Upper bounds for singular perturbation problems involving gradient fields

We prove an upper bound for the Aviles-Giga problem, which involves the minimization of the energy $E_\e(v)=\e\int_\Omega\big|\nabla^2v\big|^2dx+\frac{1}{\e}\int_\Omega\big(1-|\nabla v|^2\big)^2dx$ over $v\in H^2(\Omega)$, where $\e>0$ 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 $\{v_\e\}$ satisfying: $v_\e\to v$ in $W^{1,p}(\Omega)$ and $E_\e(v_\e)\to\frac{1}{3}\int_{J_{\nabla v}}|\nabla^+v-\nabla^-v|^3\,d{\mathcal H}^{N-1}$, as $\e$ goes to $0$.