsilon }{\int }_{\Omega }(1-|\nabla v{|}^2{)}^{2}dx$ over $v\in H^2(\Omega )$, where $\epsilon >0$ is a small parameter. Given $v\in W^{1,\infty }(\Omega )$ such that $\nabla v\in BV$ and $|\nabla v|=1$ a.e., we construct a family $\{v_{\epsilon }\}$ satisfying: $v_{\epsilon }\to v$ in $W^{1,p}(\Omega )$ and $E_{\epsilon }(v_{\epsilon })\to \frac{1}{3}{\int }_{J_{\nabla v}}|{\nabla }^{+}v-{\nabla }^{-}v{|}^{3}d{\mathcal{H}}^{N-1}$, as $\epsilon$ goes to $0$.