# Upper bounds for singular perturbation problems involving gradient fields

### Arkady Poliakovsky

Technion - Israel Institute of Technology, Haifa, Israel

## 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 $v∈H_{2}(Ω)$, where \( \e>0 \) is a small parameter. Given $v∈W_{1,∞}(Ω)$ such that $∇v∈BV$ and $∣∇v∣=1$ a.e., we construct a family \( \{v_\e\} \) satisfying: \( v_\e\to v \) in $W_{1,p}(Ω)$ 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 $0$.

## Cite this article

Arkady Poliakovsky, Upper bounds for singular perturbation problems involving gradient fields. J. Eur. Math. Soc. 9 (2007), no. 1, pp. 1–43

DOI 10.4171/JEMS/70