Upper bounds for singular perturbation problems involving gradient fields (Q866484)

Summary: We prove an upper bound for the Aviles--Giga problem, which involves the minimization of the energy \[ E_\varepsilon(v)=\varepsilon\int_\Omega |\nabla^2v|^2\,dx+\varepsilon^{-1}\int_\Omega (1-|\nabla v|^2)^2\,dx \] over \(v\in H^2(\Omega)\), where \(\varepsilon>0\) is a small parameter. Given \(v\in W^{1,\infty}(\Omega)\) such that \(\nabla v\in\text{BV}\) and \(|\nabla v| =1\) a.e., we construct a family \(\{v_\varepsilon\}\) satisfying \(v_\varepsilon\to v\) in \(W^{1,p}(\Omega)\) and \[ E_\varepsilon(v_\varepsilon)\to \frac 13 \int_{J_{\nabla v}}|\nabla^+v-\nabla^-v|^3\,d\mathcal H^{N-1} \] as \(\varepsilon\) goes to 0.