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 \( E_\e(v)=\e\int_\Omega\big|\nabla^2v\big|^2dx+\frac{1}{\e}\int_\Omega\big(1-|\nabla v|^2\big)^2dx \) 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 $0$.