Upper bounds for singular perturbation problems involving gradient fields
Arkady Poliakovsky · Journal of the European Mathematical Society · 2007
We prove an upper bound for the Aviles–Giga problem, which involves the minimization of the energy E_\varepsilon(v)=\varepsilon\int_\Omega\big| abla^2v\big|^2dx+\frac{1}{\varepsilon}\int_\Omega\big(1-| abla v|^2\big)^2dx over v\in H^2(\Omega) , where \varepsilon>0 is a small parameter. Given v\in W^{1,\infty}(\Omega) such that abla v\in BV and | abla 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{1}{3}\int_{J_{ abla v}}| abla^+v- abla^-v|^3\,d{\mathcal H}^{N-1} , as \varepsilon goes to 0 .